
B
C
Calculus
by Andrew Harrison Hill
of Aurora, IL
Table of Contents
GENERAL CALCULUS IDEAS ................................................................................................................ 6
RATES OF CHANGE ..................................................................................................................................... 6
Speeds ................................................................................................................................................... 6
Tangent Lines ........................................................................................................................................ 6
Examples ............................................................................................................................................... 6
LIMITS ........................................................................................................................................................ 6
Properties of Limits............................................................................................................................... 6
Examples ............................................................................................................................................... 7
One-sided and Two-sided Limits........................................................................................................... 7
Sandwich Theorem ................................................................................................................................ 7
Examples ............................................................................................................................................... 7
Limits Involving Infinity ........................................................................................................................ 8
Examples ............................................................................................................................................... 8
CONTINUITY ............................................................................................................................................... 8
Continuity at a Point ............................................................................................................................. 8
Types of discontinuity ........................................................................................................................... 8
Properties of Continuous Functions ..................................................................................................... 8
Composite of Continuous Functions ..................................................................................................... 9
The Intermediate Value Theorem for Continuous Functions ................................................................ 9
DERIVATIVES ............................................................................................................................................ 9
INTRO TO DERIVATIVES .............................................................................................................................. 9
Examples ............................................................................................................................................... 9
NOTATION .................................................................................................................................................. 9
DIFFERENTIABILITY ...................................................................................................................................10
Examples ..............................................................................................................................................10
RULES FOR DIFFERENTIATION ...................................................................................................................10
Rule 1: Derivative of a Constant Function ..........................................................................................10
Rule 2: Power Rule for Positive Integer Powers of x...........................................................................10
Rule 3: The Constant Multiple Rule .....................................................................................................10
Rule 4: The Sum and Difference Rule ..................................................................................................11
Rule 5: The Product Rule .....................................................................................................................11
Rule 6: The Quotient Rule....................................................................................................................11
Rule 7: Power Rule for Negative Integer Powers of x .........................................................................11
Rule 8: Chain Rule: .............................................................................................................................11
Rule 9: Power Rule for Rational Powers of x ......................................................................................11
Rule 10: Power Rule for Arbitrary Real Powers .................................................................................11
Examples ..............................................................................................................................................11
PARAMETRIC CURVES ...............................................................................................................................12
Examples ..............................................................................................................................................12
IMPLICIT DIFFERENTIATION .......................................................................................................................12
Examples ..............................................................................................................................................12
SECOND AND HIGHER ORDER DERIVATIVES .............................................................................................13
Examples ..............................................................................................................................................13
VELOCITY AND OTHER RATES OF CHANGE ...............................................................................................13
Free-fall Constants (Earth) ..................................................................................................................13
TRIGONOMETRIC FUNCTION ......................................................................................................................13
DERIVATIVES OF EXPONENTIAL AND LOGARITHMIC FUNCTIONS ..............................................................14
Derivative of ex.....................................................................................................................................14
Derivative of ax ....................................................................................................................................14
Derivative of ln x ..................................................................................................................................14
i
Derivative of logax ...............................................................................................................................14
Examples ..............................................................................................................................................14
APPLICATIONS OF DERIVATIVES ......................................................................................................15
THEOREMS ................................................................................................................................................15
Theorem 1: The Extreme Value Theorem ............................................................................................15
Theorem 2: Local Extreme Values .......................................................................................................15
Theorem 3: Mean Value Theorem for Derivatives ...............................................................................15
Theorem 4: First Derivative Test for Local Extrema ...........................................................................15
Theorem 5: Second Derivative Test for Local Extrema .......................................................................15
Theorem 6: Maximum Profit ................................................................................................................15
Theorem 7: Minimizing Average Cost .................................................................................................16
EXTREME VALUES OF FUNCTIONS .............................................................................................................16
Absolute Extreme Values .....................................................................................................................16
Local (Relative) Extreme Values ..........................................................................................................16
Critical Point .......................................................................................................................................16
Examples ..............................................................................................................................................16
MEAN VALUE THEOREM ...........................................................................................................................16
Increasing Function, Decreasing Function .........................................................................................16
Antiderivative .......................................................................................................................................16
Corollary 1: Increasing and Decreasing Functions ............................................................................16
Corollary 2: Functions with f’ = 0 are Constant .................................................................................17
Corollary 3: Functions with the Same Derivative Differ by a Constant ..............................................17
Examples ..............................................................................................................................................17
CONNECTING F’ AND F’’ WITH THE GRAPH OF F.........................................................................................17
Concavity .............................................................................................................................................17
Concavity Test......................................................................................................................................17
Point of Inflection ................................................................................................................................17
Examples ..............................................................................................................................................17
MODELING AND OPTIMIZATION.................................................................................................................18
Strategies for solving Max-Min Problems ...........................................................................................18
Examples ..............................................................................................................................................18
LINEARIZATION AND NEWTON’S METHOD ................................................................................................18
Linearization ........................................................................................................................................18
Newton’s Method .................................................................................................................................18
Differentials .........................................................................................................................................18
Differential Estimate of Change ..........................................................................................................18
Examples ..............................................................................................................................................18
RELATED RATES ........................................................................................................................................19
Examples ..............................................................................................................................................19
INTEGRALS ...............................................................................................................................................19
THEOREMS ................................................................................................................................................19
Theorem 1: The Existence of Definite Integrals...................................................................................19
Theorem 2: The Integral of a Constant ................................................................................................19
Theorem 3: The Mean Value Theorem for Definite Integrals ..............................................................19
Theorem 4: The Fundamental Theorem of Calculus, Part 1................................................................20
Theorem 4 (continued): The Fundamental Theorem of Calculus, Part 2 ............................................20
ESTIMATING WITH FINITE SUMS ................................................................................................................20
Rectangular Approximation Method (RAM) ........................................................................................20
Examples ..............................................................................................................................................20
DEFINITE INTEGRALS ................................................................................................................................20
Riemann Sums ......................................................................................................................................20
The Definite Integral as a Limit of Riemann Sums ..............................................................................21
The Definite Integral of a Continuous Function on [a, b] ...................................................................21
Notation ...............................................................................................................................................21
ii
Area Under a Curve (as a Definite Integral) .......................................................................................21
Discontinuous Integrable Functions ....................................................................................................21
Examples ..............................................................................................................................................21
DEFINITE INTEGRALS AND ANTIDERIVATIVES ...........................................................................................22
Rules for Definite Integrals ..................................................................................................................22
Average (Mean) Value .........................................................................................................................22
Differential and Integral Calculus .......................................................................................................22
Examples ..............................................................................................................................................22
TRAPEZOIDAL RULE ..................................................................................................................................23
Trapezoidal Rule ..................................................................................................................................23
Simpson’s Rule .....................................................................................................................................23
Error Bounds .......................................................................................................................................23
Examples:.............................................................................................................................................24
APPLICATIONS OF DEFINITE INTEGRALS ......................................................................................24
INTEGRAL AS NET CHANGE .......................................................................................................................24
Strategy for Modeling with Integrals ...................................................................................................24
Work .....................................................................................................................................................24
Examples:.............................................................................................................................................25
AREAS IN THE PLANE ................................................................................................................................25
Area Between Curves ...........................................................................................................................25
Area Enclosed by Intersecting Curves .................................................................................................25
Boundaries with Changing Functions ..................................................................................................25
Integrating with Respect to y ...............................................................................................................25
Examples:.............................................................................................................................................25
VOLUMES ..................................................................................................................................................26
Volumes of a Solid ...............................................................................................................................26
Solids of Revolution .............................................................................................................................26
Examples:.............................................................................................................................................27
LENGTH OF CURVES ..................................................................................................................................27
Arc Length: Length of a Smooth Curve ................................................................................................27
Vertical Tangents, Corners, and Cusps ...............................................................................................28
Examples:.............................................................................................................................................28
OTHER DEFINITIONS ..................................................................................................................................28
Probability Density Function (pdf) ......................................................................................................28
Normal Probability Density Function ..................................................................................................28
The 68-95-99.7 Rule for Normal Distributions ....................................................................................28
DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING ............................................29
ANTIDERIVATIVES AND SLOPE FIELDS ......................................................................................................29
Solving Initial Value Problems ............................................................................................................29
Slope Field or Direction Field .............................................................................................................29
Indefinite Integral ................................................................................................................................29
Integral Formulas ................................................................................................................................29
Properties of Indefinite Integrals .........................................................................................................30
Examples:.............................................................................................................................................30
INTEGRATION BY SUBSTITUTION ...............................................................................................................30
Power Rule for Integration ..................................................................................................................30
Substitution in Definite Integrals .........................................................................................................30
Separable Differential Equations.........................................................................................................30
Examples:.............................................................................................................................................31
INTEGRATION BY PARTS ............................................................................................................................31
Product Rule in Integral Form.............................................................................................................31
Examples:.............................................................................................................................................32
EXPONENTIAL GROWTH AND DECAY ........................................................................................................32
Exponential Model ...............................................................................................................................32
iii
Logistic Growth Rate ...........................................................................................................................32
Examples:.............................................................................................................................................32
NUMERICAL METHODS ..............................................................................................................................33
Euler’s Method ....................................................................................................................................33
Improved Euler’s Method ....................................................................................................................34
L’HÔPITAL’S RULE, IMPROPER INTEGRALS, AND PARTIAL FRACTIONS ...........................34
L’HÔPITAL’S RULE ...................................................................................................................................34
Theorem 1: L’Hôpital’s Rule (First Form) ..........................................................................................34
Theorem 2: L’Hôpital’s Rule (Stronger Form) ....................................................................................34
Exponential Indeterminate Forms........................................................................................................34
Indeterminate Forms ............................................................................................................................34
Examples:.............................................................................................................................................34
RELATIVE RATES OF GROWTH ..................................................................................................................35
Faster, Slower, Same-rate Growth as x∞.........................................................................................35
Transitivity of Growing Rates ..............................................................................................................35
Examples:.............................................................................................................................................35
IMPROPER INTEGRALS ...............................................................................................................................35
Improper Integrals with Infinite Integration Limits .............................................................................35
Improper Integrals with Infinite Discontinuities ..................................................................................36
Direct Comparison Test .......................................................................................................................36
Limit Comparison Test .........................................................................................................................36
Examples:.............................................................................................................................................36
PARTIAL FRACTIONS AND INTEGRAL TABLES ...........................................................................................37
Partial Fractions..................................................................................................................................37
Trigonometric Substitutions .................................................................................................................38
Examples:.............................................................................................................................................38
INFINITE SERIES ......................................................................................................................................39
POWER SERIES ...........................................................................................................................................39
Infinite Series .......................................................................................................................................39
Geometric Series ..................................................................................................................................39
Power Series ........................................................................................................................................39
Term-by-Term Differentiation..............................................................................................................40
Term-by-Term Integration ...................................................................................................................40
Examples:.............................................................................................................................................40
TAYLOR SERIES .........................................................................................................................................41
Taylor Series Generated by f at x = a ..................................................................................................41
Maclaurin Series ..................................................................................................................................41
Examples:.............................................................................................................................................41
Table of Maclaurin Series ....................................................................................................................42
TAYLOR’S THEOREM .................................................................................................................................42
Taylor’s Theorem with Remainder ......................................................................................................42
Remainder Estimation Theorem...........................................................................................................42
Examples:.............................................................................................................................................43
RADIUS OF CONVERGENCE ........................................................................................................................43
The Convergence Theorem for Power Series .......................................................................................43
The nth-Term Test for Divergence ........................................................................................................43
The Direct Comparison Test ................................................................................................................43
Absolute Convergence .........................................................................................................................44
Absolute Convergence Implies Convergence .......................................................................................44
The Ratio Test ......................................................................................................................................44
Examples:.............................................................................................................................................44
TESTING CONVERGENCE AT ENDPOINTS ...................................................................................................45
The Integral Test ..................................................................................................................................45
Harmonic Series and p-series ..............................................................................................................45
iv
The Limit Comparison Test (LCT) .......................................................................................................45
The Alternating Series Test (Leibniz’s Theorem) .................................................................................45
The Alternating Series Estimation Theorem ........................................................................................46
Rearrangements of Absolutely Convergent Series ...............................................................................46
Rearrangement of Conditionally Convergent Series............................................................................46

How to Test a Power Series
c
n 0
n
( x  a) n for Convergence ..........................................................46
Examples:.............................................................................................................................................46
PARAMETRIC, VECTOR, AND POLAR FUNCTIONS .......................................................................47
PARAMETRIC FUNCTIONS ..........................................................................................................................47
Derivative at a Point ............................................................................................................................47
Length of a Smooth Parameterized Curve ...........................................................................................47
Surface Area ........................................................................................................................................47
Examples:.............................................................................................................................................48
VECTORS ...................................................................................................................................................48
Vector, Equal Vector ............................................................................................................................48
Component Form of a Vector...............................................................................................................48
Dot Product..........................................................................................................................................48
Magnitude ............................................................................................................................................49
Angle Between Two Vectors .................................................................................................................49
Examples:.............................................................................................................................................49
Limit .....................................................................................................................................................49
Continuity are a point ..........................................................................................................................49
Component Test for Continuity at a Point ...........................................................................................49
Velocity, Speed, Acceleration, Direction of Motion .............................................................................49
Examples:.............................................................................................................................................50
MODELING PROJECTILE MOTION ...............................................................................................................50
Height, Flight Times, and Range for Ideal Projectile Motion .............................................................50
Projectile Motion with Linear Drag ....................................................................................................50
Examples:.............................................................................................................................................51
POLAR CURVES .........................................................................................................................................51
Polar Coordinates ................................................................................................................................51
Equations Relating Polar and Cartesian Coordinates ........................................................................51
Examples:.............................................................................................................................................51
Slope of the Curve r = f(θ) ...................................................................................................................51
Area in Polar Coordinates ...................................................................................................................51
Area Between Polar Curves .................................................................................................................52
Length of a Polar Curve ......................................................................................................................52
Area of a Surface of Revolution ...........................................................................................................52
Examples:.............................................................................................................................................52
v
General Calculus Ideas
Rates of Change
Speeds
Average Speed – found by dividing the distance covered by the elapsed time
16(2) 2  16(0) 2
y y 2  y1
 y=16t²,
 32

20
t
t 2  t1
Instantaneous Speed – found by calculating the speed at a specific time
Tangent Lines
The slope of the curve y = f(x) at the point P(a, f(a)) is the number
f ( a  h)  f ( a )
m  lim
, provided the limit exists. The tangent line to the curve at P is
h 0
h
the line through P with this slope.
Examples:
Let f(x) = 1/x.
(a) The slope at x = a is
1
1

f ( a  h)  f ( a )
1 a  ( a  h)
h
1
1
lim
 lim a  h a  lim 
 lim
 lim
 2
h 0
h

0
h

0
h

0
h

0
h
h
h a ( a  h)
ha(a  h)
a ( a  h) a
(b) The slope will be -1/4 if
1 1

, a 2  4, a  2
2
4
a
Limits
Limit – Let c and L be real numbers. The function f has limit l as x approaches c if,
given any positive number ε, there is a positive number δ such that for all x,
0  x  c    f ( x)  L   ... lim f ( x)  L
x c
Properties of Limits
If L, M, c, and k are real numbers and
lim f ( x)  L
and
x c
1. Sum Rule:
lim g ( x)  M
x c
then
lim ( f ( x)  g ( x))  L  M
x c
The limit of the sum of two functions is the sum of their limits.
lim ( f ( x)  g ( x))  L  M
2. Difference Rule:
x c
The limit of the difference of two functions is the difference of their limits.
lim ( f ( x)  g ( x))  L  M
3. Product Rule:
x c
The limit of a product of two functions is the product of their limits.
Page 6 of 53
lim (k  f ( x))  k  L
4. Constant Multiple Rule:
x c
The limit of a constant times a function is the constant times the limit of the
function.
f ( x) L
5. Quotient Rule:
lim

,M  0
x c g ( x )
M
The limit of a quotient of two functions is the quotient of their limits, provided the
limit of the denominator is not zero.
6. Power Rule:
lim ( f ( x)) r / s  Lr / s
x c
If r and s are integers, s ≠ 0, then (see above) provided that Lr / s is a real number.
The limit of a rational power of a function is that power of the limit of the
function, provided the latter is a real number.
Examples:
(a) lim ( x 3  4 x 2  3)  lim x 3  lim 4 x 2  lim 3
Sum & Difference Rule
xc
(b) lim
x c
xc
3
x4  x2 1

x2  5
xc
x c
 c  4c  3
lim ( x 4  x 2  1)
2
Product & Multiple Rule
x c
Quotient Rule
lim ( x 2  5)
x c

lim x 4  lim x 2  lim 1
x c
x c
2
x c
Sum & Difference Rule
lim x  lim 5
x c
x c
c4  c2 1

c2  5
Product Rule
One-sided and Two-sided Limits
Right-hand:
Left-hand:
lim f ( x )
The limit of f as x approaches c from the right.
lim f ( x )
The limit of f as x approaches c from the left.
x c 
x c 
A function f(x) has a limit as x approaches c if and only if the right-hand and left-hand
lim f ( x)  L  lim f ( x)  L and lim f ( x)  L
limits at c exists and are equal.
x c
x c
x c
Sandwich Theorem
If g(x) ≤ f(x) ≤ h(x) for all x ≠ c in some interval about c, and
lim g ( x)  lim h( x)  L , then lim f ( x)  L
x c
x c
x c
Examples:
Show that lim [ x 2 sin( 1 x)]  0
x 0
x sin (1 x)  x 2  sin (1 x)  x 2  1  x 2
2
and
 x 2  x 2 sin (1 x)  x 2
Because lim  x 2  lim x 2 , the Sandwich Theorem gives lim ( x 2 sin (1 x))  0
xo
x0
x0
Page 7 of 53
Limits Involving Infinity
The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either
lim f ( x)  b or lim f ( x)  b
x 
x  
The line x = a is a vertical asymptote of the graph of a function y = f(x) if either
lim f ( x)   or lim f ( x)  
xa
xa
The function g is a right end behavior model for f if and only if lim
x 
end behavior model for f if and only if lim
x 
f ( x)
 1 or a left
g ( x)
f ( x)
1
g ( x)
Examples:
1
(a) lim (2  )  2 has a horizontal asymptote at y = 2.
x 
x
1
(b) lim 2   has a vertical asymptote at x = 0.
x 0 x
(c) f ( x)  3x 4  2 x 3  3x 2  5x  6 g ( x)  3x 4
3x 4  2 x 3  3x 2  5 x  6
2
1
5
2
 lim (1 
 2  3  4 ) 1
4
x 
x 
3x x
3x
3x
x
lim
Continuity
Continuity at a Point
Interior Point: A function y = f(x) is continuous at an interior point c of its domain if
lim f ( x)  f (c).
x c
Endpoint: A function y = f(x) is continuous at a left endpoint a or is continuous at a
right endpoint b of its domain if lim f ( x)  f (a) or lim f ( x)  f (b) , respectively.
xa
x b
Types of discontinuity
Removable discontinuity: limit goes to a point, but the function does not exist there.
Jump discontinuity: the one-sided limits exist, but have different values.
Oscillating discontinuity: it oscillates too much to have a limit as x0.
Properties of Continuous Functions
If the functions f and g are continuous at x = c, then the following combinations are
continuous at x = c:
Sums:
f+g
Differences:
f–g
Products:
f∙g
Constant Multiples: k ∙f, for any number k
Quotients:
f/g, provided g(c) ≠ 0
Page 8 of 53
Composite of Continuous Functions
If f is continuous at c and g is continuous at f(c), then the composite g  f is continuous
at c.
The Intermediate Value Theorem for Continuous Functions
A function y = f(x) that is continuous on a closed interval [a, b] takes on every value
between f(a) and f(b). In other words, if y0 is between f(a) and f(b), then y0 = f(c) for
some c in [a, b].
Derivatives
Intro to Derivatives
The derivative of the function f with respect to the variable x is the function f’ whose
f ( x  h)  f ( x )
value at x is f ' ( x)  lim
, provided the limit exists. If f’(x) exists, we say
h 0
h
that f is differentiable at x. A function that is differentiable at every point of its domain is
a differentiable function.
f ( x)  f (a)
The derivative of the function f at the point x = a is the limit f ' (a )  lim
,
xa
xa
provided the limit exists.
A function y = f(x) is differentiable on a closed interval [a, b] if it has a derivative at
every interior point of the interval, and if the limits
f ( a  h)  f ( a )
f (b  h)  f (b)
lim
and lim
h 0
h 0
h
h
exist at the endpoints
Examples:
(a) Differentiate f(x) = x3
f ( x  h)  f ( x )
( x  h) 3  x 3
( x 3  3x 2 h  3xh2  h 3 )  x 3
f ' ( x)  lim
 lim
 lim
h 0
h 0
h 0
h
h
h
2
2
(3x  3xh  h )h
 lim
 lim (3x 2  3xh  h 2 )  3x 2
h 0
h 0
h
(b) Differentiate f(x) = x
f ' (a)  lim
x a
 lim
x a
f ( x)  f ( a )
x a
x a
x a
xa
 lim
 lim

 lim
x a
x a
xa
xa
xa
x  a xa ( x  a)( x  a )
1
x a

1
2 a
Notation
y’ 
dy

dx
“y prime”
“dy dx” or “the derivative of y with respect to x
Page 9 of 53
dy

dx
d
f (x) 
dx
“df dx” or “the derivative of f with respect to x
“d dx of f at x” or “the derivative of f at x”
Differentiability
1. corner: where one-sided derivatives differ. ( x )
2. cusp: where the slopes of the secant lines approach ∞ from one side and -∞ from
the other side (an extreme corner). (x⅔)
3. vertical tangent: where the slopes of the secant lines approach either ∞ or -∞ from
both sides. ( 3 x )
4. discontinuity
Differentiability Implies Local Linearity
Differentiability Implies Continuity: If f has a derivative at x = a, then f is continuous at
x = a.
Intermediate Value Theorem for Derivatives: If a and b are any two points in an interval
on which f is differentiable, the f’ takes on every value between f’(a) and f’(b).
Examples:
(a) Prove that
lim f ( x)  f (a) or lim f ( x)  f (a)  0
x a
xa
lim [ f ( x)  f (a )]  lim [( x  a)
x a
x a
f ( x)  f (a )
f ( x)  f (a)
]  lim ( x  a)  lim
 0  f ' (a)  0
x

a
x

a
xa
xa
Rules for Differentiation
Rule 1: Derivative of a Constant Function
If f is the function with the constant value c, then
df
d

(c )  0
dx dx
Rule 2: Power Rule for Positive Integer Powers of x
If n is a positive integer, then
d n
( x )  nx n 1
dx
Rule 3: The Constant Multiple Rule
If u is a differentiable function of x and c is a constant, then
d
du
(cu )  c
dx
dx
Page 10 of 53
Rule 4: The Sum and Difference Rule
If u and v are differentiable functions of x, then their sum and difference are
differentiable at every point where u and v are differentiable. At such points,
d
du dv
(u  v) 

dx
dx dx
Rule 5: The Product Rule
The product of two differentiable functions u and v is differentiable, and
d
dv
du
(uv)  u
v
dx
dx
dx
Rule 6: The Quotient Rule
At a point where v ≠ 0, the quotient y = u/v of two differentiable functions is
differentiable, and
du
dv
v
u
d u
dx
dx
 
2
dx  v 
v
Rule 7: Power Rule for Negative Integer Powers of x
If n is a negative integer and x ≠ 0, then
d n
( x )  nx n 1
dx
Rule 8: Chain Rule:
If f is differentiable at the point u=g(x), and g is differentiable at x, then the
composite function (f ○g)(x) = f(g(x)) is differentiable at x, and
( f  g )' ( x)  f ' ( g ( x))  g ' ( x).
In Leibniz notation, if y = f(u) and u = g(x), then
dy dy du

 ,
dx du dx
where dy/du is evaluated at u = g(x).
Rule 9: Power Rule for Rational Powers of x
If n is any rational number, then
d n
x  nx n 1 . If n < 1, then the derivative does
dx
not exist at x = 0.
Rule 10: Power Rule for Arbitrary Real Powers
If u is a positive differential function of x and n is any real number, then un is a
differentiable function of x, and
d n
du
u  nu n 1
dx
dx
Examples:
Page 11 of 53
(a) if f ( x)  6 then f ' ( x)  0
(b) if f ( x)  x 3 then f ' ( x)  3x 2
(c) if f ( x)  4 x then f ' ( x)  4
(d) if f ( x)  5 x  x 2 then f ' ( x)  5  2 x
(e) if f ( x)  (2 x  1)( x 3  2) then f ' ( x)  (2 x  1)(3x 2 )  ( x 3  2)( 2)
(f)
2x
x 3 (2)  2 x(3x 2 )
then
f
'
(
x
)

x3
(x3 )2
if f ( x) 
( g ) if f ( x)  x  4 then f ' ( x)  4 x 5
dx
du
(h) if f ( x)  sin( x 2 ) then
 cos(u ), x  sin( u ) and
 2t , u  t 2 so f ' ( x)  2 x cos( x 2 )
du
dt
2
3
3
(i) if f ( x)  sin (tan( x )) then f ' ( x)  2 sin(tan( x )) cos(tan( x 3 )) sec 2 ( x 3 )(3 x 2 )
 6 x 2 sin(tan( x 3 )) cos(tan( x 3 )) sec 2 ( x 3 )
1
(j) if f ( x)  x then f ' ( x) 
2 x
(h) if f ( x)  x
2
then f ' ( x)  2 x
2 1
Parametric Curves
If all three derivatives exist and dx/dt ≠ 0,
dy dy / dt

dx dx / dt
Examples:
x  sec t ,
y  tan t , 

2
t

2
2
dy dy / dt
sec t
sec t



 csc t
dx dx / dt sec t tan t tan t
Implicit Differentiation
Process:
1. Differentiate both sides of the equation with respect to x.
2. Collect the terms with dy/dx on one side of the equation.
3. Factor out dy/dx.
4. Solve for dy/dx.
Examples:
(a) Find dy/dx for x 2  xy  y 2  7
2 x  ( xy' y )  2 yy '  0
(2 y  x) y '  y  2 x
y  2x
y' 
2y  x
Page 12 of 53
Second and Higher Order Derivatives
The nth derivative is the derivative of the nth – 1 derivative.
Examples:
Function :
y  x 3  5x 2  2
First derivative :
y '  3 x 2  10 x
Second derivative : y ' '  6 x
Third derivative : y ' ' '  6
Fourth derivative : y ( 4 )  0
Velocity and Other Rates of Change
Instantaneous rate of change of f with respect to x at a is the derivative
f ( a  h)  f ( a )
f ' (a )  lim
h 0
h
Instantaneous velocity is the derivative of the position function s = f(t) with respect to
time. At time t the velocity is
ds
f (t  t )  f (t )
v(t ) 
 lim

t

0
dt
t
Speed is the absolute value of velocity
ds
Speed  v(t ) 
dt
Acceleration is the derivative of velocity with respect to time. If a body’s velocity at
time t is v(t) = ds/dt, then the body’s acceleration at time t is
dv d 2 s
a(t ) 

dt dt 2
Free-fall Constants (Earth)
English units:
Metric units:
ft
, s  16t 2
sec 2
m
g  9.8
, s  4.9t 2
2
sec
g  32
Trigonometric Function
Function
sin(x)
cos(x)
tan(x)
csc(x)
sec(x)
cot(x)
sin-1(x)
Derivative
cos(x)
-sin(x)
sec2(x)
-csc(x)cot(x)
sec(x)tan(x)
-csc2(x)
1
1 x2
Page 13 of 53
1
cos-1(x)
1 x2
1
1 x2
1
tan-1(x)
csc-1(x)
x x2 1
sec-1(x)
1
x x2 1
1
1 x2
cot-1(x)
Derivatives of Exponential and Logarithmic Functions
Derivative of ex
d x
e  ex ,
dx
d u
du
e  eu
dx
dx
Derivative of ax
d u
du
a  a u ln a
dx
dx
Derivative of ln x
d
1 du
ln u 
dx
u dx
Derivative of logax
d
1 du
log a u 
dx
u ln a dx
Examples:
(a) if f ( x)  e 2 x then f ' ( x)  2e 2 x
(b) if f ( x)  4 x then f ' ( x)  4 x 2 x ln 4
2x
(c) if f ( x)  ln x 2 then f ' ( x)  2
x
2
2
(d) if f ( x)  log 2 (sin x) then f ' ( x) 
1
cos x
sin( x) ln 2
Page 14 of 53
Applications of Derivatives
Theorems
Theorem 1: The Extreme Value Theorem
If f is continuous on a closed interval [a, b], then f has both a maximum value and
a minimum value on the interval.
Theorem 2: Local Extreme Values
If a function f has a local maximum value or a local minimum value at an interior
point c of its domain, and if f’ exists at c, then f’(c) = 0
Theorem 3: Mean Value Theorem for Derivatives
If y = f(x) is continuous at every point of the closed interval [a, b] and
differentiable at every point of its interior (a, b), then there is at least one point c
in (a, b) at which
f (b)  f (a )
f ' (c ) 
.
ba
Theorem 4: First Derivative Test for Local Extrema
The following tests apply to a continuous function f(x).
At critical point c:
1.
If f’ changes sign from positive to negative at c, then f has a
local maximum value at c.
2.
If f’ changes sign from negative to positive at c, then f has a
local minimum value at c.
3.
If f’ does not change sign at c, then f has no local extreme value
at c
At a left endpoint a:
If f’ < 0 (f’ > 0) for x > a, then f has a local maximum (minimum)
value at a.
At a right endpoint b:
If f’ < 0 (f’ > 0) for x < b, then f has a local minimum (maximum)
value at b.
Theorem 5: Second Derivative Test for Local Extrema
If f’(c) = 0 and f’’(x) < 0, then f has a local maximum at x = c.
If f’(c) = 0 and f’’(x) > 0, then f has a local minimum at x = c.
Theorem 6: Maximum Profit
Maximum profit (if any) occurs at a production level at which marginal revenue
equals marginal cost.
Page 15 of 53
Theorem 7: Minimizing Average Cost
The production level (if any) at which average cost is smallest is a level at which
the average cost equals the marginal cost.
Extreme Values of Functions
Absolute Extreme Values
Let f be a function with domain D. Then f(c) is the:
Absolute maximum value on D if and only if f(x) ≤ f(c) for all x in D.
Absolute minimum value on D if and only if f(x) ≥ f(c) for all x in D.
Local (Relative) Extreme Values
Let c be an interior point of the domain of the function f. Then f(c) is a:
Local maximum value at c if and only if f(x) ≤ f(c) for all x in some open interval
containing c.
Local minimum value at c if and only if f(x) ≥ f(c) for all x in some open interval
containing c.
Critical Point
A point in the interior of the domain of a function f at which f’ = 0 or f’ does not exist is a
critical point of f.
Examples:
(a) find all critical points on the curve f ( x)  x 2 / 3 on the interval [-2,3]
f ' ( x) 
2
33 x
f ' ( x)  0 or does not exists at x  0
,
Critical points at x  0,  2, 3
Abs max of 3 9 at x  3, local max of
3
4 at x  2, Abs min of 0 at x  0
Mean Value Theorem
Increasing Function, Decreasing Function
Let f be a function defined on an interval I and let x1 and x2 be any two points in I.
f increases on I if
x1 < x2

f(x1) < f(x2).
f decreases on I if
x1 < x2

f(x1) > f(x2).
Antiderivative
A function F(x) is an antiderivative of a function f(x) if F’(x) = f(x) for all x in the
domain of f. The process of finding an antiderivative is antidifferentiable.
Corollary 1: Increasing and Decreasing Functions
Let f be continuous on [a, b] and differentiable on (a, b).
If f’ > 0 at each point of (a, b), then f increases on [a, b].
If f’ < 0 at each point on (a, b), then f decreases on [a, b].
Page 16 of 53
Corollary 2: Functions with f’ = 0 are Constant
If f’(x) = 0 at each point of an interval I, then there is a constant C for which f(x) = C for
all x in I.
Corollary 3: Functions with the Same Derivative Differ by a Constant
If f’(x) = g’(x) at each point of an interval I, then there is a constant C such that f(x) = g(x)
+ C for all x in I.
Examples:
(a) The function y = x2 is decreasing on (-∞, 0] because y’ = 2x < 0 on (-∞, 0) and
increasing on [0, ∞) because y’ = 2x > 0 on (0, ∞).
(b) Find the function f(x) whose derivative is sin x and whose graph passes through
the point (0, 2).
f(x) = -cos x + C,
f(0) = 2,
-1 + C = 2
C = 3,
f(x) = -cos x + 3
(c) Find the antiderivative of f(x) = 2x + 3
F(x) = x2 + 3x
Connecting f’ and f’’ with the Graph of f
Concavity
The graph of a differentiable function y = f(x) is
Concave up on an open interval I if y’ is increasing on I.
Concave down on an open interval I if y’ is decreasing on I.
Concavity Test
The graph of a twice-differentiable function y = f(x) is
Concave up on any interval where y’’ > 0.
Concave down on any interval where y’’ < 0.
Point of Inflection
A point where the graph of a function has a tangent line and where the concavity changes
is a point of inflection.
Examples:
(a) Find the critical points for f(x) = x3 – 12x – 5.
f ' ( x)  3 x 2  12  0
x2  4
x  2
(b) Find the interval of concavity for f(x) = x2
f ' ( x)  2 x, f ' ' ( x)  2 is always positive, therefore always concave up.
(c) Find the inflection point for f(x) = x3 – 12x – 5
f ' ( x)  3x 2  12
, inflection point at x = 0.
f ' ' ( x)  6 x
Page 17 of 53
Modeling and Optimization
Strategies for solving Max-Min Problems
1. Understand the Problem Red the problem carefully. Identify the information
you need to solve the problem
2. Develop a Mathematical Model of the Problem Draw pictures and label the
parts that are important to the problem. Introduce a variable to represent the
quantity to be maximized or minimized. Using that variable, write a function
whose extreme value gives the information sought.
3. Graph the Function Find the domain of the function. Determine what values of
the variable make sense in the problem.
4. Identify the Critical Points and Endpoints Find where the derivative is zero or
fails to exist
5. Solve the Mathematical Model If unsure the result, support or confirm your
solution with another method.
6. Interpret the Solution Translate your mathematical result into the problem
setting and decide whether the result makes sense
Examples:
(a) Find two numbers whose sum is 20 and whose product is as large as possible.
f(x) = x(20 – x),
f’(x) = 20 – 2x = 0 at x = 10, #’s = 10 and 10
Linearization and Newton’s Method
Linearization
If f is differentiable at x = a, then the approximating function
L( x)  f (a)  f ' (a)( x  a) is the linearization of f at a.
Newton’s Method
1. Guess a first approximation to a solution of the equation f(x) = 0. A graph of y =
f(x) may help.
2. Use the first approximation to get a second, the second to get a third, and so on,
f ( xn )
using the formula xn1  xn 
.
f ' ( xn )
Differentials
Let y= f(x) be a differentiable function. The differential dx is an independent variable.
The differential dy is dy = f’(x)dx.
Differential Estimate of Change
Let f(x) be differentiable at x = a. The approximate change in the value of f when x
changes from a to a + dx is df = f’(a)dx.
Examples:
(a) Find the linearization of f ( x)  1  x at x = 0
L( x)  f (a)  f ' (a)( x  a)  1  .5( x  0)  1  x / 2
Page 18 of 53
(b) Find dy if y = x5 + 37x,
dy = (5x4 +37)dx
Related Rates
1. Draw a picture and name the variable and constants. Use t for time. Assume all
variables are differentiable functions of t.
2. Write down the numerical information
3. Write down what we are asked to find
4. Write an equation that relates the variable. You may have to combine two or
more equations to get a single equation that relates the variable whose rate you
want to the variables whose rates you know.
5. Differentiate with respect to t. Then express the rate you want in terms of the rate
and variables whose values you know.
6. Evaluate. Use known values to find the unknown rate.
Examples:
(a) Water runs into a conical tank at the rate of 9ft3/min. The tank stands point down
and has a height of 10 ft and a base radius of 5 ft. How fast is the water level
rising when the water is 6 ft deep?
dV
y  6 ft and
 9ft 3 / min .
dt
1
y
 3
V  x 2 y, x  , V 
y
3
2
12
dV  2 dy
 y
dt
4
dt

dy
dy 1
9  (6) 2
,

4
dt
dt 
Integrals
Theorems
Theorem 1: The Existence of Definite Integrals
All continuous functions are integrable. That is, if a function f is continuous on an
interval [a, b], then its definite integral over [a, b] exists.
Theorem 2: The Integral of a Constant
If f(x) = c, where c is a constant, on the interval [a, b], then

b
a
b
f ( x)dx   cdx  c(b  a)
a
Theorem 3: The Mean Value Theorem for Definite Integrals
If f is continuous on [a, b], then at some point c in [a, b], f (c) 
Page 19 of 53
1 b
f ( x)dx.
b  a a
Theorem 4: The Fundamental Theorem of Calculus, Part 1
x
If f is continuous on [a, b], then the function F ( x)   f (t )dt has a derivative at
a
dF
d x

f (t )dt  f ( x) .
every point x in [a, b], and
dx dx a
Theorem 4 (continued): The Fundamental Theorem of Calculus, Part 2
If f is continuous at every point of [a, b], and if F is any antiderivative of f on
[a, b], then

b
a
f ( x)dx  F (b)  F (a). This part of the Fundamental Theorem is
also called the Integral Evaluation Theorem.
Estimating with Finite Sums
Rectangular Approximation Method (RAM)
Midpoint RAM:
Evaluate the value of the function at the midpoint of each subinterval.
Left RAM:
Evaluate the value of the function at the left-hand point of each subinterval.
Right RAM:
Evaluate the value of the function at the right-hand point of each subinterval.
Examples:
(a) Approximate the area under the curve y = x2 from x = 0 to x = 3 with six intervals.
LRAM:
2
2
2
0  1    1   1   12  1    3   1   22  1    5   1   6.875
2 2 2
2 2 2
2 2 2
MRAM:
2
2
2
2
2
2
2
 1   1   3   1   5   1   7   1   9   1   11   1 
                              8.9375
4 2 4 2 4 2 4 2 4 2  4  2
RRAM:
2
2
2
2
2
2
 1   1  1  1   3   1   2   1   5   1   3   1 
                              11.375
 2   2  1  2   2   2   1   2   2   2   1   2 
Definite Integrals
Riemann Sums
Terms become infinitely small and their number infinitely large.
n
a
k 1
k
 a1  a 2  a3    a n 1  a n .
Page 20 of 53
The Definite Integral as a Limit of Riemann Sums
Let f be a function defined on a closed interval [a, b]. For any partition P of [a, b], let the
numbers ck be chosen arbitrarily in the subintervals [xk-1, xk]. If there exists a number I
such that
n
lim
p 0
 f (c
k 1
k
)x k  I
no matter how P and the ck’s are chosen, then f is integrable on [a, b] and I is the
definite integral of f over [a, b].
The Definite Integral of a Continuous Function on [a, b]
Let f be continuous on [a, b], and let [a, b] be partitioned into n subintervals of equal
length x  (b  a ) / n . Then the definite integral of f over [a, b] is given by
n
lim
n 
 f (c
k 1
k
) x ,
Where each ck is chosen arbitrarily in the kth subinterval.
Notation
n
lim
n 
 f (c
k 1
b
k
)x   f ( x)dx
a
Area Under a Curve (as a Definite Integral)
If y = f(x) is nonnegative and integrable over a closed interval [a, b], then the area under
b
the curve y = f(x) from a to b is the integral of f from a to b, A   f ( x)dx.
a
b
If y = f(x) is negative then A   f ( x)dx.
a
If y = f(x) is both negative and nonnegative then

b
a
f ( x)dx  (area above x - axis) - (area below the x - axis)
Discontinuous Integrable Functions
A bounded function that has a finite number of points of discontinuity on an interval
[a, b] will still be integrable on the interval if it is continuous everywhere else.
Examples:
(a) Evaluate the integral

2
2
4  x 2 dx
This is also the function for a semi circle with radius of 2, therefore
1
1
Area = r 2   2 2  2
2
2
(b)Evaluate the integral
9
 75dt
7
This is a constant function defined at 75, therefore
Area = 75(9  7)  150
Page 21 of 53
Definite Integrals and Antiderivatives
Rules for Definite Integrals
1. Order of Integration:
2. Zero:
3. Constant Multiple:
4. Sum and Difference:
5. Additivity:
a
b
b
a
a
 f ( x)dx   f ( x)dx
 f ( x)dx  0
 kf ( x)dx  k  f ( x)dx
  f ( x)dx   f ( x)dx
 ( f ( x)  g ( x))dx   f ( x)dx  
 f ( x)dx   f ( x)dx   f ( x)dx
a
b
b
a
a
b
b
a
a
b
b
b
a
b
a
a
c
c
a
b
a
g ( x)dx
6. Max-Min Inequality: If max f and min f are the maximum and minimum values
b
of f on [a, b], then min f  (b  a)   f ( x)dx  max f  (b  a).
a
7. Domination:
b
b
a
a
f ( x)  g ( x) on [a, b]   f ( x)dx   g ( x)dx
b
f ( x)  0 on [a, b]   f ( x)dx  0
a
Average (Mean) Value
If f is integrable on [a, b], its average (mean) value on [a, b] is
1 b
f ( x)dx.
av(f)=
b  a a
Differential and Integral Calculus
The derivative with respect to x of the integral of f from a to x is simply f.
d x
f (t )dt  f ( x).
dx a
This means that the integral is an antiderivative of f, a fact we can exploit in the follwing
way. If F is any antiderivative of f, then

x
a
f (t )dt  F ( x)  C
Examples:
(a) Find the average value of f(x) = 4 – x2 on [0, 3].
1 b
av( f ) 
f ( x)dx
b  a a
1 3

(4  x 2 )dx
3  0 0
1
 3
3
1
Page 22 of 53
(b) Find


0
sin xdx


0
sin xdx   cos   ( cos 0)
 (1)  (1)
2
Trapezoidal Rule
Trapezoidal Rule
To approximate

b
a
f ( x)dx , use
h
 y 0  2 y1  2 y 2    3 y n1  y n  ,
2
where [a, b] is partitioned into n subintervals of equal length h = (b – a)/n. Equivalently,
LRAM n  RRAM n
,
T
2
where LRAMn and RRAMn are the Riemann sums using the left and right endpoints,
respectively, for f for the partition.
T
Simpson’s Rule
To approximate

b
a
f ( x)dx , use
h
 y 0  4 y1  2 y 2  4 y3    2 y n2  4 y n1  y n  ,
3
where [a, b] is partitioned into an even number n subintervals of equal length
h = (b – a)/n.
S
Error Bounds
If T and S represent the approximations to

b
a
f ( x)dx given by the Trapezoidal Rule and
Simpson’s Rule, respectively, then the errors ET and ES satisfy
ba 2
ba 4
ER 
h M f '' and E R 
h Mf f ( 4 ) .
12
180
Page 23 of 53
Examples:

(a) Use the Trapezoidal Rule with n = 4 to estimate
2
1
x 2 dx .
h
 y 0  2 y1  2 y 2  2 y 3  y 4 
2

1
 25 
 36 
 49 
 1  2   2   2   4 
8
 16 
 16 
 16 

75

32
T 
(b) Use Simpson’s Rule with n = 4 to approximate
2
 5x dx .
4
0
h
 y 0  4 y1  2 y 2  4 y3  y 4 
3

1
5
 405 
  0  4   25  4
  80 
6
 16 
 16 

S

385
12
Applications of Definite Integrals
Integral as Net Change
In many applications, the integral is viewed as net change over time.
Strategy for Modeling with Integrals
1. Approximate what you want to find as a Riemann sum of values of a continuous
function multiplied by interval lengths. If f(x) is the function and [a, b] the
interval, and you partition the interval into subintervals of length Δx, the
approximating sums will have the form  f (ck )x with ck a point in the kth
subinterval.
2. Write a definite integral, here

b
a
f ( x)dx , to express the limit of these sums as the
norms of the partitions go to zero.
3. Evaluate the integral numerically or with an antiderivative.
Integrals can find the net change and total accumulation of pretty much everything.
Whenever you want to find the cumulative effect of a varying rate of change, integrate it.
Work
When a body moves a distance d along a straight line as a result of the action of a force of
constant magnitude F in the direction of motion, the work done by the force is
W = Fd.
The equation W = Fd is the constant-force formula for work.
Page 24 of 53
Hooke’s Law for springs says that the force it takes to stretch or compress a spring x
units from its natural length is a constant times x. In symbols, F = kx, where k, measured
in force unites per unit length, is a characteristic of the spring call the force constant.
Examples:
(a) Find the total distance traveled given the velocity v(t )  t 2 

5
0
5
v(t ) dt   t 2 
0
8
.
(t  1) 2
8
dt
(t  1) 2
 42.59
(b) It takes a force of 10 N to stretch 2 m beyond its natural length. How much work is
done in stretching the spring 4m from its natural length?
F (2)  10  2k  k  5

4
0
4
F ( x)dx   5 x dx  5
0
x2
2
4
]
0
 40 Nm
Areas in the Plane
Area Between Curves
If f and g are continuous with f(x) ≥ g(x) throughout [a, b], then the area between the
curves y = f(x) and y = g(x) from a to b is the integral of [f – g] from a to b.
b
A   [ f ( x)  g ( x)]dx .
a
Area Enclosed by Intersecting Curves
When a region is enclosed by intersecting curves, the intersection points give the limits of
integration.
Boundaries with Changing Functions
If a boundary of a region is defined by more than one function, we can partition the
region into subregions that correspond to the function changes and proceed as usual.
Integrating with Respect to y
Sometimes the boundaries of a region are more easily described by functions of y than by
functions of x. We can use approximating rectangles that are horizontal rather than
vertical and the resulting basic formula has y in place of x.
Examples:
(a) Find the area of the region between y = sec2x and y = sin x from x = 0 to x =  /4.
A
 /4
0
[sec 2 x  sin x]dx
 /4
 [tan x  cos x ]
0

2
2
Page 25 of 53
(b) Find the area of the region enclosed by the parabola y = 2 – x2 and the line y = -x.
2  x 2   x  x  1, 2
2
A   [2  x 2  ( x)]dx
1
x3 x2 2
9
 [2 x 
 ] 1 
3
2
2
(c) Find the area of the region R in the first quadrant that is bounded above by
y  x and below by the x-axis and the line y = x – 2.
R
2
4
x dx   [ x  ( x  2)]dx
0
2
2 3/ 2 2 2 3/ 2 x2
x ]0  [ x 
 2 x]42
3
3
2
10

3
(d) Find the area of the region in Example c by integrating with respect to y.
x  y  2,
x  y2

2
R   ( y  2  y 2 )dy
0
[
y2
y3
10
 2 y  ]02 
2
3
3
Volumes
Volumes of a Solid
The volume of a solid of known integrable cross section area A(x) from x = a to x = b is
b
the integral of A from a to b, V   A( x)dx.
a
How to Find Volume by the Method of Slicing
1. Sketch the solid and a typical cross section.
2. Find a formula for A(x).
3. Find the limits of integration.
4. Integrate A(x) to find the volume.
Solids of Revolution
When a function is rotated about an axis creating circular cross sections then the solid
created is a solid of revolution. There are three different ways to find a solid by
revolution:
1. Disks: when a single function that is entirely to one side of an axis is rotated,
b
V    f ( x) 2 dx
a
2. Washers: when two functions are rotated together about an axis,
b
V    [ f ( x) 2  g ( x) 2 ]dx , where f(x) is dominate over g(x)
a
3. Shells: When the axis of rotation is perpendicular to the axis of integration,
Page 26 of 53
b
V  2  [ x  f ( x)]dx
a
Examples:
(a) The base of a solid is the region between the curve y  2 sin x and the interval
[0,  ] on the x-axis. The cross sections are perpendicular to the x-axis are squares with
bases running from the x-axis to the curve.
a( x)  (2 sin x ) 2

b
V   A( x)dx   4 sin x dx
a
0

 4 cos x ]
0
 4  (1)  8
(b) The region between the function f ( x)  2  x cos x and the x-axis over the interval [2, 2] is revolved about the x-axis to generate a solid. Find the volume of the solid.
b
V    f ( x) 2 dx
a
2
   (2  x cos x) 2 dx  52.43
2
(c) The region in the first quadrant enclosed by the y-axis and the graphs of y = cos x and
y = sin x is revolved about the x-axis to form a solid. Find its volume.
b
V    [ f ( x) 2  g ( x) 2 ]dx
a

 /4
0
(cos 2 x  sin 2 x)dx   
 /4
0
 /4
 sin 2 x 

 2  0

cos 2 x dx

2
(d) The region bounded by the curve y  x , the x-axis, and the line x = 4 is revolved
about the x-axis to generate a solid. Find the volume of the solid.
b
V  2  (radius)(he ight )dy
a
2
 2  ( y )( 4  y 2 )dy  8
0
Length of Curves
Arc Length: Length of a Smooth Curve
If a smooth curve begins at (a, c) and ends at (b, d), a < b, c < d, then the length of the
curve is
L
b
a
2
 dy 
1    dx if y is a smooth function of x on [a, b];
 dx 
Page 27 of 53
2
L
d
c
 dx 
1    dy if x is a smooth function of y on [c, d].
 dy 
Vertical Tangents, Corners, and Cusps
Sometimes a curve has a vertical tangent, corner, or cusp where the derivative we need to
work with is undefined.
Examples:
(a) Find the exact length of the curve y 
4 2 3/ 2
x  1 for 0 ≤ x ≤ 1.
3
dy
 2 2 x1 / 2
dx
1
L   1  (2 2 x 1 / 2 ) 2 dx
0
1
  1  8 x dx
0

1
1
13
(1  8 x) 3 / 2 ] 
0
12
6
Other Definitions
Probability Density Function (pdf)
A probability density function is a function f(x) with domain all reals such that
f ( x)  0 for all x and



f ( x)dx  1.
Then the probability associated with an interval [a, b] is

b
a
f ( x)dx.
Normal Probability Density Function
The normal probability density function (Gaussian curve) for a population with mean
μ and standard deviation σ is
2
2
1
f ( x) 
e ( x   ) /(2 ) .
 2
The 68-95-99.7 Rule for Normal Distributions
Given a normal curve.
 68% of the area will lie within σ of the mean μ,
 95% of the area will lie within 2σ of the mean μ,
 99.7% of the area will lie within 3σ of the mean μ.
Page 28 of 53
Differential Equations and Mathematical Modeling
Antiderivatives and Slope Fields
Solving Initial Value Problems
An equation that contains a derivative is a differential equation. The problem of finding
a function y of x when we are given its derivative and its value at a particular point is
called an initial value problem. The value of f for one value of x is the initial condition
of the problem. When we find all the functions y that satisfy the differential equation we
have solved the differential equation. When we the find the particular solution that
fulfills the initial condition, we have solved the initial value problem.
Slope Field or Direction Field
A slope field or direction field for the first order differential equation
dy
 f ( x, y )
dx
is a plot of short line segments with slopes f(x, y)
for a lattice of points (x, y) in the plane.
Plot the solution curves of the differential
equation
2 xy
y'  
1 x2
Indefinite Integral
The set of all antiderivatives of a function f(x) is
the indefinite integral of f with respect to x and is
denoted by  f ( x) dx.
 f ( x)dx  F ( x)  C
Integral Formulas
1. (a)
n
 x dx 
x n 1
 C , n  1
n 1
dx
 ln x  C
x
e kx
C
2.  e kx dx 
k
cos kx
C
3.  sin kx dx  
k
sin kx
C
4.  cos kx dx 
k
(b)

Page 29 of 53
5.  sec 2 x dx  tan x  C
6.  csc 2 xdx   cot x  C
7.  sec x tan xdx  sec x  C
8.  csc x cot xdx   csc x  C
Properties of Indefinite Integrals
Let k be a real number.
Constant Multiple Rule:
Sum and Difference Rule:
 kf ( x)dx  k  f ( x)dx
 [ f ( x)  g ( x)]dx   f ( x)dx   g ( x)dx
Examples:
(a) Suppose $100 is invested in a n account that pays 5.6% interest compounded
continuously. Find a formula for the amount in the account at any time t.
dy
 .056 y and y (0)  100.
dt
dy
 .056 y (t ), y (t )  Ce.056t
dt
100  C , y (t )  100e .056t
x6
c
6
e 3 x
C
(c)  e 3 x dx 
3
(b)
5
 x dx 
Integration by Substitution
Power Rule for Integration
If u is any differentiable function of x, then  u n du 
u n 1
 C, n  1
n 1
Substitution in Definite Integrals
Substitute u = g(x), du = g’(x)dx, and integrate with respect to u from u = g(a) to u = g(b)

b
a
f ( g ( x))  g ' ( x)dx  
g (b )
g (a)
f (u )du
Separable Differential Equations
A differential equation y’ = f(x, y) is separable if f can be expressed as a product of a
function of x and a function of y. The differential equation then has the form
dy
 g ( x)h( y ).
dx
Page 30 of 53
Examples:
(a) Evaluate  ( x  2) 5 dx.
 ( x  2)
5
dx   u 5 du u  x  2, du  d ( x  2)  dx
u6

C
6
( x  2) 6

C
6
(b) Evaluate
 /4

0
tan x sec 2 x dx.
u  tan x, du  sec 2 x dx
u2
2
1
1
0 
2
2
2
dy
 2 x(1  y 2 )e x
(c) Solve the differential equation
dx
2
1
x
 1  y 2 dy   2 xe dx
1
1
 1  y 2 dy  tan y  C
1

0
udu 
1
]
0

x
u
 2 xe dx   e du
2
u  x 2 , du  2 x dx
 eu  C
tan 1 y  e x  C
2
y  tan( e x  C )
2
Integration by Parts
Product Rule in Integral Form
d
dv
du
uv  u
v
dx
dx
dx
 dv 
d

 du 
  u dx dx    dx uv dx    v dx dx
 du 
 uv    v dx
 dx 
Integration by Parts Formula:
 udv  uv   vdu
Sometimes integration by parts must be used more than once when integrating. Solving
for the unknown integral requires two integration by parts, followed by solving for the
unknown integral. If many steps are required in integration by parts, then time is saved by
using tabular integration.
Page 31 of 53
Examples:
(a) Evaluate
 x cos x dx
ux
dv  cos x
du  dx
v  sin x

(b) Evaluate
x
2
x cos x dx  x sin x   sin x dx
 x sin x  cos x  C
e x dx
x2
+
ex
2x
-
ex
2
+
ex
ex
0
x
2
e x dx  x 2 e x  2 xe x  2e x  C
Exponential Growth and Decay
Exponential Model
Because population consists of only whole numbers, it is commonly expressed with a
differentiable function P growing at a rate proportional to the size of the population. This,
for some constant k,
dP
 kP
dt
K is the relative growth rate.
Logistic Growth Rate
In realistic models, there is a maximum population M, the carrying capacity, that the
environment is capable of sustaining in the long run. If we assume the relative growth
rate is proportional to 1-(P/M) with positive proportionality constant k, then
dP / dt
P

 k 1   or
P
 M
dP
k

P( M  P).
dt M
The solution to this logistic differential equation is called the logistic growth model.
Examples:
(a) Find an initial value problem model for world population and use it to predict the
population in the year 2010. Let t = 0 represent 1986, therefore t = 24 will represent
2010. If k = .0178 then what is the estimated population in 2010?
Page 32 of 53
dP
 .0178P,
P(0)  4936
dt
P  4936e .0178t , P(24)  7567
(b) A national park is known to be capable of supporting no more than 100 grizzly bears.
Ten bears are in the park at present. We model the population with a logistic differential
equation with k = .01. Find a logistic growth model P(t) for the population and when will
the bear population reach 50?
dP .01

P(100  P)  .001P(100  P),
P(0)  10
dt 100
1
dP
1  dP
1
 .001   
 .01

P(100  P) dt
 P 100  P  dt
100  P
100  P
 .1t  C 
 Ce .1t
P
P
100
P
1  Ae .1t
100
10 
 A9
1  Ae 0
100
100
P
,
50 
.1t
1  9e
1  9e .1t
1  9e .1t  2
ln
e .1t  9
ln 9
t
 22 years
.1
Numerical Methods
Euler’s Method
If we are given a differential equation dy/dx = f(x, y) and an initial condition y(x0) = y0,
we can approximate the solution y = y(x) by its linearization. The linearization will give a
good approzimation to the solution y(x) in a short interval about x0. The basis of Euler’s
method is to patch together a string of linearizations to approximate the curve over a
longer stretch.
(a) Find the first three approximations y1, y2, y3 using Euler’s method for the initial value
problem
y’ = 1 + y,
y(0) = 1,
starting at x0 = 0 with dx = .1.
Page 33 of 53
y1  y 0  f ( x0 , y 0 )dx
 1  (1  1)(. 1)  1.2
y 2  y1  f ( x1 , y1 )dx
 1.2  (1  1.2)(. 1)  1.42
y3  y 2  f ( x2 , y 2 )
 1.42  (1  1.42)(. 1)  1.662
Improved Euler’s Method
We can improve on Euler’s method by taking an average of two slopes. We first estimate
yn as in the original Euler method, but denote it by zn. We then take the average of
f(xn-1, yn-1) and f(xn, zn) in place of f(xn-1, yn-1) in the next step. Thus
z n  y n 1  f ( x n 1 , y n 1 )dx,
 f ( x n 1 , y n 1 )  f ( x n , z n ) 
y n  y n 1  
 dx.
2


L’Hôpital’s Rule, Improper Integrals, and Partial Fractions
L’Hôpital’s Rule
Theorem 1: L’Hôpital’s Rule (First Form)
Suppose that f(a) = g(a) = 0, that f’(a) and g’(a) exist, and that g’(a) ≠ 0. Then
f ( x) f ' (a )
lim

.
x a g ( x)
g ' (a)
Theorem 2: L’Hôpital’s Rule (Stronger Form)
Suppose that f(a) = g(a) = 0, that f and g are differentiable on an open interval I
containing a, and that g’(x) ≠ 0 on I if x ≠ a. Then
f ( x)
f ' ( x)
lim
 lim
x a g ( x)
x a g ' ( x)
Exponential Indeterminate Forms
lim ln f ( x)  L  lim f ( x)  lim e ln f ( x )  e L
xa
xa
x a
Here a can be finite or infinite.
Indeterminate Forms
0
,
0

,   0,   , 1 , 0 0 ,  0

Examples:
(a) lim
x 0
1  cos x 0
 L' H
0 
x  x2
lim
x 0
sin x 0
 0
1  2x 1
Page 34 of 53
Relative Rates of Growth
Faster, Slower, Same-rate Growth as x∞
Let f(x) and g(x) be positive for x sufficiently large.
1. f grows faster than g (and g grows slower than f) as x∞ if
f ( x)
a. lim

x  g ( x )
g ( x)
b. lim
0
x  f ( x )
2. f and g grow at the same rate as x∞ if
f ( x)
a. lim
L0
x  g ( x )
Transitivity of Growing Rates
If f grows at the same rate as g as x∞ and g grows at the same rate as h as x∞, then f
grows at the same rate as h as x∞.
Examples:
(a) Show that ln x grows slower than x as x∞
ln x
1/ x
1
lim
 lim
 lim  0
x  x
x  1
x  x
(b) Show that f ( x)  x 2  5 and g ( x)  (2 x  1) 2 grow at the same rate as x∞
x2  5
5
 lim 1  2  1
x 
x
x
lim
x 
2
 2 x 1

(2 x  1) 2
1 
  lim  2 
lim
 lim 
2  4
x 
x 
x 
x
x 
x


 lim
x 
 f h
f
1 1
 lim     1  
g x  h g 
4 4
Improper Integrals
Improper Integrals with Infinite Integration Limits
Integrals with infinite limits of integration are improper integrals.
1. If f(x) is continuous on [a, ∞), then


a
f ( x)dx  lim

b
f ( x)dx.
b  a
2. If f(x) is continuous on (-∞, b], then

b

f ( x)dx  lim

b
a  a
f ( x)dx.

c



c
3. If f(x) is continuous on (-∞, ∞), then  f ( x)dx   f ( x)dx   f ( x)dx.
Page 35 of 53
Improper Integrals with Infinite Discontinuities
Integrals of functions that become infinite at a point withing the interval of integration are
improper integrals.
1. If f(x) is continuous on (a, b], then

b
a
b
f ( x)dx  lim  f ( x)dx.
c a
c
2. If f(x) is continuous on [a, b), then

b
a
c
f ( x)dx  lim  f ( x)dx.
c b
a
3. If f(x) is continuous on [a, c) U (c, b], then

b
a
c
b
a
c
f ( x)dx   f ( x)dx   f ( x)dx.
Direct Comparison Test
Let f and g be continuous on [a, ∞) with 0 ≤ f(x) ≤ g(x) for all x ≥ a. Then

2. 
1.

a

a

f ( x)dx converges if
g ( x)dx diverges if


a

a
g ( x)dx converges.
f ( x)dx diverges.
Limit Comparison Test
If the positive functions f and g are continuous on [a, ∞) and if
f ( x)
lim
 L,
0  L  ,
x  g ( x )
then


a
f ( x)dx
and


g ( x)dx
a
both converge or both diverge.
Examples:
(a) Does the improper integral


1


1
dx
converge or diverge?
x
b dx
dx
 lim 
x b  1 x
b
 lim ln x ]  lim (ln b  ln 1)  
b 
1
b 
Thus, the integral diverges.
4 dx
.
(b) Evaluate 
1 x2
Page 36 of 53
2 dx
4 dx
dx


1 x2
1 x2
2 x2
2 dx
c dx
1 x  2  clim
 1
2
x2

4
c
 lim ln x  2 ]
c2
1
 lim (ln c  2  ln  1 )  
c 2
Thus, because one side diverges, the entire integral diverges.
sin 2 x
(c) 
dx converges because
1
x2
 1
sin 2 x 1
0
 2 on [1, ∞) and  2 dx converges
2
a x
x
x

 dx
dx
(d) Show that 
converges
by
comparison
with
1 x 2
1 1 x2
f ( x)
1/ x 2
1 x2
lim
 lim

lim
x  g ( x )
x  1 /(1  x 2 )
x 
x2
2x
 lim
1
x  2 x

Partial Fractions and Integral Tables
Partial Fractions
Success in writing a rational function f(x)/g(x) as a sum of partial fractions depends on
two things:
 The degree of f(x) must be less than the degree of g(x). That is,
the fraction must be proper. If it isn’t, divide f(x) by g(x) and
work with the remainder term.
 We must know the factors of g(x). In theory, any polynomial with
real coefficients can be written as a product of real linea factors
and real quadratic factors.
Method of Partial Fractions
1. Let x – r be a linear factor of g(x). Suppose (x – r)m is the highest power of x – r that
divides g(x). Then, to this factor, assign the sum of the m partial fractions:
Am
A1
A2


.
2
x  r (x  r)
(x  r) m
Do this for each distinct linear factor of g(x).
2. Let x 2  px  q be a quadratic factor of g(x). Suppose ( x 2  px  q )n is the highest
power of this factor that divides g(x). Then, to this factor, assign the sum of the n partial
fractions:
B x  Cn
B1 x  C1
B x  C2
 2 2
 2 n
.
2
2
x  px  q ( x  px  q)
( x  px  q) n
Page 37 of 53
Do this for each distinct quadratic factor of g(x) that cannot be factored into linear factors
with real coefficients.
3. Set the original fraction f(x)/g(x) equal to the sum of all these partial fractions. Clear
the resulting equation of fractions and arrange the terms in decreasing powers of x.
4. Equate the coefficients of corresponding powers of x and solve the rsulting equation
for the undetermined coefficients.
Trigonometric Substitutions
Trigonometric Substitutions enable us to replace binomials of the form
a 2  x 2 , a 2  x 2 , and x 2  a 2 by single squared terms, and thereby transform a number of
integrals into ones we can evaluate directly or find in a table of integrals.
Common Substitutions:
1. x  a tan  replaces a 2  x 2 with a 2 sec 2 
2. x  a sin  replaces a 2  x 2 with a 2 cos 2 
3. x  a sec replaces x 2  a 2 with a 2 tan 2 
Examples:
(a) Evaluate
6x  7
 ( x  2)
2
dx
 6
6x  7
5
dx

 ( x  2) 2
  x  2  ( x  2) 2

dx

dx
 5 ( x  2)  2 dx
x2
 6 ln x  2  5( x  2) 1  C
 6
(b) Evaluate

x 3 dx
9  x2
x  3 sin  , dx  3 cos d ,
9  x 2  9  9 sin 2   9(1  sin 2  )  9 cos 2 

27 sin 3  3 cos d

 27  1  cos 2  sin d
2
3 cos 
9 x

x 3 dx
 27 cos   9 cos 3   C
 9  x2
 27

3

 9 9  x
  9  x2
  9
 
3
 
9  x 

2 3/ 2
2
3
3

 C


C
Page 38 of 53

Infinite Series
Power Series
Infinite Series
An infinite series is an expression of the form
a1  a2  a3    an   ,

or
a
k 1
k
The numbers a1, a2, … are the terms of the series; an is the nth term.
The partial sums of the series form a sequence
s1  a1
s 2  a1  a 2
s 3  a1  a 2  a 3

n
sn   ak
k 1

of real numbers, each defined as a finite sum. If the sequence of partial sums has a limit S
as n∞, we say the series converges to a sum S, and we write

a1  a 2  a3    a n     a k  S
k 1
Otherwise, we say the series diverges.
Geometric Series

a  ar  ar 2  ar 3    ar n 1     ar n 1
n 1
Converge to the sum
a
if r  1 , and diverges if r  1 .
1 r
Power Series
An expression of the form

c
n 0
n
x n  c0  c1 x  c 2 x 2    c n x n  
is a power series centered at x = 0. An expression of the form

c
n 0
n
( x  a) n  c0  c1 ( x  a )  c 2 ( x  a) 2    c n ( x  a) n  
is a power series centered at x = a. The term c n ( x  a) n is the nth term; the number a is
the center.
Page 39 of 53
Term-by-Term Differentiation

If f ( x)   c n ( x  a) n  c0  c1 ( x  a)  c 2 ( x  a ) 2    c n ( x  a) n  
n 0
converges for x  a  R , then the series

 nc
n 0
n
( x  a) n 1  c1  2c 2 ( x  a)  3c3 ( x  a) 2    nc n ( x  a) n 1   ,
obtained by differentiating the series for f term by term, converges for x  a  R and
represents f’(x) on that interval. If the series for f converges for all x, then so does the
series f’.
Term-by-Term Integration

If f ( x)   c n ( x  a) n  c0  c1 ( x  a)  c 2 ( x  a ) 2    c n ( x  a) n  
n 0
converges for x  a  R , then the series
( x  a) n1
( x  a) 2
( x  a) 3
( x  a) n1
 c0 ( x  a)  c1
 c2
   cn
 ,
n 1
2
3
n 1
n 0
obtained by integrating the series for f term by term, converges for x  a  R and

 cn
represents

x
a
f (t )dt on that interval. If the series for f converges for all x, then so does
the series for the integral.
Examples:
3
3
3
3


   n   converge?
10 100 1000
10
1
In decimal form: 0.3, 0.33, 0.333, 0.3333, … =
YES
3
(b) Decide if the series converges and if so, to what sum.
(a) Does the series

k
1
5
3

YES
  

3 2
k 0  5 
1
5
1
1
(c) Find a power series that represents 
on (0, 2).
x 1  ( x  1)
1  ( x  1)  ( x  1) 2  ( x  1) 3    ( x  1) n  
1
1
 1  x  x 2    x n   , find a power series to represent
(d) Given that
.
1 x
(1  x ) 2
d  1  d
(1  x  x 2  x 3    x n  )


dx  1  x  dx
1
 1  2 x  3 x 2  4 x 3    nx n 1  
2
(1  x)
Page 40 of 53
1
 1  x  x 2  x 3    ( x) n  
1 x
find a power series to represent ln (1 + x).
x 1
x
2
3
n n
0 1  t dt  0 (1  t  t  t    (1) t  )dt
(e) Given that
x
 t2 t3 t4

t n 1
ln( 1  t ) ]  t       (1) n
 
0
2 3 4
n 1

0
x
n 1
x2 x3 x4
n x
ln( 1  x)  x 


   (1)

x
3
4
n 1
Taylor Series
Taylor Series Generated by f at x = a
Let f be a function with derivatives o all orders throughout some open interval containing
a. Then the Taylor series generated by f at x = a is
f ' ' (a)
f ( n ) (a)
f ( k ) (a)
f (a)  f ' (a)( x  a) 
( x  a) 2   
( x  a) n    
( x  a) k .
2!
n!
k!
The partial sum
n
f ( k ) (a)
Pn ( x)  
( x  a) k
k
!
k o
Is the Taylor polynomial of order n for f at x = a.
Maclaurin Series
A Maclaurin series is a Taylor series that is centered around x= 0.
Examples:
(a) Find the fourth order Taylor polynomial that approximates y = cos 2x near x = 0.
x2 x4
x 2n
cos x  1 

   (1) n

2! 4!
(2n)!
cos 2 x  1 
(2 x) 2 (2 x) 4
(2 x) 2 n

   (1) n

2!
4!
(2n)!
(2 x) 2 (2 x) 4
2

 1  2x 2  x 4 .
2!
4!
3
(b) Find the Taylor series generated by f(x) = ex at x = 2.

e2
e2
e2
e x  e 2  e 2 ( x  2)  ( x  2) 2    ( x  2) n     ( x  2) k
2!
n!
k 0 k!
P4 ( x)  1 
Page 41 of 53
Table of Maclaurin Series

1
 1 x  x 2  x n    x n
1 x
n0
( x  1)

1
 1  x  x 2    ( x) n     (1) n x n
1 x
n 0
e x  1 x 

x2
xn
xn

  
2!
n!
n  0 n!
( x  1)
(all real x)

x3 x5
x 2 n 1
x 2 n 1
n
n
sin x  x 

   (1)
    (1)
(all real x)
3! 5!
(2n  1)!
(2n  1)!
n0
cos x  1 

x2 x4
x 2n
x 2n

   (1) n
    (1) n
(all real x)
2! 4!
(2n)!
(2n)!
n0
ln( 1  x)  x 
tan 1 x  x 

x2 x3
xn
xn

   (1) n 1
    (1) n 1
2
3
n
n
n 1
(1  x  1)

x3 x5
x 2 n 1
x 2 n 1

   (1) n
    (1) n
( x  1)
3
5
2n  1
2n  1
n 0
Taylor’s Theorem
Taylor’s Theorem with Remainder
If f has derivatives of all orders in an open interval I containing a, then for each positive
integer n and for each x in I,
f ' ' (a)
f ( n ) (a)
f ( x)  f (a)  f ' (a)( x  a) 
( x  a) 2   
( x  a ) n  Rn ( x ) ,
2!
n!
where
f n1c
Rn ( x) 
( x  a) n1
(n  1)!
for some c between a and x.
Remainder Estimation Theorem
If there are positive constants M and r such that f
n 1
t  Mr n 1 for all t between a and x,
then the remainder Rn(x) in Taylor’s Theorem satisfies the inequality
Rn ( x)  M
r n1 x  a
n 1
.
(n  1)!
If these conditions hold for every n and all other conditions of aylor’s Theorem are
satisfied by f, then the series converges to f(x).
Page 42 of 53
Examples:

(a) Prove that the series
 (1) k
k 0
Rn ( x ) 

x 2 k 1
converges to sin x for all real x.
(2k  1)!
f n 1c
( x  0) n 1
(n  1)!
f
n 1
c
(n  1)!
x n 1
n 1
x
1

x n 1 
.
(n  1)!
(n  1)!
(b) The approximation ln( 1  x)  x  ( x 2 / 2) is used when x is small. Use the Remainder
Estimation Theorem to get a bound for the maximum error when x  0.1 .
M 
2
2000

3
729
(1  (0.1))
3
2000 x
2000 0.13
R2 ( x ) 



 4.6  10 4.
729
3!
729
3!
Radius of Convergence
The Convergence Theorem for Power Series
There are three possibilities for


n 0
cn ( x  a) n with respect to convergence:
1. There is a positive number R such that the series diverges for |x – a| > R but
converges for |x –a| < R. The series may or may not converge at either of the
endpoints x = a – R and x = a + R.
2. The series converges for every x (R = ∞).
3. The series converge at x = a and diverge elsewhere (R = 0).
The nth-Term Test for Divergence


n 1
a n diverges if lim a n fails to exist or is different from zero.
n
The Direct Comparison Test
a
(a)  a
Let
n
be a series with no negative terms.
converges if there is a convergent series  c n with an ≤ cn for all n > N, for
some integer N.
(b)  an diverges if there is a divergent series  d n or nonnegative terms with an ≥ dn
for all n > N, for some integer N.
n
Page 43 of 53
Absolute Convergence
If the series
a
n
of absolute values converge, then
a
n
converges absolutely.
Absolute Convergence Implies Convergence
If
a
converges, then
n
a
n
converges.
The Ratio Test
Let
a
n
be a series with positive terms, and with
a n 1
L
n  a
n
lim
Then,
(a) the series converges if L < 1,
(b) the series diverges if L > 1,
(c) the test is inconclusive if L = 1.
Examples:
(a) Find the radius of convergence of the series


n 0
n! x n .
If |x| ≥ 1, the terms n!xn grows without bound as x∞
diverge
If 0 < |x| < 1, then
n!
lim n! x n  lim
diverge
n 
n  (1 / x ) n
If x = 0, the series does converge. The radius of convergence is R = 0, and the
interval of convergence is {0}.

x 2n
(b) Prove that 
2
n 0 ( n!)
x 2n
x 2n

n!
(n!) 2

2
2
(x 2 )n
 e x which converges. Since e x dominates

n!
n 0

x 2n
, the latter series must converge.

2
n  0 ( n!)

(c) Show that
(sin x) n
converges for all x.

n!
n 0


| sin x | n
1

which converges


n!
n 0
n  0 n!
 the first series converges absolutely by direct comparison
Page 44 of 53

(d) Find the radius of convergence of
nx n
 10
n 0
lim
a n 1
n 
 lim
(n  1) x n 1
10 n 1
n 
an

n
.
10 n
n xn
x
 n 1 x
 lim 


n 
 n  10 10
 10  x  10,
R  10
Testing Convergence at Endpoints
The Integral Test
Let {an} be a sequence of positive terms. Suppose that an = f(n), where f is a continuous,
positive, decreasing function of x for all x ≥ N (N a positive integer). Then the series

a
n N
n
and the integral


N
f ( x)dx either both converge or both diverge.
Harmonic Series and p-series

Any series of the form
 (1 / n
p
) , p a real constant, is called a p-series. The p-series with
n 1
p = 1 is the harmonic series, and it is probably the most famous divergent series in
mathematics.
The Limit Comparison Test (LCT)
Suppose that an > 0 and bn > 0 for all n ≥ N (N a positive integer).
a
1. If lim n  c , 0 < c < ∞, then  an and  bn both converge or both diverge.
n  b
n
an
2. If lim
 0 and  bn converges, then  an converges.
n  b
n
a
3. If lim n   and  bn diverges, then  an diverges.
n  b
n
The Alternating Series Test (Leibniz’s Theorem)

The series
 (1)
n 1
n 1
u n  u1  u 2  u 3  u 4   converges is all three of the following
conditions are satisfied:
1. each un is positive;
2. un ≥ un+1 for all n ≥ N, for some integer N;
3. lim n u n  0 .
Page 45 of 53
The Alternating Series Estimation Theorem
If the alternating series


n 1
(1) n1 u n satisfied the conditions of Theorem 12, then the
truncation error for the nth partial sum is less than un+1 and has the same sign as the
unused tem.
Rearrangements of Absolutely Convergent Series
If
a
n
converges absolutely, and if b1, b2, b3, …, bn, … is any rearrangement of the
sequence {an}, then
b
n
converges absolutely and


b  n1 an .

n 1 n
Rearrangement of Conditionally Convergent Series
If  an converges conditionally, then the terms can be rearranged to form a divergent
series. The terms can also be rearranged to form a series that converges to any
preassigned sum.

How to Test a Power Series
c
n 0
( x  a) n for Convergence
n
1. Use the Ratio Test to find the values of x for which the series converges absolutely.
Ordinarily, this is an open interval
aR xaR
In some instances, the series converges for all values of x. In rare cases, the series
converges only at x = a.
2. If the interval of absolute convergence is finite, test for convergence or divergence at
each endpoint. The Ratio Test fails at these points. Use a comparison test, the Integral
test, or the Alternating Series Test.
3. If the interval of absolute convergence is a  R  x  a  R , conclude that the series
diverges for |x – a| > R, because for those values of x the nth term does not approach zero.
Examples:

(a) Does
x
n 1
1
x


1
converge?
1
x x
 lim

k
k  1

x 3 / 2 dx
 lim  2 x 1 / 2
k 

k
1
Integral converges  series converges.
 2

 lim  
 2   2
k 
k


Page 46 of 53

3 5 7
9
2n  1
converge?
  
  
2
4 9 16 25
n 1 (n  1)
2n  1
2
behaves like
2
n
(n  1)
(b) Does the series
(2n  1) /( n  1)
2n  1 n
 lim

n 
n  ( n  1) 2
1/ n
1
lim
2n 2  n
2
n  n 2  2n  1
 lim
Since the limit is positive and
 (1/ n) diverges,

2n  1
 (n  1)
n 1
2
also diverges.
Parametric, Vector, and Polar Functions
Parametric Functions
Derivative at a Point
A parameterized curve x = f(t), y = g(t), a ≤ t ≤ b, has a derivative at t = t0 if f and g have
derivatives at t = t0.
d  dy 
dy
2
dx
dy dt
d y dt  dx 
whenever
≠ 0.


2
dx
dt
dt dx
dx
dt
dt
Length of a Smooth Parameterized Curve
If a smooth curve x = f(t), y = g(t), a ≤ t ≤ b, is traversed exactly once as t increases from
a to b, the curve’s length is
L
b
a
2
 dx   dy 
    2dt .
 dt   dt 
Surface Area
If a smooth curve x = f(t), y = g(t), a ≤ t ≤ b, is traversed exactly once as t increases from
a to b, then the areas of the surface generated by revolving the curve about the coordinate
axes are as follows.
1. Revolution about the x-axis (y ≥ 0):
2
2
2
2
b
 dx   dy 
S   2y      dt
a
 dt   dt 
2. Revolution about the y-axis (x ≥ 0):
 dx   dy 
S   2x      dt
a
 dt   dt 
b
Page 47 of 53
Examples:
(a) Find d2y/dx2 as a function of t if x = t – t2, y = t – t3.
dy 1  3t 2

dx 1  2t
dy ' 2  6t  6t 2

dt
(1  2t ) 2
d 2 y (2  6t  6t 2 ) /(1  2t ) 2 1  6t  6t 2


1  2t
dx 2
(1  2t ) 3
(b) Find the length of the curve x = cos t, y = sin t., 0 ≤ t ≤ 2π
dy
dx
 cos t
  sin t
dt
dt
L
2
( sin t ) 2  (cos t ) 2 dt
0
2
  1dt
0
2
 t ]  2
0
(c) The standard parameterization of the circle of radius 1 centered at the point (0, 1) in
the xy-plane is
x = cos t,
y = 1 + sin t, 0 ≤ t ≤ 2π
dy
dx
 cos t
  sin t
dt
dt
2
S  2  (1  sin t ) ( sin t ) 2  (cos t ) 2 dt
0
2
 2  (1  sin t )dt
0
 2 t  cos t 0  4 2
2
Vectors
Vector, Equal Vector
A vector in the plane is represented by a directed line segment. Two vectors are equal if
they have the same length and direction.
Component Form of a Vector
If v is a vector in the plane equal to the vector with initial point (0, 0) and terminal point
(v1, v2), then the component form of v is v  v1 ,v2 .
Dot Product
The dot product u ∙ v of vectors u = u1 ,u 2 and v = v1 ,v2
u  v  u1v1  u2 v2 .
Page 48 of 53
Magnitude
The magnitude of v is:
v  v1  v2 .
2
2
Angle Between Two Vectors
The angle between nonzero vectors u and v is
 uv 
.
  cos 1 

 u v 
Examples:
(a) Find the angle between the vectors u  3,5 and v  5,2
 35  5 2 

 34 29 
  cos 1 
 25 
 cos 1 
  .6499 radians or 37.2346 o
 986 
Limit
Let r(t) = f(t)i + g(t)j, then
lim f (t )  L1
t c
lim g (t )  L2
and
t c
then the limit of r(t) as t approaches c is
lim r (t )  L  L1i  L2 j .
t c
Continuity are a point
A vector function r(t) is continuous at a point t = c in its domain if
lim r (t )  r (c) .
t c
Component Test for Continuity at a Point
The vector function r(t) = f(t)i + g(t)j is continuous at t = c if and only if f and g are
continuous at t = c.
Velocity, Speed, Acceleration, Direction of Motion
r (t )  f (t )i  g (t ) j
r ' (t )  v(t )  f ' (t )i  g ' (t ) j
Speed: |v(t)|
r ' ' (t )  v' (t )  a(t )  f ' ' (t )i  g ' ' (t ) j
v
Direction of Motion:
v
Position vector:
Velocity vector:
Acceleration vector:
Page 49 of 53
Examples:
(a) If r(t) = (cos t)i + (sin t)j, then lim r (t )  ( lim cos t )i  ( lim sin t ) j 
t  / 4
t  / 4
t  / 4
2
2
i
j.
2
2
(b) Find the velocity and acceleration vectors if r (t )  x i  x 2 j
4
v(t )  4 x 3 i  2 xj
a(t )  12 x 2 i  2 j
Modeling Projectile Motion
Height, Flight Times, and Range for Ideal Projectile Motion
For ideal projectile motion when an object is launched from the origin over a horizontal
surface with initial speed v0 and launch angle α:
(v sin  ) 2
Maximum height:
y max  0
2g
2v sin 
t 0
Flight time:
g
2
Range:
R
v0
sin 2
g
Projectile Motion with Linear Drag
Equations for the motion of a projectile with linear drag force launched from the origin
over a horizontal surface at t = 0:
Vector form:
v
g
v

r  0 (1  e kt )(cos  )i   0 (1  e kt )(sin  )  2 (1  kt  e kt ) j
k
k
k

Parametric form:
v
x  0 (1  e  kt ) cos 
k
v
g
y  0 (1  e  kt ) sin   2 (1  kt  e  kt )
k
k
Where the drag coefficient k is a positive constant representing resistance due to air
density, v0 and α are the projectile’s initial speed and launch angle, and g is the
acceleration of gravity.
Page 50 of 53
Examples:
(a) Find the maximum height, flight time, and range of a projectile fired from the origin
over horizontal ground at an initial speed of 500 m/sec and a launch angle of 60°.
(500 sin 60  ) 2
Maximum height =
 9566.33m
2(9.8)
2(5000) sin 60 
Flight Time =
 88.37 sec
9.8
500 2 sin 120 
 22,092.48m
Range =
9.8
Polar Curves
Polar Coordinates
P(r, θ), r is the distance from O to P and θ is the angle from initial ray to ray OP.
Equations Relating Polar and Cartesian Coordinates
x  r cos  ,
y  r sin  ,
x2  y2  r 2 ,
y
 tan 
x
Examples:
(a) Find a Cartesian equivalent for the polar equation r 
r (2 cos   sin  )  4
2r cos   r sin   4
2x  y  4
y  2x  4
4
2 cos   sin 
Slope of the Curve r = f(θ)
dy
dy / d
f ' ( ) sin   f ( ) cos 


|
(
r
,

)
dx
dx / d
f ' ( ) cos   f ( ) sin 
provided dx/dθ ≠ 0 at (r, ≠ θ).
Area in Polar Coordinates
The area of the region between the origin and the curve r = f(θ), α ≤ θ ≤ β, is
1
A   r 2 d
 2
Page 51 of 53
Area Between Polar Curves
The area of the region 0 ≤ r1(θ) ≤ r2(θ), α ≤ θ ≤ β, is
1
1 2
1
2
2
2
A   r2 d   r1 d   (r2  r1 )d .
 2
 2
 2
Length of a Polar Curve
If r = f(θ) has a continuous first derivative for α ≤ θ ≤ β and if the point P(r, θ) traces the
curve r = f(θ) exactly once as θ runs from α to β, then the length of the curve is
L


2
 dr 
r 
 d .
 d 
2
Area of a Surface of Revolution
If r = f(θ) has a continuous first derivative for α ≤ θ ≤ β and if the point P(r, θ) traces the
curve r = f(θ) exactly once as θ runs from α to β, then the areas of the surfaces generated
by revolving the curve about the x- and y-axes are given by the following formulas.
Revolution about the x-axis (y ≥ 0):
2
 dr 
S   2r sin  r  
 d

 d 
Revolution about the y-axis (x ≥ 0):

2
2
 dr 
S   2r cos  r  
 d

 d 

2
Examples:
(a) Find the area of the region in the plane enclosed by the cardioid r  2(1  cos  ) .
2 1
A
4(1  cos  ) 2 d
0 2
2
  2(1  2 cos   cos 2  )d
0
2
  (3  4 cos   cos 2 )d
0
2
sin 2 

 3  4 sin  
 6  0  6
2  0

(b) Find the area of the region that lies inside r = 1 and outside r = 1 – cosθ
 /2 1
A  2
(1  (1  2 cos   cos 2  )) d
0
2

 /2
0
(2 cos   cos 2  )d
 1.215
Page 52 of 53
(c) Find the length of the cardioid r = 1 – cosθ
dr
r  1  cos  ,
 sin 
d
2
 dr 
r 
  2  2 cos 
 d 
2
L
2
0
2  2 cos  d  8
(d) Find the area of the surface generated by revolving the right-hand loop of the
lemniscate r 2  cos 2 about the y-axis.
dr
sin 2
r  cos 2 ,

d
cos 2
sin 2 2
1
 dr 
r 

  cos 2 
cos 2
cos 2
 d 
2
2
S
 /4
 / 4
2 cos 2 cos 
1
d
cos 2
 2  cos d  2 sin   / 4
 /4
 2 2  8.886
Page 53 of 53