MATH-2410 Review Concepts (Haugen)
Unit 1: Limits
Exam 1 – Sections 1.2-1.5
How do we evaluate a limit of a function, lim f  x  ?
x c
Graphically
Numerically (table of values of x near c)
Analytically (using algebra or previous calculus theorems)
Under what conditions will the limit of a function fail to exist?
Unbounded behavior
Oscillatory behavior
The function approaches different values as we approach c from the left and the right
Limits and Limit Properties
Scalar Multiple, Sum or Difference, Product Rule, Quotient Rule, and Power Rule (p. 59)
Limits of continuous functions (direct substitution)
Limits of Polynomial Functions
Limits of Rational Functions
Limits of Radical Functions
Limits of Trigonometric Functions
Strategies for Finding Limits
Dividing out (factoring)
Rationalizing
Squeeze Theorem
sin  x 
1  cos  x 
Special Trigonometric Limits: lim
 1 and lim
0
x 0
x
One-sided Limits: lim f  x  and lim f  x 
xc
x 0
xc
Continuity
Three conditions for a function to be continuous at a point c:
1. f  c  is defined
2. lim f  x  exists
x c
3. lim f  x   f  c 
xc
Discontinuities
Removable and Nonremovable
Infinite Limits (unbounded behavior near a point c)
lim f  x    or lim f  x   
xc
xc
Vertical Asymptotes
x
Unit 2: Differentiation
Exam 2 – Sections 2.1-2.6
Definition of the Derivative
Slope of the tangent line to the graph of f at x = c
f   x   lim
x 0
f  x  x   f  x 
x
and
f   c   lim
x c
f  x  f c
xc
Vertical Tangent Lines
Differentiation Rules:
d n
 x   nx n 1
dx
d
d
d
Sum/Difference Rule
 f  x   g  x   
 f  x     g  x 
dx
dx
dx
Product Rule  f  g   f  g   g  f  and  f  g  h   f   g  h  f  g   h  f  g  h
Power Rule
 f  g  f   f  g 
Quotient Rule   
g2
g

Differentiating Composite Functions (Chain Rule)  f  g  x     f   g  x    g   x 
Implicit Differentiation
Related Rates
Relationship between Differentiability and Continuity:
If a function is differentiable at a point c, it is continuous at c. (differentiability implies continuity)
If a function is continuous at a point c, it may not be continuous at c.
(continuity does not necessarily imply differentiability)
Physics Applications
1 2
gt  v0t  s0
2
Velocity Function v  t   s  t 
Position Function s  t  
Unit 3: Applications of the Derivative
Exam 3 – Sections 3.1-3.9
Extrema on an Interval
Global Extrema (Global or Absolute Maximum / Minimum)
Relative Extrema (Relative or Local Maximum / Minimum)
Extreme Value Theorem
Guarantees that every continuous function will have both a global maximum and a global
minimum on a closed interval
Critical Numbers
Let f be defined at c. If f   x   0 or f is not differentiable at c, then c is a critical number
of f . (Relative extrema only occur at critical numbers)
Rolle’s Theorem (helps us find extrema on the interior of [a, b] )
Mean Value Theorem
Identifying Intervals where f  x  is:
Increasing / Decreasing
First Derivative Test
Identifying Intervals where f  x  is:
Concave up / Concave down
Points of Inflection
Second Derivative Test
Limits at Infinity
lim f  x  and
x
lim f  x 
x 
Rational Functions
Radical Functions
Trigonometric Functions (Squeeze Theorem usually comes in handy)
Optimization Problems
Primary and Secondary Equations
Newton’s Method
Differentials
Unit 4: Integration
Exam 4 – Sections 4.1-4.6
Antiderivatives F   x   f  x  for all x in an interval I
Indefinite Integrals
 f  x dx  F  x   C
Differential Equations
General Solution
Particular Solution (initial value problems)
Summation
Summation Notation and Formulas
Partition (subdivide) the closed interval [a, b]
ba
Width of ith subinterval: x 
(Regular Partition)
n
Upper and Lower Sums
 f  M  x
n
S  n 
i
 f m  x
n
and s  n  
i 1
i
i 1
If f  x  is continuous and nonnegative on [a, b], then the area below f  x  above the x-axis between the
 f c  x
n
lines x = a and x = b is given by: lim S  n   lim s  n   lim
n
n
n
i
i 1

n
Riemann Sum =
f  ci  xi
i 1
Width of each subinterval may vary
f  x  is defined (not necessarily continuous) at each x in [a, b].
Definite Integrals
b
 f c  x   f  x  dx
n
lim
 0
i
i
i 1
(A definite integral is a limit of Riemann Sums)
a
Evaluating Definite Integrals
b
Fundamental Theorem of Calculus:
 f  x  dx  F b  F  a 
a
x

d 
Second Fundamental Theorem of Calculus:
f  t  dt   f  x 

dx 
a

Integrating Composite Functions
Substitution and Change of Variables

Numerical Methods (Trapezoidal Rule and Simpson’s Rule)
Unit 5: Transcendental Functions
Exam 5 – Sections 5.1-5.5
Natural Logarithmic Function f  x   ln  x 
Properties of Logarithms
Product Rule, Quotient Rule, and Power Rule
Derivative of Natural Logarithmic Functions
d
1
ln  x   
dx
x
d
1
 ln x  
dx
x
d
u
ln  u   
dx
u
d
u
ln u  
( u  g  x )
dx
u
Integrals of Natural Logarithmic Functions

1
dx  ln x  C
x

1
du  ln u  C
u
Inverse Functions
Domain of f 1 = Range of f
Range of f 1 = Domain of f
One-to-One Functions
Strictly Monotonic Functions
1
Derivative of the inverse function at the point a:  f 1  a   
f   f 1  a  
Natural Exponential Function f  x   e x
Derivatives and Integrals of the Natural Exponential Function
d x
e   e x
dx  
d u
 e   eu  u 
dx  


e x dx  e x  C
eu du  eu  C
Logarithmic Functions and Exponential Functions are inverses!!
y  loga  x  if and only if a y  x
Bases other than e
Derivatives and Integrals
d
 a x    ln a  a x
dx
d u
 a    ln a  a u  u 
dx


a x dx 
ax
C
ln a
au du 
au
C
ln a
d
1
log a  x  
dx
 ln a  x
d
u
log a  u  
dx
 ln a  u
Derivatives and Indefinite Integrals:
d
 kx   k
dx
d n
 x   nx n 1
dx  
d
sin  x    cos  x 
dx 
d
cos  x     sin  x 
dx 
d
 tan  x    sec 2  x 
dx 
d
csc  x     csc  x  cot  x 
dx 
d
sec  x    sec  x  tan  x 
dx 
d
cot  x     csc 2  x 
dx
d
1
ln  x   
dx
x
d x
e   e x
dx
d
1
log a  x  
dx
 ln a  x
d
 a x    ln a  a x
dx
 kdx  kx  C
x
x dx 
 n 1  C, n  1
 cos  x  dx  sin  x   C
 sin  x  dx   cos  x   C
 sec  x dx  tan  x   C
 csc  x  cot  x  dx   csc  x   C
 sec  x  tan  x  dx  sec  x   C
 csc  x  dx   cot  x   C
1
 x dx  ln x  C
 e dx  e  C
n 1
n
2
2
x





a x dx 
x
ax
C
ln a
sec  x  dx  ln sec  x   tan  x   C
csc  x  dx   ln csc  x   cot  x   C
tan  x  dx   ln cos  x   C
cot  x  dx  ln sin  x   C