Advanced Placement Calculus
Exam Review
D. Kramer
Terms & Concepts
1-1 Instantaneous Rate of Change
Average rate of change
Instantaneous rate of change Derivative
Limit
1-2 Instantaneous Rate of Change
 Equation
 Graph
 Table
1-3 Definite Integral: Counting Boxes
Definite Integral  x  y  x  f ( x)
Area under curve
Speed versus Velocity (#11)
1-4 Definite Integral: Trapezoids
Trapezoid rule, Tn
Limit as n gets large, Δx gets small
Program Trapezoid Rule
2-2
Limits: Graphically
Cusps
2-3 Limits: Algebraically
limit theorems
indeterminate form: 0/0
 Factor & cancel
 Long division
 Synthetic division
2-4 Continuity and Discontinuity
Discontinuities
 removable discontinuity (hole)
 step discontinuity
 infinite discontinuity
Limits:
 one-sided limits
 piecewise functions
Definition of continuity
2-5
Limits Involving Infinity
 Infinite limits
 End Behavior, limits at infinity
 Vertical asymptote
 Horizontal asymptote
Supp Limits: Algebraically
Name___________________________
Notes to Self
2-6
The Intermediate Value Theorem.
If f is cont on [a,b] and y* is any value between f(a)
and f(b), then there exists x*  [a,b] such that
f(x*)=y*.
The Extreme Value Theorem
If f is cont on [a,b] then there exist
such that f(
[a,b].
c1 , c2  [a,b]
c1 ) is a max and f( c2 ) is a min of f on
3-1
Graphical Interpretation of Derivative
3-2
Definition of Derivative
Difference Quotient
f ( x )  f (c ) f ( x  h )  f ( x )
xc
h
,
,
Def of Derivative:
f ( x )  f (c )
x c
xc
f ( x  h)  f ( x )
f '( x)  lim
h 0
h
f '(c)  lim
Equation of Tangent Line
3-3
Derivative Function
Sketching f’(x) from graph of f(x)
Math #8 NDeriv(function, x, x)
Calc #6 dy/dx
3-4
Derivative of Power Functions
Terms: differentiate, differentiable
dy d
d

Notation: y’, dx , dx y, dx

If f(x) = xn then f’(x) = _________
Theorems:
Dx(f(x) + g(x)) = Dx f(x) + Dx g( x)
Dx (k f(x)) = k Dx f(x)
Dx ( c) = 0
Second Derivatives and Notation
3-5
Motion Problems
Displacement and Distance
Velocity and Speed
Acceleration
speeding up / slowing down
Max & Min, relative & absolute
3-6 Sine and Cosine Functions
Dx (sin x) = _________
Dx (cos x) = __________
3-7 Derivatives of Composite Functions
The Chain Rule:
Dx f(g(x)) = __________
3-8 Derivatives of Sine and Cosine
Writing equations for sine and cosine
3-9A
Exponential and Log Functions Properties :
ln ab  ______
ln a / b  _____
ln1  ___
ln e  ___
ln a n  ______
ln e x  ___
eln x  ___
3-9B Exponential and Log Functions
Derivatives:
Dx ln x  _______
Dx e x  ________
Dxb x  ________
4-2 Derivative of a Product
Product Rule:
Dx f(x)g(x) = ______________________
Simplifying
4-3 Derivative of a Quotient
Quotient rule:
f ( x)
Dx g ( x ) = ___________________
Simplifying
4-4 Derivatives of Other Trig Functions
Dx sin x =___________________
Dx cos x =___________________
Dx tan x =___________________
Dx cot x = ___________________
Dx sec x =___________________
Dx csc x =___________________
4-5 Derivs of InverseTrig Functions
Inverse Trig Summary, page 149
Dx sin-1 x =___________________
Dx cos-1 x =___________________
Dx tan-1 x =___________________
Dx cot-1 x = ___________________
Dx sec-1 x =___________________
Dx csc-1 x =___________________
Derivative of Inverse Property
4-6 Differentiabiltiy and Continuity
Def: Differentiability
Theorem & Sketches
Supp: Conic Sections
4-7 Derivatives of Parametric Functions
Parametric Equations, Parameters
First Derivatives (Parametric Chain Rule)
dy
dx = ___________
Second Derivatives
d2y
dx 2 = _________________
4-8 Implicit Relations
Implicit Differentiation
4-9 Related Rates
5-2 Differential
y  dy
x  dx ,
y  f ( x  x)  f ( x)
dy  l ( x  x)  l ( x)  f '( x)dx
error  y  dy
5-3
Antiderivative, Indefinite Integral
 Definition
 Notation, Integrand, Operator
 Constants, Sums, Differences
Products
5-4
Riemann Sums: Left, Right, Midpt, Upper, Lower
Definite Integral
b
n
f ( x)dx  lim  f ( xk ) xk

n 
a
k 1
if this limit exists.
5-5
Integrability
 if continuous then integrable
 if bounded with finite number of discontinuities,
then integrable
Mean Value Theorem
Rolle’s Theorem
5-6
Fundamental Theorem of Calculus
 FTOC
 Integral versus Area
Exact value of an integral:

b
a
n
f ( x)dx  lim  f (k  x)  x
n 
k 1
5-7 Properties of the Definite Integral
Same integrand, different order of bounds
Same integrand, different bounds
Even & Odd functions
Substitution
5-8
Area Between Curves
Horizontal & Vertical Slices
5-9
Volume of Solids
 Solids of revolution
 Solids with Known Cross-Sections
6-2
Integrals Using ln

ln x  
x
1
1
dt
t , where x  0
1
 dx  ln x  C
Fact: x
Second Fundamental Theorem
x

Dx  f (t )dt  f ( x)
a
,
Dx 
g ( x)
a
f (t )dt  f ( g ( x))  g '( x)
6-3
Derivatives of Log Functions
D log x 
x
b

6-4
Derivatives and Integrals of Logs & Exponents
Logarithmic Differentiation
6-5
Limit Review
L’Hospital’s Rule for 0/0 and  / 
Indeterminate Form  - 
Indeterminate Exponential Forms
7-2
Exponential Growth & Decay
General/Particular Solution
Initial Condition
Other differential equations
7-4
Slope Fields
Initial Conditions
8-2
First Derivative Test for Extrema
Critical Point of f
Increasing/Decreasing
Relative/Absolute Max/Min
Plateau Point
Second Derivative Test for Inflection
Critical Points of f '
Concavity, Inflection
Cusps & Corners
Average value of a function