AP CALCULUS AB LAST MINUTE REVIEW PACKET
LIST OF TOPICS
Limits/Continuity
3 conditions for a limit to exist
left/right handed limits
finding limits from a graph
finding limits from a table
finding limits algebraically
trig limits
horizontal asymptotes (limits as x -> ∞)
vertical asymptotes (limits that equal ∞)
comparing growth rates
3 conditions for continuity
removable vs non-removable
Derivatives
definitions
product, quotient, chain rules
ex, ax, lnx, logax
higher order derivatives
implicit differentiation
derivative of an inverse function
estimating from a table
tangent/normal lines
linear approximation
trig and inverse trig
average vs instantaneous rates of change
Curve Sketching
increasing, decreasing
relative extrema
concavity
points of inflection
2nd derivative test
because statements
making sign lines from a graph
absolute extrema on closed and open intervals
Related Rates and Optimization
Reimann Sums
left, right, middle
trapezoid rule
equal vs non-equal widths
Area/Volume
disc, washer, cross section
Integrals
power rule
U-substitution
indefinite and definite
ex, ax
integration by parts
slope fields
differential equations
exponential growth/decay
trig and inverse trig
finding an antiderivative at a single point
PVA
distance vs displacement
at rest vs changing direction
speed
because statements
Theorems
IVT, EVT, Fermat's MVT, Rolle's
Squeeze
Average Value
THEOREMS
Intermediate Value Theorem: if f is continuous on a closed interval [a, b] and k is any number
between f (a) and f (b), then there is at least one number c in [a, b] such that f (c) = k.
In plain English: if a function is continuous on an interval, then it has to hit all of the y values in
between the endpoints somewhere in the interval.
Squeeze Theorem: If h(x)  f (x)  g(x) for all x in an open interval containing c, and if
lim h( x)  L  lim g ( x) , then lim f ( x)  L .
x c
x c
x c
Extreme Value Theorem: if f is continuous on [a, b], then it has both a minimum and a maximum on
that interval.
Fermat's Theorem: if a function is continuous on a closed interval, then the absolute extrema will
either be at the critical numbers or at an endpoint.
Mean Value Theorem: If a function, f(x), is continuous on [a, b] and differentiable on (a, b), then
there must be at least one point, c in (a, b) where the slope of the tangent (derivative) is equal to the
𝑓(𝑏)−𝑓(𝑎)
slope of the secant. 𝑓 ′ (𝑐) =
𝑏−𝑎
Rolle's Theorem: If a function, f (x), is continuous on [a, b], differentiable on (a, b), and f (a) = f (b),
then there must be at least one point on the function where the slope of the tangent (derivative) is 0.
sin x
x
 lim
1
x 0
x  0 sin x
x
lim
Trig Limits:
1  cos x
0
x 0
x
lim
3 Conditions that must be met for a function to be continuous at x = c
1. f (c) must be defined
2. lim f ( x ) must exist ( lim f ( x)  lim f ( x) )
x c
x c
x c
3. lim f ( x)  f (c)
x c
If the limit exists, then the discontinuity is REMOVABLE.
Product rule:
𝑑
𝑑𝑥
Quotient rule:
Chain rule:
𝑑
𝑑𝑥
𝑑
[𝑓(𝑥) ∙ 𝑔(𝑥)] = 𝑓 ′ (𝑥)𝑔(𝑥) + 𝑓(𝑥)𝑔′ (𝑥)
𝑓(𝑥)
[
]=
𝑑𝑥 𝑔(𝑥)
𝑓 ′ (𝑥)𝑔(𝑥)−𝑓(𝑥)𝑔′ (𝑥)
(𝑔(𝑥))2
[𝑓(𝑔(𝑥))] = 𝑓 ′ (𝑔(𝑥)) ∙ 𝑔′ (𝑥)
′
1
Derivative of an inverse: (𝑓 −1 (𝑥)) = 𝑓′ (𝑓−1 (𝑥))
DERIVATIVES OF BASIC FUNCTIONS
If y =
Then y' =
If y =
Then y' =
xn
nx n 1
sinx
cosx
arcsinx
ln x
1
x
cosx
-sinx
arccosx
ex
ex
tanx
sec2x
arctanx
logax
1
x ln a
secx
secxtanx
arcsecx
ax
axlna
cscx
-cscxcotx
arccscx
cotx
-csc2x
arccotx
INTEGRALS
Power rule for integrals:
x
n
Integrals to memorize:
∫ sin 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶
∫ cot 𝑥 𝑑𝑥 = ln|sin 𝑥| + 𝐶

If y =
dx
a2  x2
𝑥
 arcsin
dx 
1
n 1
Then y' =
1
1 x2
1
1 x2
1
1 x2
1
| x | x2 1
1
| x | x2 1
1
1 x2
x n 1  C
∫ cos 𝑥 𝑑𝑥 = sin 𝑥 + 𝐶
 sec xdx = ln|secx + tanx| + C
x
C
a
a
2
dx
1
x
 arctan  C
2
a
a
x
∫ tan 𝑥 𝑑𝑥 = − ln|cos 𝑥| + 𝐶
 csc xdx = -ln|cscx + cotx| + C
x
ax
 a dx  ln a  C
𝑥
x
∫ 𝑒 𝑑𝑥 = 𝑒 + 𝐶
dx
x2  a2

1
| x|
arc sec
C
a
a
1
 x dx = ln |x| + C
Integration by Parts: ∫ 𝑢 𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣 𝑑𝑢
𝑏
1
Average Value of f (x) on the interval from a to b = 𝑏−𝑎 ∫𝑎 𝑓(𝑥) 𝑑𝑥
b
Volume by disc method =   r 2 (dx or dy)
Volume by washer method = 
a
b
R
2
 r 2 (dx or dy)
a
horizontal axis –> dx, vertical axis –> dy
R = (outer function – axis) or (axis – outer function) (top – bottom or right – left)
r = (inner function – axis) or (axis – inner function) (top – bottom or right – left)
b
Volume by cross section =  (area of cross section)(d x or dy)
a
cross sections perpendicular to x-axis –> dx, cross sections perpendicular to y-axis –> dy
Chapter 3 Review Chart
Absolute Extrema on a Closed
Interval
1) find first derivative
2) find critical numbers
3) test critical numbers and endpoints
by plugging them into the original
function
First Derivative Test: use for finding
Increasing/Decreasing intervals and Relative Extrema
1) find first derivative
2) find critical numbers
3) make a sign line
4) Write because statements
 f (x) is increasing on (a, b) because f '(x) > 0
 f (x) is decreasing on (a, b) because f '(x) < 0
 a relative maximum occurs at (a, f (a)) because f '(x)
changes from positive to negative
 a relative minimum occurs at (a, f (a)) because f '(x)
changes from negative to positive
Concavity and Points of Inflection
2nd Derivative Test: use for finding relative extrema
(use only when you can't do the first derivative test)
1) find first derivative
2) find critical numbers
3) find second derivative
4) plug critical numbers from first derivative into
second derivative
5) Interpret results
 if answer is positive then a relative minimum occurs
at that x-value
 if answer is negative then a relative maximum
occurs at that x-value
 if answer is 0 or undefined then you need to go back
and make a first derivative sign line
6) write because statements
 a relative maximum occurs at (a, f (a)) because
f '(a) = 0 and f ''(a) < 0
 a relative minimum occurs at (a, f (a)) because
f '(a) = 0 and f ''(a) > 0
1)
2)
3)
4)



1)



2)
3)
4)
find first derivative
find second derivative
make a second derivative sign line
write because statements
f (x) is concave up on (a, b) because
f ''(x) > 0
f (x) is concave down on (a, b)
because f ''(x) < 0
a point of inflection occurs at (a, f
(a)) because f ''(x) changes sign
Rolle's Theorem
check that it applies
continuous on [a, b]
differentiable on (a, b)
f (a) = f (b)
find the derivative
set the derivative = 0 and solve for x
answer is only the x-values in (a, b)
Mean Value Theorem
1) check that it applies
 continuous on [a, b]
 differentiable on (a, b)
f (b)  f (a )
2) set the derivative =
and
ba
solve for x
3) answer is only the x-values in (a, b)
1)
2)
3)
4)


Absolute Extrema on an Open Interval
find first derivative
find critical number (there will only be 1)
make a sign line
Write because statements
an absolute max occurs at (a, f (a)) because f '(x) > 0
for all x < a and f '(x) < 0 for all x > a
an absolute min occurs at (a, f (a)) because f '(x) < 0
for all x < a and f '(x) > 0 for all x > a
Slope fields
1. Let dy/dx = 2xy2. Draw a slope field at the indicated points on that graph. (0, 0), (1, 1), (2, 3),
(-1, 0), (-1, -1), (-4, 2).
2. Let dy/dx = xy. Graph the slope fields for this differential equation. Then find the solution of
the differential equation that passes through (0, 1) and state its domain.
Exponential Growth/Decay
1. The rate of increase of the population of a certain city is proportional to the population. If the
population in 1930 was 50,000 and in 1960 it was 75,000, what was the expected population in
1990?
Graphs of Derivatives
1. (18 points) To the right is the graph of g', the derivative of a
continuous function, g. If the domain of g is [-4, 4]
and the range of g is [-2, 5], and if g(-4) = 2,
g(0) = 1, and g(4) = 0,
y = g '(x)
1
1
a)
b)
c)
d)
Find the interval(s) where g is increasing
Find the interval(s) where g is concave downward
Find the x-coordinates of every relative maximum, relative minimum, and point of inflection
Sketch the graph of the function y = g(x)
Continuity
1. Prove that the following is discontinuous at x = 2. Is it removable? If so, redefine f (2) to make
the function continuous.
 x2  4

x2
f ( x)   x  2 ,
 x  5 ,
x2
Trig Derivatives
1. Write the derivatives of each of the following functions:
a. y  sin x
b. y  cos x
c. y  tan x
f. y  cot x
g. y  arcsin x
h. y  arcsecx
Derivatives of inverse functions
1. if f ( x)  x 3  2 x , find ( f
Logs of bases other than e
d 3x
2
1. Find
dx
2. If y  log 4 (3x  1) , find y'
1
)' (2)
d. y  sec x
e. y  csc x
i. y  arctanx
Product, Quotient, and Chain Rules
f (x)
g (x)
f '(x)
x=2
3
-1
-4
x = -1
5
-3
2
1. The derivative of f (x)  g (x) when x = -1 is
f ( x)
2. The derivative of
when x = 2 is
g ( x)
3. If h(x) = f (g(x)), find h '(2).
g '(x)
-2
1
b
First fundamental theorem of calculus: if F ' ( x)  f ( x) , then
 f ( x)dx  F (b)  F (a)
a
2
(-3, 1)
-3
1
-2
1
-1
2
3
-1
-2
(4, -2)
Let f be a function defined on the closed interval -3 ≤ x ≤ 4 with f (0) = 3. The graph of f ', the
derivative of f, consists of one line segment and a semicircle, as shown above.
a) On what intervals, if any, is f increasing? Justify your answer.
b) Find the x-coordinate of each point of inflection on the graph of f on the open interval
-3 < x < 4. Justify your answer.
c) Find an equation for the line tangent to the graph of f at the point (0, 3).
d) Find f (-3) and f (4). Show the work that leads to your answers.
2nd Derivative Test
1. If f ''(x) = x2 – 9 and f (x) has critical numbers at x = 2, 3, 10, find the x-values of the relative
max and relative min for f (x).
Particle Motion
Pierre, the mountain climbing ant, is taking it easy and moving along the y-axis with velocity
measured in inches per second given by v(t )  t sin t 2 for t  0.
a.
b.
c.
d.
e.
f.
g.
In which direction is Pierre moving at time t = 1.5? Why?
Find Pierre's acceleration at time t = 1.5.
Is his velocity increasing or decreasing at t = 1.5? Why?
Is his speed increasing or decreasing at t = 1.5? Why?
Given that y(t) is his position at time t and that y(0) = 3, find y(2).
Find Pierre's displacement from t = 0 to t = 2.
Find the total distance traveled by Pierre from t = 0 to t = 2.
Related Rates
5 feet
y
9 feet
9 4
x
625
9 4
x from x
625
= 0 to x = 5 about the y-axis, where x and y are measured in feet. Oil flowed into an initially empty
tank at a constant rate of 8 cubic feet per minute.
An oil storage tank has the shape shown above, obtained by revolving the curve y 
a) Find the volume of the tank. Indicate units of measure.
b) To the nearest minute, how long would it take to fill the tank if the tank was empty initially?
c) Let h be the depth, in feet, of the oil in the tank. How fast was the depth of the oil in the
tank increasing when h = 4? Indicate units of measure.
l
y
P
Q
1
x2
R
1
1 

at point P, with coordinates  w, 2  ,
2
x
 w 
where w > 0. Point Q has coordinates (w, 0). Line l crosses the x-axis at R, with coordinates (k, 0).
In the figure above, line l is tangent to the graph of y 
a) Find the value of k when w = 3.
b) For all w > 0, find k in terms of w.
c) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is
the rate of change of k with respect to time?
d) Suppose that w is increasing at the constant rate of 7 units per second. When w = 5, what is
the rate of change of the area of PQR with respect to time? Determine whether the area is
increasing or decreasing at this instant.
Riemann Sums
t (hours)
0
3
6
9
12
15
18
21
24
R(t) (gallons per hour)
9.6
10.4
10.8
11.2
11.4
11.3
10.7
10.2
9.6
The rate at which water flows out of a pipe, in gallons per hour, is given by a differentiable function
R of time t. The table above shows the rate as measured every 3 hours for a 24-hour time period.
24
a) Use a midpoint Riemann sum with 4 subdivisions of equal length to approximate
 R(t )dt .
0
Using correct units, explain the meaning of your answer in terms of water flow.
b) Is there some time t, 0 < t < 24, such that R '(t) = 0? Justify your answer.
1

768  23t  t 2  . Use Q(t) to
c) The rate of water flow R(t) can be approximated by Q(t ) 
79
approximate the averate rate of water flow during the 24-hour time period. Indicate units of
measure.