AP Calculus – Final Review Sheet solutions
When you see the words ….
1. Find the zeros
4. Show that f x  is even
This is what you think of doing
Set function = 0, factor or use quadratic equation if
quadratic, graph to find zeros on calculator
Take derivative - f a   m and use
y  y1  mx  x1 
1
Same as above but m 
f a 
Show that f  x  f x - symmetric to y-axis
5. Show that f x  is odd
Show that f  x   f x - symmetric to origin
2. Find equation of the line tangent to f x  on a, b
3. Find equation of the line normal to f x  on a, b
6. Find the interval where f x  is increasing
Find f x , set both numerator and denominator to
zero to find critical points, make sign chart of f x 
and determine where it is positive.
7. Find interval where the slope of f x  is increasing Find the derivative of f x   f x , set both
numerator and denominator to zero to find critical
points, make sign chart of f x  and determine where
it is positive.
8. Find the minimum value of a function
Make a sign chart of f x , find all relative minimums
and plug those values back into f x  and choose the
smallest.
9. Find the minimum slope of a function
Make a sign chart of the derivative of f x   f x ,
find all relative minimums and plug those values back
into f x  and choose the smallest.
10. Find critical values
Express f x  as a fraction and set both numerator
and denominator equal to zero.
11. Find inflection points
Express f x  as a fraction and set both numerator
and denominator equal to zero. Make sign chart of
f x  to find where f x  changes sign. (+ to – or –
to +)
12. Show that lim f x  exists
Show that lim f x lim f x 
x a
13. Show that f x  is continuous
x a
x a
Show that 1) lim f x  exists ( lim f x lim f x )
x a

2) f a  exists
3) lim f x   f a 
xa
x a
x a
14. Find vertical asymptotes of f x 
 set denominator = 0
factor/cancel f x  and
15. Find horizontal asymptotes of f x 
Find
16. Find the average rate of change of f x  on a, b
Find
lim f x  and lim f x 
x
f b  f a 
ba
x  
17. Find instantaneous rate of change of f x  at a
18. Find the average value of f x  on a, b
Find f a 
19. Find the absolute maximum of f x  on a, b
Make a sign chart of f x , find all relative
maximums and plug those values back into f x  as
well as finding f a and f b and choose the largest.
First, be sure that the function is continuous at x  a .
Take the derivative of each piece and show that
lim f x   lim f x 
20. Show that a piecewise function is differentiable
at the point a where the function rule splits
b
1
Find
f x  dx
b-a a
xa 
x a
21. Given st  (position function), find vt 
Find vt   st 
b
22. Given vt , find how far a particle travels on a, b
Find  vt dt
23. Find the average velocity of a particle on a, b
24. Given vt , determine if a particle is speeding up
at t  k
25. Given vt  and s0,find the position, st 
 26. Show that Rolle’s Theorem holds on a, b

27. Show that Mean Value Theorem holds on a, b
28. Find domain of f x 
29. Find range of f x  on a, b

30. Find range of f x  on  , 
a
1
sb   sa 
vt  dt 

ba a
ba
b
Find
Find vk and ak . Multiply their signs. If both
positive, the particle is speeding up, if different signs,
then the particle is slowing down.
s
t   v 
t dt  C Plug in t = 0 to find C
Show that f is continuous and differentiable on the
interval. If f a f b, then find some c in a, b
such that f c 0.
Show that f is continuous and differentiable on the
interval.
Then find some c such that

f b f a 
f c 
.
ba
Assume domain is ,. Restrictable domains:
denominators ≠ 0, square roots of only non negative
numbers, log or ln of only positive numbers.
Use max/min techniques to rind relative max/mins.

Then examine
f a, f b
Use max/min techniques to rind relative max/mins.
Then examine lim f x .
x 
31. Find f x  by definition
f x  h  f x 

f x  lim
or
h0
h

f x  f a
f x  lim
x a
xa
32. Find derivative of inverse to f x  at x  a
dy
implicitly (in terms
dx
of y). Plug your x value into the inverse relation and
dy
solve for y. Finally, plug that y into your
.
dx
Interchange x with y. Solve for

33. y is increasing proportionally to y
34. Find the line x  c that divides the area under
f x  on a, b to two equal areas
dy
 ky translating to y  Ce kt
dt
c

a
x
d
f t dt 
dx a
d g ( x)
36.
f(t) dt
dx a
37. The rate of change of population is …
b
f x dx   f x dx
c
35.
2nd FTC: Answer is f x 
38. The line y  mx  b is tangent to f x  at x1 , y1 
39. Find area using left Riemann sums
2nd FTC: Answer is f g ( x)g ' ( x)
dP
 ...
dt
Two relationships are true. The two functions share
the same slope ( m  f x ) and share the same y value
at x1 .
A   f ( x0 )x  f ( x1 )x  ...  f ( xn1 )x
40. Find area using right Riemann sums
A   f ( x1 )x  f ( x2 )x  ...  f ( xn )x
41. Find area using midpoint rectangles
42. Find area using trapezoids
A   f ( x1 )x  f ( x3 )x  ...  f ( xn1 )x
Typically done with a table of values. Be sure to use
only values that are given. If you are given 6 sets of
points, you can only do 3 midpoint rectangles.
f ( x1 )  f ( x 2 )
 f ( x0 )  f ( x1 )


x

x  

2
2
A

...  f ( x n 1 )  f ( x n ) x



2
This formula only works when the base is the same. If not,
you have to do individual trapezoids.
43. Solve the differential equation …
x
44. Meaning of  f t dt
a
45. Given a base, cross sections perpendicular to the
x-axis are squares
Separate the variables – x on one side, y on the other.
The dx and dy must all be in the numerator.
The accumulation function – accumulated area under
the function f x  starting at some constant a and
ending at x.
The area between the curves typically is the base of
your square. So the volume is
46. Find where the tangent line to f x  is horizontal
47. Find where the tangent line to f x  is vertical
48. Find the minimum acceleration given vt 
49. Approximate the value of f 0.1 by using the
tangent line to f at x  0
 base dx
b
2
a
Write f x  as a fraction. Set the numerator equal to
zero.
Write f x  as a fraction. Set the denominator equal
to zero.
First find the acceleration at   vt . Then minimize
the acceleration by examining at .
Find the equation of the tangent line to f using
y  y1  mx  x1  where m  f 0 and the point is
50. Given the value of F a  and the fact that the antiDerivative of f is F, find F b 
0, f 0. Then plug in 0.1 into this line being sure to
use an approximate sign.
Usually, this problem contains an antiderivative you
cannot take. Utilize the fact that if F x is the
b
antiderivative of f, then F b   F a    f x dx . Use
a
51. Find the derivative of f g x
52. Given
b
b
a
a
 f xdx , find  f x k dx
53. Given a picture of f x , find where f x  is
Increasing
54. Given vt  and s0, find the greatest distance
from the origin of a particle on a, b
55. Given a water tank with g gallons initially being
filled at the rate of F t  gallons/min and emptied
at the rate of E t  gallons/min on t1 ,t 2 , find
a) the amount of water in the tank at m minutes
56. b) the rate the water amount is changing at m
the calculator.
f g x  g x 
b
b
b
a
a
a
 f x k dx   f x dx   kdx
Make a sign chart of f x  and determine where
f x  is positive.
Generate a sign chart of vt  to find turning points.
Then integrate vt  using s0 to find the constant to
find st . Finally, find s(all turning points) which will
give you the distance from your starting point. Adjust
for the origin.
t2
g   F t  E t dt
t
m
d
F t   E t dt  F m   E m 
dt a
57. c) the time when the water is at a minimum
F m Em=0, testing the endpoints as well.
58. Given a chart of x and f x  on selected values
between a and b, estimate f c where c is
between a and b.
Straddle c, using a value k greater than c and a value h
f k  f h 
less than c. so f c  
k h
59. Given
dy
, draw a slope field
dx
60. Find the area between curves f x , g x  on a, b
dy
, drawing
dx
little lines with the indicated slopes at the points.
Use the given points and plug them into
b
A   f x  g x dx , assuming that the f curve is
a
b
above the g curve. Or A   f x   g x dx
a
61. Find the volume if the area between f x , g x  is
rotated about the x-axis
b


A     f x   g x  dx assuming that the f curve
2
a
is above the g curve.
2
62. Find lim
x 

f (x)
if lim f (x)  lim g(x)  0
x 
x 
g(x)

Use L’Hopital’s Rule.
BC Problems
63. Find


0
h
f (x) dx
lim
h
 dP k
 P 
dP
64.
 kP1
 P(M  P) or
 
 M 
dt
dt M

65. Find
x
2
dx
where x2  ax  b

 ax  b

factors
66. The position vector of a particle moving
in the plane is r(t 
)  x(t ), y(t )

a) Find the velocity.
67. b) Find the acceleration.

68. c) Find the speed.
 slope of the tangent line to the
73. Find the
polar curve r  f () .


74. Use Euler’s method to approximate
dy
, x 0 , y 0   (1,1) , and
f (1.2) given

dx
x  0.1
75. Is the Euler’s approximation an
underestimate
or an overestimate?


xe
n ax
Signals logistic growth.
dP
lim
 0  M  P max growth at M/2
t  dt
Factor denominator and use partial fraction technique.
v(t )  x (t ), y (t )
a(t )  x (t ), y (t )
v(t ) 
69. a) Given the velocity vector
v(t )  x(t ), y(t )

and position at time 0, find the position
vector.
70. b) When does the particle stop? 


71. c) Find the slope of the tangent line to
the vector at t1.
72. Find the area inside the polar curve
r  f () .
76. Find
0



 f xdx
dx where a, n are integers
x (t )  y (t )
2
2
t dt   y
t dt  C
 x
Use s0 to find C, remembering it is a vector.
s(t ) 
v(t )  0  x 
t  0 AND y
t 0
This is the acceleration vector at t1.

A
 
2
1 2
f   d

2 1

dy
dy d
x  r cos , y  r sin  

dx dx
d
dy 
dy
x , y new  y old  dy
dx
𝑓(𝑥𝑛 ) = 𝑓(𝑥𝑛−1 ) + 𝑓 ′ (𝑥𝑛−1 )∆𝑥
dy
d2y
and
in the interval. This gives you
dx
dx 2
increasing/decreasing/concavity. Draw picture to ascertain
answer.
Look at sign of

Integration by parts,
 u dv  uv   v du  C
77. Write a series for x n cos x where n is an
integer

78. Write a series for ln(1 x) centeredat
x  0.



79.
n
1
p
converges if…..

n1
x2 x 4 x6
cos x  1


 ...
2! 4! 6!
Multiply each term by x n
Find Maclaurin polynomial:
n
f 0 2 f 0 3
f  0 n

Pn x  f 0 f 0x 
x 
x 
x
2!
3!
n!
p 1
80. If f (x)  2  6x  18x 2  54 x 3  ..., find
 1 

f  
 2 
Plug in and factor. This will be a geometric series:

a
 ar n  1 r
n0
81. Find the interval of convergence of a
series.

Use a test (usually the ratio) to find the interval and then test
convergence at the endpoints.
82. Let S4 be the sum of the first 4 terms of
an
alternating series for f (x).
Approximate
f (x)  S4

(n 1) n!
83. Suppose f (n ) (x)
. Write the
 
2n
first four terms and the general term of a

series for f (x) centered at x  c
This is the error for the 4th term of an alternating series which
is simply the 5th term. It will be positive since you are looking
for an absolute value.



84. Given a Taylor series, find the Lagrange

th
form
 for the n term
 of the remainder
where n is an integer at x = c.
85. f x  1 x 

5
87.
1 x  ...
x
x
x
f x  1


 ... 
2! 4! 6!
2n !
2

4
6
89. Given x  f t , y  g t , find
f x  f c f cx  c
n


2!
x  c   
2
f  c
n
n!
x  c
You need to determine the largest value of the 5th derivative of
f at some value of z. Usually you are told this. Then:
Rn x 
n1
z x  c
 
n  1 !
f
f x sin x
f x cos x
2n
dy 
dx
f c

f x  e x
x2 x3

 ...
2! 3!

n 2n1
1
x
   ...
x
x
86. f x  x    ... 
3! 5!
2n  1 !
3
You are being given a formula for the derivative of f (x) .
dy
dy dt

dx dx
dt
n1
n
d2y
90. Given x  f t , y  g t , find
dx 2
 Given f x , find arc length on a, b
91.

92. x  f 
t , y  g 
t , find arc length on
 t1 ,t2 






d 2 y d dy 
  
dx 2 dx dx 
L
b


d dy 
 
dt dx 
dx
dt

1 f x  dx
2
a
dx 2 dy 2
L        dt
dt  dt 
t1
x  r cos , y  r sin 
Find where r sin   0 where r cos  0
t2

93. Find horizontal tangents to a polar curve
r  f 

94. Find vertical tangents to a polar curve
r  f 

95. Find the volume when the area between

y  f x, x  0, y  0 is rotated about the
y-axis.
x  r cos , y  r sin 
Find where r cos   0 where r sin   0
b
Shell method: V  2  radius  height dx where b is the root.
0
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