AP Calculus – Final Review Sheet
When you see the words …. This is what you think of doing
1. Find the zeros
2. Find equation of the line tangent to ( )xf at ( )ba,
3. Find equation of the line normal to ( )xf at ( )ba,
4. Show that ( )xf is even
5. Show that ( )xf is odd
6. Find the interval where ( )xf is increasing
7. Find interval where the slope of ( )xf is increasing
8. Find the minimum value of a function
9. Find the minimum slope of a function
10. Find critical values
11. Find inflection points
12. Show that ( )xf
ax→
lim exists
13. Show that ( )xf is continuous
14. Find vertical asymptotes of ( )xf
15. Find horizontal asymptotes of ( )xf
16. Find the average rate of change of ( )xf on [ ]ba,
17. Find instantaneous rate of change of ( )xf at a
18. Find the average value of ( )xf on [ ]ba,
19. Find the absolute maximum of ( )xf on [ ]ba,
20. Show that a piecewise function is differentiable
at the point a where the function rule splits
21. Given ( )ts (position function), find ( )tv
22. Given ( )tv , find how far a particle travels on [ ]ba,
23. Find the average velocity of a particle on [ ]ba,
24. Given ( )tv , determine if a particle is speeding up
at
t = k
25. Given ( )tv and ( )0s , find ( )ts
26. Show that Rolle’s Theorem holds on [ ]ba,
27. Show that Mean Value Theorem holds on [ ]ba,
28. Find domain of ( )xf
29. Find range of ( )xf on [ ]ba,
30. Find range of ( )xf on ( )∞∞− ,
31. Find ( )xf ′ by definition
32. Find derivative of inverse to ( )xf at ax =
33. y is increasing proportionally to y
34. Find the line cx = that divides the area under
( )xf on [ ]ba, to two equal areas
35. ( ) =∫ dttfdx
d x
a
36.
d
dx
f t( )
a
u
∫ dt
37. The rate of change of population is …
38. The line bmxy += is tangent to ( )xf at ( )ba,
39. Find area using left Riemann sums
40. Find area using right Riemann sums
41. Find area using midpoint rectangles
42. Find area using trapezoids
43. Solve the differential equation …
44. Meaning of ( )dttf
x
a
∫
45. Given a base, cross sections perpendicular to the
x-axis are squares
46. Find where the tangent line to ( )xf is horizontal
47. Find where the tangent line to ( )xf is vertical
48. Find the minimum acceleration given ( )tv
49. Approximate the value of ( )1.0f by using the
tangent line to f at 0=x
50. Given the value of ( )af and the fact that the anti-
derivative of f is F, find ( )bF
51. Find the derivative of ( )( )xgf
52. Given ( )dxxf
b
a
∫ , find ( )[ ]dxkxf
b
a
∫ +
53. Given a picture of ( )xf ′ , find where ( )xf is
increasing
54. Given ( )tv and ( )0s , find the greatest distance
from the origin of a particle on [ ]ba,
55. Given a water tank with g gallons initially being
filled at the rate of ( )tF gallons/min and emptied
at the rate of ( )tE gallons/min on [ ]21 , tt , find
a) the amount of water in the tank at m minutes
56. b) the rate the water amount is changing at m
57. c) the time when the water is at a minimum
58. Given a chart of x and ( )xf on selected values
between a and b, estimate ( )cf ′ where c is
between a and b.
59. Given
dx
dy
, draw a slope field
60. Find the area between curves ( ) ( )xgxf , on [ ]ba,
61. Find the volume if the area between ( ) ( )xgxf , is
rotated about the x-axis
BC Problems
62. Find lim
x→∞
f (x)
g(x) if limx→∞ f (x) = limx→∞ g(x) = 0
63. Find f (x)
0
∞
∫ dx
64. dP
dt
=
k
M
P(M − P) or dP
dt
= kP 1− P
M
65. Find
dx
x2 + ax+ b∫
where
x2 + 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.
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 (θ).
73. Find the slope of the tangent line to the polar
curve r = f (θ).
74. Use Euler’s method to approximate f (1.2) given
dy
dx
, x0, y0( )= (1,1) , and ∆x = 0.1
75. Is the Euler’s approximation an underestimate or
an overestimate?
76. Find
xneaxdx∫ where a, n are integers
77. Write a series for
xn cosx where n is an integer
78. Write a series for ln(1+ x) centered at x = 0 .
79. 1
n p
n=1
∞
∑ converges if…..
80. If f (x) = 2 + 6x +18x 2 + 54 x 3 + ..., find f − 1
2
81. Find the interval of convergence of a series.
82. Let S4be the sum of the first 4 terms of an
alternating series for f (x). Approximate
f (x) − S4
83. Suppose f (n )(x) = (n +1) n!
2n
. Write the first four
terms and the general term of a series for f (x)
centered at
x = c
84. Given a Taylor series, find the Lagrange form of
the remainder for the 4th term.
85.
f x( )=1+ x + x22! +
x3
3!
+ ...
86.
f x( )= x − x33 ! +
x5
5 !
−
x7
7 !
+ ...+
−1( )n x2n+1
2n +1( ) ! + ...
87.
f x( )=1− x22 ! +
x4
4 !
−
x6
6 !
+ ...+
−1( )n x2n
2n( ) ! + ...
88. Find
sinx( )∫ m cosx( )ndx where m and n are
integers
89. Given
x = f t( ), y = g t( ), find dydx
90. Given
x = f t( ), y = g t( ), find d2ydx2
91. Given
f x( ), find arc length on
a,b[ ]
92.
x = f t( ), y = g t( ), find arc length on t1,t2[ ]
93. Find horizontal tangents to a polar curve
r = f θ( )
94. Find vertical tangents to a polar curve
r = f θ( )
95. Find the volume when the area between
y = f x( ), x = 0, y = 0 is rotated about the y-axis.
96. Given a set of points, estimate the volume under
the curve using Simpson’s rule.
AP Calculus – Final Review Sheet
When you see the words …. This is what you think of doing
1. Find the zeros
Set function = 0, factor or use quadratic equation if
quadratic, graph to find zeros on calculator
2. Find equation of the line tangent to ( )xf on [ ]ba,
Take derivative - ( ) maf =′ and use
( )11 xxmyy −=−
3. Find equation of the line normal to ( )xf on [ ]ba, Same as above but ( )afm ′
−
=
1
4. Show that ( )xf is even Show that ( ) ( )xfxf =− - symmetric to y-axis
5. Show that ( )xf is odd Show that ( ) ( )xfxf −=− - symmetric to origin
6. Find the interval where ( )xf is increasing Find ( )xf ′ , set both numerator and denominator to
zero to find critical points, make sign chart of ( )xf ′
and determine where it is positive.
7. Find interval where the slope of ( )xf is increasing Find the derivative of ( ) ( )xfxf ′′=′ , set both
numerator and denominator to zero to find critical
points, make sign chart of ( )xf ′′ and determine where
it is positive.
8. Find the minimum value of a function Make a sign chart of ( )xf ′ , find all relative minimums
and plug those values back into ( )xf and choose the
smallest.
9. Find the minimum slope of a function Make a sign chart of the derivative of ( ) ( )xfxf ′′=′ ,
find all relative minimums and plug those values back
into ( )xf ′ and choose the smallest.
10. Find critical values Express ( )xf ′ as a fraction and set both numerator
and denominator equal to zero.
11. Find inflection points Express ( )xf ′′ as a fraction and set both numerator
and denominator equal to zero. Make sign chart of
( )xf ′′ to find where ( )xf ′′ changes sign. (+ to – or –
to +)
12. Show that ( )xf
ax→
lim exists Show that
lim
x→a−
f x( )= lim
x→a+
f x( )
13. Show that ( )xf is continuous Show that 1) ( )xf
ax→
lim exists (
lim
x→a−
f x( )= lim
x→a+
f x( ))
2) ( )af exists
3) ( ) ( )afxf
ax
=
→
lim
14. Find vertical asymptotes of ( )xf Do all factor/cancel of ( )xf and set denominator = 0
15. Find horizontal asymptotes of ( )xf Find ( )xf
x ∞→
lim and ( )xf
x −∞→
lim
16. Find the average rate of change of ( )xf on [ ]ba, Find ( ) ( )
ab
afbf
−
−
17. Find instantaneous rate of change of ( )xf at a Find ( )af ′
18. Find the average value of ( )xf on [ ]ba,
Find
( )
b-a
dxxf
b
a
∫
19. Find the absolute maximum of ( )xf on [ ]ba, Make a sign chart of ( )xf ′ , find all relative
maximums and plug those values back into ( )xf as
well as finding ( )af and ( )bf and choose the largest.
20. Show that a piecewise function is differentiable
at the point a where the function rule splits
First, be sure that the function is continuous at ax = .
Take the derivative of each piece and show that
( ) ( )xfxf
axax
′=′
+→→ −
limlim
21. Given ( )ts (position function), find ( )tv Find ( ) ( )tstv ′=
22. Given ( )tv , find how far a particle travels on [ ]ba,
Find ( )∫
b
a
dttv
23. Find the average velocity of a particle on [ ]ba,
Find
v t( )
a
b
∫ dt
b− a
=
s b( )− s a( )
b− a
24. Given ( )tv , determine if a particle is speeding up
at
t = k
Find ( )kv and ( )ka . Multiply their signs. If both
positive, the particle is speeding up, if different signs,
then the particle is slowing down.
25. Given ( )tv and ( )0s , find ( )ts
s t( )= v t( )∫ dt + C Plug in t = 0 to find C
26. Show that Rolle’s Theorem holds on [ ]ba, 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.
27. Show that Mean Value Theorem holds on [ ]ba, Show that f is continuous and differentiable on the
interval. Then find some c such that
′ f c( )= f b( )− f a( )b − a .
28. Find domain of ( )xf Assume domain is
−∞,∞( ). Restrictable domains:
denominators ≠ 0, square roots of only non negative
numbers, log or ln of only positive numbers.
29. Find range of ( )xf on [ ]ba, Use max/min techniques to rind relative max/mins.
Then examine
f a( ), f b( )
30. Find range of ( )xf on ( )∞∞− , Use max/min techniques to rind relative max/mins.
Then examine
lim
x→±∞
f x( ).
31. Find ( )xf ′ by definition
′ f x( )= lim
h→0
f x + h( )− f x( )
h
or
′ f x( )= lim
x →a
f x( )− f a( )
x − a
32. Find derivative of inverse to ( )xf at ax = Interchange x with y. Solve for
dx
dy implicitly (in terms
of y). Plug your x value into the inverse relation and
solve for y. Finally, plug that y into your
dx
dy
.
33. y is increasing proportionally to y ky
dt
dy
= translating to ktCey =
34. Find the line cx = that divides the area under
( )xf on [ ]ba, to two equal areas ( ) ( )dxxfdxxf
b
c
c
a
∫∫ =
35. ( ) =∫ dttfdx
d x
a
2nd FTC: Answer is ( )xf
36.
d
dx
f t( )
a
u
∫ dt
2nd FTC: Answer is ( )
dx
du
uf
37. The rate of change of population is …
...=
dt
dP
38. The line bmxy += is tangent to ( )xf at ( )11 , yx Two relationships are true. The two functions share
the same slope ( ( )xfm ′= ) and share the same y value
at 1x .
39. Find area using left Riemann sums [ ]1210 ... −++++= nxxxxbaseA
40. Find area using right Riemann sums [ ]nxxxxbaseA ++++= ...321
41. Find area using midpoint rectangles 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.
42. Find area using trapezoids [ ]nn xxxxxbaseA +++++= −1210 2...222
This formula only works when the base is the same. If
not, you have to do individual trapezoids.
43. Solve the differential equation … Separate the variables – x on one side, y on the other.
The dx and dy must all be upstairs.
44. Meaning of ( )dttf
x
a
∫
The accumulation function – accumulated area under
the function ( )xf starting at some constant a and
ending at x.
45. Given a base, cross sections perpendicular to the
x-axis are squares
The area between the curves typically is the base of
your square. So the volume is ( )dxbaseb
a
∫
2
46. Find where the tangent line to ( )xf is horizontal Write ( )xf ′ as a fraction. Set the numerator equal to
zero.
47. Find where the tangent line to ( )xf is vertical Write ( )xf ′ as a fraction. Set the denominator equal
to zero.
48. Find the minimum acceleration given ( )tv First find the acceleration ( ) ( )tvta ′= . Then minimize
the acceleration by examining ( )ta′ .
49. Approximate the value of ( )1.0f by using the
tangent line to f at 0=x
Find the equation of the tangent line to f using
( )11 xxmyy −=− where ( )0fm ′= and the point is
( )( )0,0 f . Then plug in 0.1 into this line being sure to
use an approximate ( )≈ sign.
50. Given the value of ( )aF and the fact that the anti-
derivative of f is F, find ( )bF 1
Usually, this problem contains an antiderivative you
cannot take. Utilize the fact that if ( )xF is the
antiderivative of f, then ( ) ( ) ( )aFbFdxxF
b
a
−=∫ . So
solve for ( )bF using the calculator to find the definite
integral.
51. Find the derivative of ( )( )xgf ( )( ) ( )xgxgf ′⋅′
52. Given ( )dxxf
b
a
∫ , find ( )[ ]dxkxf
b
a
∫ +
( )[ ] ( ) dxkdxxfdxkxf
b
a
b
a
b
a
∫∫∫ +=+
53. Given a picture of ( )xf ′ , find where ( )xf is
increasing
Make a sign chart of ( )xf ′ and determine where
( )xf ′ is positive.
54. Given ( )tv and ( )0s , find the greatest distance
from the origin of a particle on [ ]ba,
Generate a sign chart of ( )tv to find turning points.
Then integrate ( )tv using ( )0s to find the constant to
find ( )ts . Finally, find s(all turning points) which will
give you the distance from your starting point. Adjust
for the origin.
55. Given a water tank with g gallons initially being
filled at the rate of ( )tF gallons/min and emptied
at the rate of ( )tE gallons/min on [ ]21 , tt , find
a) the amount of water in the tank at m minutes
( ) ( )( )dttEtFg
t
t
∫ −+
2
56. b) the rate the water amount is changing at m
( ) ( )( ) ( ) ( )mEmFdttEtF
dt
d m
t
−=−∫
57. c) the time when the water is at a minimum ( ) ( )mEmF − =0, testing the endpoints as well.
58. Given a chart of x and ( )xf on selected values
between a and b, estimate ( )cf ′ where c is
between a and b.
Straddle c, using a value k greater than c and a value h
less than c. so ( ) ( ) ( )
hk
hfkf
cf
−
−
≈′
59. Given
dx
dy
, draw a slope field Use the given points and plug them into
dx
dy
, drawing
little lines with the indicated slopes at the points.
60. Find the area between curves ( ) ( )xgxf , on [ ]ba, ( ) ( )[ ]dxxgxfA
b
a
∫ −= , assuming that the f curve is
above the g curve.
61. Find the volume if the area between ( ) ( )xgxf , is
rotated about the x-axis ( )( ) ( )( )[ ]dxxgxfA
b
a
∫ −=
22
assuming that the f curve is
above the g curve.
BC Problems
62. Find lim
x→∞
f (x)
g(x) if limx→∞ f (x) = limx→∞ g(x) = 0
Use L’Hopital’s Rule.
63. Find f (x)
0
∞
∫ dx
lim
h→∞
f x( )
0
h
∫ dx
64. dP
dt
=
k
M
P(M − P) or dP
dt
= kP 1− P
M
Signals logistic growth.
lim
t →∞
dP
dt
= 0⇒ M = P
65. Find
dx
x2 + ax+ b∫
where
x2 + ax+ b
factors
Factor denominator and use Heaviside partial fraction
technique.
66. The position vector of a particle moving
in the plane is
r (t ) = x(t ), y(t )
a) Find the velocity.
v(t ) = ′ x (t ), ′ y (t )
67. b) Find the acceleration.
a(t ) = ′ ′ x (t ), ′ ′ y (t )
68. c) Find the speed.
v(t ) = ′ x (t )[ ]2 + ′ y (t )[ ]2
69. a) Given the velocity vector
v(t ) = x(t ), y(t )
and position at time 0, find the position
vector.
s(t ) = x t( )∫ dt + y t( )∫ dt + C
Use
s 0( ) to find C, remembering it is a vector.
70. b) When does the particle stop?
v(t ) = 0 → x t( )= 0 AND y t( )= 0
71. c) Find the slope of the tangent line to
the vector at
t1.
This is the acceleration vector at
t1.
72. Find the area inside the polar curve
r = f (θ).
A = 1
2
f θ( )[ ]2
θ1
θ 2
∫ dθ
73. Find the slope of the tangent line to the
polar curve r = f (θ).
x = r cosθ, y = r sinθ ⇒ dy
dx
=
dy
dθ
dx
dθ
74. Use Euler’s method to approximate
f (1.2) given
dy
dx
, x0, y0( )= (1,1) , and
∆x = 0.1
dy = dy
dx
∆x( ), ynew = yold + dy
75. Is the Euler’s approximation an
underestimate or an overestimate? Look at sign of
dy
dx
and d
2y
dx2
in the interval. This gives you
increasing.decreasing/concavity. Draw picture to ascertain
answer.
76. Find
xneaxdx∫ where a, n are integers Integration by parts,
u dv∫ = uv− v du∫ + C
77. Write a series for
xn cosx where n is an
integer
cosx =1− x
2
2 !
+
x4
4 !
−
x6
6 !
+ ...
Multiply each term by
xn
78. Write a series for ln(1+ x) centered at
x = 0.
Find Maclaurin polynomial:
Pn x( )= f 0( )+ ′ f 0( )x + ′ ′ f 0( )2 ! x2 +
′ ′ ′ f 0( )
3 !
x3 +…+
f n( ) 0( )
n !
xn
79. 1
n p
n=1
∞
∑ converges if…..
p >1
80. If f (x) = 2 + 6x +18x 2 + 54 x 3 + ..., find
f − 1
2
Plug in and factor. This will be a geometric series:
ar n
n=0
∞
∑ = a1− r
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 S4be the sum of the first 4 terms of an
alternating series for f (x). Approximate
f (x) − S4
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.
83. Suppose f (n )(x) = (n +1) n!
2n
. Write the
first four terms and the general term of a
series for f (x) centered at
x = c
You are being given a formula for the derivative of f (x) .
f x( )= f c( )+ ′ f c( ) x − c( )+ ′ ′ f c( )2 ! x − c( )
2
+…+
f n( ) c( )
n !
x − c( )n
84. Given a Taylor series, find the Lagrange
form of the remainder for the nth term
where n is an integer at 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( )= f
n+1( ) z( )
n +1( ) ! x − c( )
n+1
85.
f x( )=1+ x + x22! +
x3
3!
+ ...
f x( )= ex
86.
f x( )= x − x33 ! +
x5
5 !
+ ...
本文档为【ap calculus bc review worksheets】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。