AP Calculus – Final Review Sheet

AP Calculus – Final Review Sheet
When you see the words ….
1. Find the zeros
This is what you think of doing
2. Find equation of the line tangent to f (x ) at (a, b )
3. Find equation of the line normal to f (x ) at (a, b )
4. Show that f (x ) is even
5. Show that f (x ) is odd
6. Find the interval where f (x ) is increasing
7. Find interval where the slope of f (x ) 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 lim f (x ) exists
x! a
13. Show that f (x ) is continuous
14. Find vertical asymptotes of f (x )
15. Find horizontal asymptotes of f (x )
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16. Find the average rate of change of f (x ) on [a, b]
17. Find instantaneous rate of change of f (x ) at a
18. Find the average value of f (x ) on [a, b]
19. Find the absolute maximum of f (x ) on [a, b]
20. Show that a piecewise function is differentiable
at the point a where the function rule splits
21. Given s (t ) (position function), find v(t )
22. Given v(t ), find how far a particle travels on [a, b]
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 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 (" !, ! )
31. Find f !(x ) by definition
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32. Find derivative of inverse to f (x ) at x = a
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
x
d
35.
f (t )dt =
dx !a
d u
36.
" f (t ) dt
dx a
37. The rate of change of population is …
!
38. The line y = mx + b is tangent to f (x ) at (a, b )
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 …
x
44. Meaning of ! f (t )dt
a
45. Given a base, cross sections perpendicular to the
x-axis are squares
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 )
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49. Approximate the value of f (0.1) by using the
tangent line to f at x = 0
50. Given the value of f (a ) and the fact that the antiderivative of f is F, find F (b )
51. Find the derivative of f (g (x ))
b
52. Given
b
! f (x )dx , find ! [f (x )+ k ]dx
a
a
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
57. c) the time when the water is at a minimum
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.
dy
59. Given
, draw a slope field
dx
60. Find the area between curves f (x ), g (x ) on [a, b]
61. Find the volume if the area between f (x ), g (x ) is
rotated about the x-axis
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BC Problems
62. Find lim
x "#
63. Find
!
#
f (x)
if lim f (x) = lim g(x) = 0
x "#
x "#
g(x)
"
0
f (x) dx
!
#
dP
P&
dP k
= kP%1" (
64.
= P(M " P) or
\$ M'
dt
dt M
!
dx
where x 2 + ax + b factors
+ ax + b
!
66. The position vector of a particle moving in the
! ), y(t )
plane is r(t ) = x(t
!
a) Find the velocity.
67. b) Find the acceleration.
65. Find
!
"x
2
!
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 (" ) .
!
73. Find the slope of the tangent line to the polar
!
curve r = f (" ) .
!
74. Use Euler’s method to approximate f (1.2) given
dy
, ( x 0 , y 0 ) = (1,1) , and "x = 0.1
!
dx
75. Is the Euler’s approximation
an underestimate or
!
an overestimate?
!
!
n ax
76. Find " x e dx where a, n are integers
77. Write a series for x n cos x where n is an integer
!
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!
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78. Write a series for ln(1+ x) centered at x = 0 .
"
79.
1
#n
p
converges
if…..
!
!
n=1
# 1&
80. If f (x) = 2 + 6x + 18x 2 + 54 x 3 + ... , find f % " (
\$ 2'
!
81. Find the interval of convergence of a series.
!
!
82. Let S4 be 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! . Write the first four
2n
!
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.
!
!
x2 x3
85. f ( x ) = 1+ x + + + ...
2! 3!
n
!
"1) x 2n+1
(
x3 x5 x 7
86. f ( x ) = x " + " + ... +
+ ...
3! 5! 7!
(2n + 1) !
n
!
"1) x 2n
(
x2 x 4 x6
87. f ( x ) = 1" +
" + ... +
+ ...
2! 4! 6!
(2n) !
88. Find
m
" (sin x ) (cos x )
n
dx where m and n are
integers
!
dy
89. Given x = f (t ), y = g (t ), find
dx
!
d2y
90. Given x = f (t ), y = g (t ), find
dx 2
!
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 (" )
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!
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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.
96. Given a set of points, estimate the volume under
the curve using Simpson’s rule.
97. Find the dot product: u1 , u2 " v1 ,v2
98. Multiply two vectors:
!
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AP Calculus – Final Review Sheet
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
6. Find the interval where f (x ) is increasing
12. Show that lim f (x ) exists
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.
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.
Make a sign chart of f !(x ), find all relative minimums
and plug those values back into f (x ) and choose the
smallest.
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.
Express f !(x ) as a fraction and set both numerator
and denominator equal to zero.
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 +)
Show that lim# f ( x ) = lim+ f ( x )
13. Show that f (x ) is continuous
Show that 1) lim f (x ) exists ( lim# f ( x ) = lim+ f ( x ) )
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]
7. Find interval where the slope of f (x ) 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
x! a
x "a
x "a
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 )
Do all factor/cancel of! f (x ) and set denominator = 0
15. Find horizontal asymptotes of f (x )
Find
16. Find the average rate of change of f (x ) on [a, b]
Find
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lim f (x ) and lim f (x )
x "!
x # !"
f (b )! f (a )
b!a
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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 )
b
Find
! f (x )dx
a
b-a
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 )
19. Find the absolute maximum of f (x ) on [a, b]
20. Show that a piecewise function is differentiable
at the point a where the function rule splits
x"a +
x"a
21. Given s (t ) (position function), find v(t )
22. Given v(t ), find how far a particle travels on [a, b]
Find v(t ) = s !(t )
b
Find
! v(t )dt
a
23. Find the average velocity of a particle on [a, b]
b
Find
24. Given v(t ), determine if a particle is speeding up
at t = k
!
25. Given v(t ) and s (0 ), find s (t )
! 26. Show that Rolle’s Theorem holds on [a, b]
a
=
s ( b) # s ( a )
b#a
b#a
Find v(k )and a(k ). Examine 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]
!
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]
" v (t ) dt
!
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 ) .
30. Find range of f (x ) on (" !, ! )
x "±#
31. Find f !(x ) by definition
!
f ( x + h) \$ f ( x)
f "( x ) = lim
or
h#0
h
!
f ( x ) \$ f ( a)
f "( a ) = lim
x #a
x\$a
32. Find derivative of inverse to f (x ) at x = a
!
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dy
implicitly (in terms
dx
of y). Plug your x value into the inverse relation and
Interchange x with y. Solve for
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solve for y. Finally, plug that y into your
33. y is increasing proportionally to y
dy
= ky translating to y = Ce kt
dt
34. Find the line x = c that divides the area under
f (x ) on [a, b] to two equal areas
c
dy
.
dx
b
! f (x )dx = ! f (x )dx
a
c
x
35.
d
f (t )dt =
dx !a
36.
d
dx
u
" f (t ) dt
a
37. The rate of change of population is …
!
2nd FTC: Answer is f (x )
2nd FTC: Answer is f (u )
du
dx
39. Find area using left Riemann sums
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 = base[x0 + x1 + x 2 + ... + x n !1 ]
40. Find area using right Riemann sums
A = base[x1 + x 2 + x3 + ... + x n ]
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.
base
[x0 + 2 x1 + 2 x2 + ... + 2 xn!1 + xn ]
A=
2
This formula only works when the base is the same. If
not, you have to do individual trapezoids.
Separate the variables – x on one side, y on the other.
The dx and dy must all be upstairs.
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
38. The line y = mx + b is tangent to f (x ) at (x1 , y1 )
42. Find area using 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
b
your square. So the volume is
! (base )dx
2
a
46. Find where the tangent line to f (x ) is horizontal
Write f !(x ) as a fraction. Set the numerator equal to
zero.
47. Find where the tangent line to f (x ) is vertical
Write f !(x ) as a fraction. Set the denominator equal
to zero.
48. Find the minimum acceleration given v(t )
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
49. Approximate the value of f (0.1) by using the
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tangent line to f at x = 0
50. Given the value of F (a ) and the fact that the antiderivative of f is F, find F (b )1
y ! y1 = m(x ! x1 ) where m = f !(0 ) and the point is
(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 (x )dx = F (b )! F (a ). So
a
solve for F (b ) using the calculator to find the definite
integral.
f !(g (x ))" g !(x )
51. Find the derivative of f (g (x ))
b
52. Given
b
! f (x )dx , find ! [f (x )+ k ]dx
a
b
b
b
! [f (x )+ k ]dx = ! f (x )dx + ! kdx
a
a
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]
a
a
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.
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
g + ! (F (t )" E (t ))dt
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
t2
t
m
d
(F (t )! E (t ))dt = F (m )! E (m )
dt t"
dy
, drawing
dx
little lines with the indicated slopes at the points.
Use the given points and plug them into
60. Find the area between curves f (x ), g (x ) on [a, b]
b
A = ! [f (x )" g (x )]dx , assuming that the f curve is
a
above the g curve.
61. Find the volume if the area between f (x ), g (x ) is
rotated about the x-axis
b
2
2'
\$
A = " * & f ( x ) # g ( x ) ) dx assuming that the f curve is
%
(
a
above the g curve.
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!
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BC Problems
f (x)
if lim f (x) = lim g(x) = 0
x "#
x "#
x "# g(x)
62. Find lim
63. Find
!
#
"
0
h
f (x) dx
!
lim \$ f ( x ) dx
h"#
! dP k
#
dP
P&
= kP%1"
64.
= P(M " P) or
! (
\$ M'
dt
dt M
!
65. Find
Use L’Hopital’s Rule.
"x
2
dx
where x 2 + ax + b
!
+ ax + b
!
factors
66. The position vector of a particle moving
) = x(t ), y(t )
in the plane is r(t !
!
a) Find the velocity.
67. b) Find the acceleration.
!
68. c) Find the speed.
0
Signals logistic growth.
dP
lim
=0\$ M =P
t "# dt
Factor denominator and use Heaviside partial fraction
technique.
v(t ) = x "(t ), y "(t )
a(t ) = x ""(t ), y ""(t )
!
2
[ 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 (" ) .
!
!
73. Find the slope of the tangent line to 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?!
!
!
2
" x (t ) dt + " y (t ) dt + C
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 ), ynew = yold + dy
dx
Look at sign of
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dy
d2y
in the interval. This gives you
and
dx
dx 2
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!
n ax
76. Find
"xe
dx where a, n are integers
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.
increasing.decreasing/concavity. Draw picture to ascertain
Integration by parts, " u dv = uv # " v du + C
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!
!
"
!
79.
1
#n
p
converges if…..
!
n=1
!
!
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 This is the error for the 4th term of an alternating series which
th
alternating series for f (x). Approximate is simply the 5 term. It will be positive since you are looking
for an absolute value.
f (x) " S4
!
(n + 1) n!
(n )
83. Suppose f !
. Write the
(x) =
2n
!
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 nth term
!
! at x = c.
where
n is an integer
!
85. f ( x ) = 1+ x +
x2 x3
+ + ...
2! 3!
!
!
!
86. f ( x ) = x "
n
5
m
" (sin x ) (cos x )
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!
n
2!
( x # c)
2
+…+
f
( n)
(c)
( x # c)
!
f ( x ) = sin x
n
f ( x ) = cos x
dx where m and n
!
n
n!
!
You need to determine the largest value of the 5th derivative of
f at some value of z. Usually you are told this. Then:
( n+1)
f
( z) x " c n+1
Rn ( x ) =
( )
(n + 1) !
2n+1
"1) x 2n
(
x2 x 4 x6
! + ...
87. f ( x ) = 1" +
" + ... +
2! 4! 6!
(2n) !
88. Find
f ( x ) = f (c) + f "(c)( x # c) +
f ""(c)
f ( x) = ex
("1) x !+ ...
x
x
+ + ... +
3! 5!
(2n + 1) !
3
You are being given a formula for the derivative of f (x) .
If m is odd and positive, save a sine and convert everything
else to cosine. The sine will be the du. If n is odd and positive,
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are integers
89. Given x = f (t ), y = g (t ), find
dy
dx
!
2
! Given x = f (t ), y = g (t ), find d y
90.
dx 2
!
91.
! Given f ( x ) , find arc length on [ a, b]
save a cosine and convert everything else to sine. The cosine
will be the du. Otherwise use the fact that:
1" cos2x
1+ cos2x
sin 2 x =
and cos2 x =
2
2
dy
dy dt
=
dx dx
dt
d " dy %
\$ '
d 2 y d " dy % dt # dx &
= \$ '=
dx
dx 2 dx # dx &
dt
b
L=
#
a
!
92. x = f (t ), y = g (t ) , find arc length on
! [t1 ,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.
96. Given a set of points, estimate the volume
under the curve using Simpson’s rule on
[a, b] .
97. Find the dot product: u1 , u2 " v1 ,v2
98. Multiply two vectors:
!
[
]
2
1+ f "( x ) dx
" dx %2 " dy %2
L = ( \$ ' + \$ ' dt
# dt & # dt &
t1
x = r cos" , y = r sin "
t2
Find where r sin " = 0 where r cos" # 0
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
A"
b#a
[ y0 + 4 y1 + 2 y2 + 4 y3 + 2 y4 + ... + 4 yn#1 + yn ]
3n
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u1 , u2 " v1 ,v2 = u1v1 + u2v2
You get a scalar.
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Stu Schwartz