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
( )
x
f
at
(
)
b
a
,
3.
Find equation of the line normal to
( )
x
f
at
(
)
b
a
,
4.
Show that
( )
x
f
is even
5.
Show that
( )
x
f
is odd
6.
Find the interval where
( )
x
f
is increasing
7.
Find
interval where the slope of
( )
x
f
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
( )
x
f
a
x
→
lim
exists
13. Show that
( )
x
f
is continuous
14. Find vertical asymptotes of
( )
x
f
15. Find horizontal asymptotes of
( )
x
f
16. Find the average rate of change of
( )
x
f
on
[
]
b
a
,
17. Find instantaneous rate of change of
( )
x
f
at
a
18. Find the average value of ( )xfon []ba,
19. Find the absolute maximum of
( )
x
f
on
[
]
b
a
,
20. Show that a piecewise function is differentiable
at the point
a
where the function rule splits
21. Given
( )
t
s
(position function), find
( )
t
v
22. Given
( )
t
v
, find how far a particle travels on
[
]
b
a
,
23. Find the average velocity of a particle on []ba,
24.
Given
( )
t
v
, determine if a particle is speeding up
at
t
=
k
25.
Given
( )
t
v
and
( )
0
s
, find
( )
t
s
26.
Show that Rolle’s Theorem holds on
[
]
b
a
,
27.
Show that
Mean Value Theorem holds on
[
]
b
a
,
28.
Find domain of
( )
x
f
29.
Find range of
( )
x
f
on
[
]
b
a
,
30.
Find range of
( )
x
f
on
(
)
∞
∞

,
31. Find
( )
x
f
′
by definition
32. Find derivative of inverse
to
( )
x
f
at
a
x
=
33.
y
is increasing proportionally to
y
34. Find the line
c
x
=
that divides the area under
( )
x
f
on
[
]
b
a
,
to two equal areas
35.
( )
=
∫
dt
t
f
dx
d
x
a
36.
d
dx
f t
()
a
u
∫
dt
37. The rate of change of population is …
38.
The line
b
mx
y
+
=
is tangent to
( )
x
f
at
(
)
b
a
,
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
( )
dt
t
f
x
a
∫
45. Given a base, cross sections perpendicular to the
x
axis are squares
46. Find where the tangent line to
( )
x
f
is horizontal
47. Find where the tangent line to
( )
x
f
is vertical
48. Find the minimum acceleration given
( )
t
v
49. Approximate the value of
(
)
1
.
0
f
by using the
tangent line to
f
at
0
=
x
50. Given the value of
( )
a
f
and the fact that the anti
derivative of
f
is
F
, find
( )
b
F
51. Find the derivative of
( )
(
)
x
g
f
52.
Given
( )
dx
x
f
b
a
∫
, find
( )
[
]
dx
k
x
f
b
a
∫
+
53. Given a picture of
( )
x
f
′
, find where
( )
x
f
is
increasing
54. Given
( )
t
v
and
( )
0
s
, find the greatest distance
from the origin of a particle on
[
]
b
a
,
55.
Given a water tank with
g
gallons initially being
filled at the rate of
( )
t
F
gallons/min and emptied
at the rate of
( )
t
E
gallons/min on
[
]
2
1
,
t
t
, 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
( )
x
f
on selected values
between
a
and
b
, estimate
( )
c
f
′
where
c
is
between
a
and
b.