Math 2413 Calculus 1 Exam 2 Review
1
1. Find the derivative of the function.
f t( ) =8+2t( )
7
5
2. Find the derivative of the function.
f x( ) =x
5
8−7x
3. Find the derivative of the function.
g x( ) =x+2
x2+5
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
6
4. Find the derivative of the function
y= −7sin2x
.
5. Find the derivative of the function.
y=cos 3x4−4
Ê
Ë
Á
Á
Áˆ
¯
˜
˜
˜
6. Find the derivative of the function.
f
θ
( ) =7
5sin22
θ
7. Evaluate the derivative of the function
f t( ) =7t2+4
3t−1
at the point
4, 116
11
Ê
Ë
Á
Á
Á
Á
Á
Á
ˆ
¯
˜
˜
˜
˜
˜
˜
.
8. Find an equation to the tangent line for the graph of f at the given point.
f x( ) =4x5+5
Ê
Ë
Á
Á
Áˆ
¯
˜
˜
˜
2
,
1,81
Ê
Ë
Á
Áˆ
¯
˜
˜
9. Find the second derivative of the function
f x( ) =sin3x4
.
10. Find
dy
dx
by implicit differentiation.
x2+4x+13xy −y2=16
Name: ________________________ ID: A
2
11. Find
dy
dx
by implicit differentiation.
sinx+9 cos 9y=5
12. Find the slope of the tangent line
10 −x
( )
y
2
=x
3
at the given point
5,5
Ê
Ë
Á
Áˆ
¯
˜
˜
. Round your answer to two
decimal places.
13. Use implicit differentiation to find an equation of the tangent line to the ellipse
x
2
2+y
2
162 =1
at
1,9
Ê
Ë
Á
Áˆ
¯
˜
˜
.
14. Find
d
2
y
d
2
x
in terms of x and y given that
7x
2
+3y
2
=7
. Use the original equation to simplify your answer.
15. Find the points at which the graph of the equation has a vertical or horizontal tangent line.
2x
2
+2y
2
−4x+12y+1=0
16. Assume that x and y are both differentiable functions of t. Find
dx
dt when x= 11 and dy
dt =−2
for the equation
xy =88
.
17. A spherical balloon is inflated with gas at the rate of
300
cubic centimeters per minute. How fast is the
radius of the balloon increasing at the instant the radius is
60
centimeters?
18. All edges of a cube are expanding at a rate of
8
centimeters per second. How fast is the volume changing
when each edge is
3
centimeters?
19. A conical tank (with vertex down) is
20
feet across the top and
26
feet deep. If water is flowing into the tank
at a rate of
16
cubic feet per minute, find the rate of change of the depth of the water when the water is
6
feet
deep.
20. Determine whether Rolle’s Theorem can be applied to the function
f x( ) =x+2( ) x+3( )
2
on the closed
interval
−3,−2
È
Î
Í
Í
͢
˚
˙
˙
˙
. If Rolle’s Theorem can be applied, find all numbers c in the open interval
−3,−2
Ê
Ë
Á
Áˆ
¯
˜
˜
such
that
f′c( ) =0
.
21.
Determine whether the Mean Value Theorem can be applied to the function
f x( ) =x
2
on the closed
interval [
1
,
11
]. If the Mean Value Theorem can be applied, find all numbers c in the open interval
(
1
,
11
) such that
f′c( ) =f11( ) −f1( )
11 −1( )
.
Name: ________________________ ID: A
3
22. The height of an object t seconds after it is dropped from a height of 450 meters is
s t( ) = −4.9t
2
+450.
Find
the time during the first 11 seconds of fall at which the instantaneous velocity equals the average velocity.
23. Identify the open intervals where the function
f x( ) =x30 −x
2
is increasing or decreasing.
24. A ladder
25
feet long is leaning against the wall of a house (see figure). The base of the ladder is pulled away
from the wall at a rate of
2
feet per second. How fast is the top of the ladder moving down the wall when its
base is
14
feet from the wall? Round your answer to two decimal places.
25. A man 6 feet tall walks at a rate of
14
feet per second away from a light that is 15 feet above the ground (see
figure). When he is
13
feet from the base of the light, at what rate is the length of his shadow changing?
Name: ________________________ ID: A
26. Find the value of the derivative (if it exists) of the function
f x( ) =x
2
x
2
+144
at the extremum point
0,0
Ê
Ë
Á
Áˆ
¯
˜
˜
.
27. Find the value of the derivative (if it exists) of the function
f x( ) =5−x| |
at the extremum point
0, 5
Ê
Ë
Á
Áˆ
¯
˜
˜
.
28. Find any critical numbers of the function
g t( ) =t7−t, t <7
.
π
