MATH 2011 C
Quiz 5 Solution
13 February 2019
Name: Dr. Terry
1. Find the derivative of each of the following functions.
1
a.
b.
√3
c.
sec
4
2
3
2
1
2
⁄
∙3
1
√
sec tan
2. A cylindrical tank holding 100,000 gallons of water can be drained in an hour through a valve at the
bottom of the tank. The volume V of water in the tank t minutes after the valve has been opened is
given by the equation
100,000 1
,0
60.
a. What is the volume of the water in the tank 35 minutes after the valve has been opened?
35
100,000 1
17,361 gallons
b. At what rate is the water leaving the tank 35 minutes after the valve has been opened?
100,000 ∙ 2 1
35
,
1
3. Consider the equation
,
∙
1
gal/min
1389 gal/min
4
13.
a. Find all points on the graph of this equation that have an x‐coordinate of
When you substitute
2, you get the quadratic equation
Solving this quadratic using your favorite method, you find
Thus the points on the graph are 2, 9 and 2,1 .
8
9
9
0.
1.
2.
b. Find the slope of the line tangent to the graph of the above equation at each point whose x‐
coordinate is
2.
Take an implicit derivative of the equation:
4
2
Solving for
4
13
2
4
0
gives:
2
4
∙
At the point 2, 9 :
4. Consider the curve
∙
∙
3.2
∙
∙
At the point 2,1 :
4
2
∙
∙
0.8
∙
at the point
,0 .
a. Find ′.
sin
2 sin cos
2 sin cos
b. Find the slope of the tangent to
2 sin cos
at the point
sin
sin
1 sin
sin
,0 .
2 ∙0∙
1
0
0
0.1 √
5. The position of a toy car on a track is described by the equation
seconds.
a. Find an equation for the velocity of the toy car.
0.1 1 ∙
1
⁄
b. What is the velocity of the car at
3
3
0.1 √3
1
Then
sin ∙
cos ∙
∙
0.1 √
1
√
ft/s
3 seconds?
0.1 2
√
0.75
sin cos
6. Find the derivative of the function
This is a double chain rule. Set
⁄
1
and
cos ∙
cos
cos
0.275 ft/s
.
cos .
∙
1∙
∙
1 feet, where t is in