Math 161

Sample First Midterm

Spring 2005

Dr. Wilson

To see a complete worked solution for a problem, click on the problem number.

1. Find

when

2. A drag racer accelerating from a complete stop, covered 1/4 mile in 5 seconds. A frame by frame analysis of a video tape of the event resulted in the perparation of the following data.

t

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

s

0

12.7

29,9

54.9

84.5

118.1

155.2

195.6

238.9

285.1

333.9

385.3

439.0

t

2.6

2.8

3.0

3.2

3.4

3.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

s

495.0

553.2

613.5

675.8

740.2

806.4

874.6

944.5

1016.2

1089.7

1164.8

1241.6

1320.0

where s denotes the distance traveled in feet and t denotes the time in seconds.

a) Draw the graph. What can you tell about the velocity and acceleration by looking at the graph?

b) The output from a radar gun was included in the video tape. It said that the instantaneous velocity at 4 sec was 241.5 mi/hr. Does this agree with the data above? Compute the average velocity between 3.6 sec. and 4 sec, 3.8 sec. and 4 sec., 4sec. and 4.4 sec., and 4 sec. and 4.2 sec. (Don't forget to change ft./sec. into mi./hr.) What can you say about the acceleration from these results?

c) In order to get a better approximation for the instantaneous velocity, one would need to use smaller increments of time. The video tape has 50 frames per second, so we can get the following refinement of the data.

t

3.80

3.82

3.84

3.86

3.88

3.90

3.92

3.94

3.96

3.98

4.00

s

874.6

881.5

888.4

895.4

902.3

909.3

916.3

923.3

930.4

937.4

944.5

t

4.00

4.02

4.04

4.06

4.08

4.10

4.12

4.14

4.16

4.18

4.20

s

944.5

951.6

958.7

965.8

973.0

980.2

987.3

994.5

1001.7

1009.0

1016.2

What is the best estimate for the instantaneous velocity at 4 sec. that you can get from this data?

d) Notice that in part b) the instantaneous velocity at 4 sec. was fairly close to the average between the average velocity between 3.8 and 4 sec., and between 4 and 4.2 sec. Use this technique to find the instantaneous velocities at 1, 2, and 3 sec. Use these data together with your instantaneous velocity at 4 sec. and the fact that you can figure that the instantaneous velocity when t = 0 is 0, to draw a graph of the velocity as a function of time. What can you say about the acceleration?

 

3. Given the following graphs of f(x), sketch the graphs of f'(x) and f"(x).

a)

b)

c)

4. Find the first and second derivative.

a)

b)

c)

d)

e)

f)

g)

h)

5. Find the slope of the line which is tangent to the curve consisting of points in the plane satisfying the following equation

xy2 - y = x + 1.

Find the slope of the line which is tangent to the curve at the following points.

a)

(1, 2)

b)

(-1/2, -1)

6. The formula for the height of a ball thrown straight up in the air is

h(t) = -16t2 + 60t + 8

7. An electric guitarist gets enough feedback into his guitar so that the amplitude of string playing a note stays constant. The formula for the displacement of the string from its equilibrium point over the pickup as a function of time is given by the formula

where the displacement is measured in millimeters. Find the formula for the velocity and the acceleration of the string.

8. Find the best linear approximation for ln x which is tangent to the graph of the function when x = 1. Use this to approximate ln 1.05. Check your answer with a calculator.

9. Use L'Hopital's rule to evaluate