Chapter 3: Worksheet 2 Jack K. Cohen Colorado School of Mines
The Derivative as the Rate of Change Function
Suggested Problems
Section 3.1: 22, 24, 26
- Find the rate of change of the area of a circle with respect to its radius.
- A stone dropped into a pond causes an expanding circular ripple. What is the rate of change of the area of a circle with respect to its radius when the radius is 3.5 meters? Give the answer correct to two decimal places.
- (3.1.31) Find the rate of change of the area of a circle with respect to its circumference.
- A stone dropped into a pond makes an expanding circular ripple. What is the rate of change of the area of a circle with respect to its circumference when the circumference is 3.5 meters?
- (3.1.32) A stone dropped into a pond causes a circular ripple that travels out from the point of impact at 5 ft/sec. At what rate (in ft2/sec) is the area within the circle increasing when t = 10. Hint: Find a formula for r in terms of t.
- (Generalization of Example 6)
- Find the maximum height attained by a ball that is thrown straight up from the ground with initial velocity v0. Assume a gravitational constant g.
- Find the velocity with which it hits the ground upon its return.
- One benefit of using parameters instead of specific numbers is the ability to deduce general laws. Illustrate this by stating a general connection between the initial and final velocities.
- Another benefit of using parameters is the ability to check that answers have the correct dimensions. For example, if somehow, we got the answer
stop =
, we could see that the dimensions of the purported answer were
÷
= T when they should have been L. Dimensional checks can save a lot of embarrassment and every good scientist uses them to check calculations. So don't you be embarrassed—give a dimensional check for your answers to parts (a) and (b).
- The treatment in the last problem assumed that the force of gravity was a constant. Actually it falls off like 1/r2 where r is the distance of the body (ball) from the center of the Earth (this is called the ``inverse square law'') . Taking this variable force law into account gives the equation
s =

,
where R is the mean radius of the Earth and s = r - R is the distance measured upward from the Earth's surface.
- Is the final velocity still the same as the initial velocity when the inverse square law is taken into account?
- What is the maximum height attained when the inverse square law is taken into account?
- Using the numbers: v0 = 96 ft/sec, g = 32 ft/sec2, R = 3960 miles, discuss whether it is worthwhile to consider the more complicated inverse square law model for this problem. Remark: Air resistence is another effect we might consider. Mathematical modeling is always a question of making things ``as simple as possible, but no simpler.''
- (3.1.34) A water bucket containing 10 liters of water develops a leak at time t = 0, and the volume V of water in the bucket t seconds later is given by
until the bucket is empty at time t = 100.
- At what rate is water leaking from the bucket after exactly one minute has passed?
- When is the instantaneous rate of change of V equal to the average change of V from t = 0 to t = 100?
- Does the given formula for V make sense when t = 200?
- (3.1.36) The following data describe the growth of the population P (in thousands) of a certain city during the 1970s. Use the graphical method of Example 4 to estimates its rate of growth in 1975.
Year |
1970 |
1972 |
1974 |
1976 |
1978 |
1980 |
P |
265 |
293 |
324 |
358 |
395 |
437 |