The document contains 10 multi-part exercises involving calculating rates of change, finding maximums and optima, and approximating changes in functions. The exercises involve concepts like linear price-demand functions, surface area and volume relationships for geometric objects, and force functions related to physics concepts like gravity and electric force.

1. EXERCISE SET 4.2
1. A store can sell 100 bags of ice at a price of $2 per bag and 80 bags at $2.50. Assume the
price-demand function is linear. (a) Find the price-demand function. (b) Find the revenue
function. (c) Find the revenue and rates of change of revenue if x = $2 and x = $3. Explain the
results. (d) Find the price that would produce the maximum revenue and the maximum
revenue.
2. A hardware store can sell 60 saws at a price of $8 per saw and 70 saws at $6. Assume the
price-demand function is linear. (a) Find the price-demand function. (b) Find the revenue
function. (c) Find the revenue and rates of change of revenue if x = $7 and x = $14. Explain the
results. (d) Find the price that would produce the maximum revenue and the maximum
revenue.
3. (a) Express the volume V of a sphere as a function of its surface area S. (b) Find dV/dS and
evaluate it at S = 16π and S = 144π. (c) What can you conclude?
4. (a) Express the area A of a circle as a function of its circumference C. (b) Find dA/dC and
evaluate it at C = 2π and C = 10π. (c) What can you conclude?
5. (a) Express the volume V of a rectangular parallelepiped, whose dimensions are x by x by 2x,
as a function of its surface area S. (b) Find dV/dS and evaluate it at S = 10 and S = 90. (c) What
can you conclude?
6. (a) Express the volume V of a rectangular parallelepiped, whose dimensions are x by 2x by
3x, as a function of its surface area S. (b) Find dV/dS and evaluate it at S = 22 and S = 88. (c)
What can you conclude?

2. 7. A ladder 20 feet long leans on a house touching it h feet above the ground. Let be the
angle the ladder makes with the ground. (a) Express h in terms of . Find the rate of change of
h with respect to . What are the units of this rate of change? What is the rate of change when
= 60 ? (b) Approximate the height gained when the angle increases from 60 to 62 .
8. A ladder 8 meters long leans on a house touching it h meters above the ground. Let be the
angle the ladder makes with the house. (a) Express h in terms of . Find the rate of change of h
with respect to . What are the units of this rate of change? What is the rate of change when
= 60 ? (b) Approximate the height lost when the angle increases from 30 to 33 .
9. The electric force between two charged particles separated by x meters is given by
24
F= where F is in Newtons. If the particles are 5 meters apart, find the approximate change
x2
in the force if they are moved 10 centimeters closer.
10. The force of gravity between two masses separated by x kilometers is given by
36
F= where F is in pounds. If the masses are 2 km apart, find the approximate change in the
x2
force if they are moved 20 meters closer.