4. Applications of Derivatives

Explain the difference between the average rate of change and the instantaneous rate of change of a function f.

A balloon and a bicycle A balloon is rising vertically above a level, straight road at a constant rate of 1 ft/sec. Just when the balloon is 65 ft above the ground, a bicycle moving at a constant rate of 17 ft/sec passes under it. How fast is the distance s(t) between the bicycle and the balloon increasing 3 sec later?

Interpreting the derivative Find the derivative of each function at the given point and interpret the physical meaning of this quantity. Include units in your answer.

When a faucet is turned on to fill a bathtub, the volume of water in gallons in the tub after t minutes is V(t)=3t. Find V′(12).

Watching an elevator An observer is 20 m above the ground floor of a large hotel atrium looking at a glass-enclosed elevator shaft that is 20 m horizontally from the observer (see figure). The angle of elevation of the elevator is the angle that the observer’s line of sight makes with the horizontal (it may be positive or negative). Assuming the elevator rises at a rate of 5 m/s, what is the rate of change of the angle of elevation when the elevator is 10 m above the ground? When the elevator is 40 m above the ground? <IMAGE>

The sides of a square decrease in length at a rate of 1 m/s.

b. At what rate are the lengths of the diagonals of the square changing?

The lateral surface area S of a right circular cone is related to the base radius r and height h by the equation S = πr√(r² + h²). 

a. How is dS/dt related to dr/dt if h is constant?

The perimeter of a rectangle is fixed at . If the length $L$ is increasing at a rate of , for what value of $L$ does the area start to decrease? Hint: the rectangle's area starts to decrease when the rate of change for the area is less than 0.

{Use of Tech} A mixing tank A 500-liter (L) tank is filled with pure water. At time t=0, a salt solution begins flowing into the tank at a rate of 5 L/min. At the same time, the (fully mixed) solution flows out of the tank at a rate of 5.5 L/min. The mass of salt in grams in the tank at any time t≥0 is given by M(t) = 250(1000−t)(1−10−³⁰(1000−t)¹⁰) and the volume of solution in the tank is given by V(t) = 500-0.5t.

b. Graph the volume function and verify that the tank is empty when t=1000 min. 

Right circular cylinder The total surface area S of a right circular cylinder is related to the base radius r and height h by the equation S = 2πr² + 2πrh.

c. How is dS/dt related to dr/dt and dh/dt if neither r nor h is constant?

Diagonals If x, y, and z are lengths of the edges of a rectangular box, then the common length of the box’s diagonals is s = √(x² + y² + z²).

a. Assuming that x, y, and z are differentiable functions of t, how is ds/dt related to dx/dt, dy/dt, and dz/dt?

Economics

Marginal cost Suppose that the dollar cost of producing x washing machines is c(x) = 2000 + 100x − 0.1x².

c. Show that the marginal cost when 100 washing machines are produced is approximately the cost of producing one more washing machine after the first 100 have been made, by calculating the latter cost directly.

Given the equation below, find $dy/dt$ when $dx/dt=12$ and $x=\frac{15}{2}$.

$y=\sqrt{2x+1}$

Consider the following cost functions.

a. Find the average cost and marginal cost functions.

C(x) = 1000+0.1x, 0≤x≤5000, a=2000

A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.

c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?

Given the equation $e^x-y=x{y}^3+e^2-18$, what is the value of $\frac{dy}{dx}$ at the point $(2,2)$?