4. Applications of Derivatives

Two cars leave the same intersection and drive in perpendicular directions. Car A travels east at a speed of $60\(\frac{mi}{hr}\)$, Car B travels north at a speed of $40\(\frac{mi}{hr}\)$. Car A leaves the intersection at $2pm$, while Car B leaves at $2:30pm$. Determine the rate at which the distance between the two cars is changing at $3pm$.

Which of the following gives the equations of both lines through the point $\left(2,,,,-,3\right)$ that are tangent to the parabola $y=x^2+x$?

A 12-ft ladder is leaning against a vertical wall when Jack begins pulling the foot of the ladder away from the wall at a rate of 0.2 ft/s. What is the configuration of the ladder at the instant when the vertical speed of the top of the ladder equals the horizontal speed of the foot of the ladder?

A 15-foot plank leans against a vertical pole. The top of the plank begins to slide down the pole at a steady speed of 2 inches per second. How fast is the bottom of the plank moving away from the pole when it is 8 feet away from the base of the pole (in inches per second)?

Define the acceleration of an object moving in a straight line.

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.

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

Suppose the cost of producing x lawn mowers is C(x) = −0.02x²+400x+5000. 

b. Interpret the meaning of your results in part (a).

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

a. At what rate is the area of the square changing when the sides are 5 m long?

Given the equation $x+3y{y}^{13}$=$y$, what is $\frac{dy}{dx}$ at the point $(2,8)$?

A sliding ladder

A 13-ft ladder is leaning against a house when its base starts to slide away. By the time the base is 12 ft from the house, the base is moving at the rate of 5 ft/sec.

b. At what rate is the area of the triangle formed by the ladder, wall, and ground changing then?

Suppose w(t) is the weight (in pounds) of a golden retriever puppy t weeks after it is born. Interpret the meaning of w'(15) = 1.75.

If x²y³ = 4/27 and dy/dt = ¹/₂, then what is dx/dt when x = 2?

Once Kate’s kite reaches a height of 50 ft (above her hands), it rises no higher but drifts due east in a wind blowing 5 ft/s. How fast is the string running through Kate’s hands at the moment when she has released 120 ft of string?

The edges of a cube increase at a rate of 2 cm/s. How fast is the volume changing when the length of each edge is 50 cm?

Economics

Marginal revenue

Suppose that the revenue from selling x washing machines is

r(x) = 20000(1 − 1/x) dollars.

c. Find the limit of r'(x) as x → ∞. How would you interpret this number?