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)?

Checking the Mean Value Theorem

Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.

f(x) = {x² − x, −2 ≤ x ≤−1

2x² − 3x − 3, −1 < x ≤ 0

A sphere is growing at a rate of $50\frac{\operatorname{\mathrm{cm}}^3}{s}$. At what rate is the radius of the sphere increasing when the radius is $5\operatorname{\mathrm{cm}}$?

A right tringle has a base of $10\operatorname{\mathrm{cm}}$ and a height of $12\operatorname{\mathrm{cm}}$. The height of the right triangle is decreasing at a rate of $0.4\operatorname{\frac{\mathrm{cm}}{s}}$, at what rate is the area of the triangle decreasing?

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.

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 $3pm$.

Given the equation below, find $dz/dt$ when $dx/dt=3$, $dy/dt=2$, $x=2$, $y=4,$ and $z=1$.

$x^2+y^2-3z^2=125$

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

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

Demand and elasticity The economic advisor of a large tire store proposes the demand function D(p) = 1800/p-40, where D(p) is the number of tires of one brand and size that can be sold in one day at a price p.

c. Find the elasticity function on the domain of the demand function.