Unit 4 test: Contextual Applications of Differentiation

Let $N(t)$ be the number of visitors in a museum $t$ hours after it opens. Which is the best interpretation of $N'(2) = -40$?

The amount of caffeine in a person's bloodstream, $C(t)$ milligrams, is measured $t$ hours after drinking coffee.

Use the data to estimate $C'(2)$, in milligrams per hour.

The volume of air in a balloon is $B(t)$ cubic centimeters, $t$ seconds after inflation begins. What are the units of $B''(t)$?

A particle moves along the $x$-axis with position $x(t) = t^3 - 12t + 1$. Find its velocity at $t = 3$.

A particle has position $x(t) = t^3 - 6t^2 + 2$. At what time $t > 0$ is the particle at rest?

A particle has position $x(t) = t^3 - 6t^2 + 9t$. At $t = 1.5$, the particle is

A ball is thrown upward from a 48-foot platform, and its height after $t$ seconds is $h(t) = -16t^2 + 32t + 48$ feet. Find its velocity, in feet per second, when it hits the ground.

An oil spill spreads in a circle whose radius increases at 3 meters per minute. How fast is the area of the spill increasing when the radius is 20 meters? Give an exact answer in square meters per minute.

A 25-foot ladder leans against a wall. The bottom slides away from the wall at 3 feet per second. Find the rate of change of the height of the top of the ladder, in feet per second, when the bottom is 7 feet from the wall.

Water flows at 8 cubic feet per minute into a tank shaped like an inverted cone with height 9 feet and top radius 3 feet. How fast is the water level rising when the water is 6 feet deep? Give the exact value or a decimal rounded to three places, in feet per minute.

A differentiable function $f$ has $f(4) = 3$ and $f'(4) = -0.5$. Use the tangent line at $x = 4$ to approximate $f(4.2)$.

The tangent line to $f(x) = \sqrt{x}$ at $x = 25$ is used to approximate $\sqrt{26}$. Which is correct?

Evaluate $\lim\limits_{x \to 0} \dfrac{\sin(5x)}{2x}$.

Functions $f$ and $g$ have continuous derivatives. The table gives selected values.

Find $\lim\limits_{x \to 3} \dfrac{f(x)}{g(x)}$.

Evaluate $\lim\limits_{x \to 0} \dfrac{x - \sin x}{x^3}$.