Question 1

A body moves along a straight line with velocity $v = F(t)$ as shown in the graph. At which time interval is the body moving in the negative direction?

Accepted Answer

$t = 2$ to $t = 5$

Question 2

Given the velocity function $v = F(t)$, when is the body at its maximum speed?

Accepted Answer

When $|F(t)|$ is at its maximum value

Question 3

If the length $l$ of a rectangle is decreasing at a rate of 2 inches per second while the width $w$ is increasing at a rate of 2 inches per second, what happens to the area $A$ of the rectangle?

Accepted Answer

$A$ remains constant.

Question 4

Find $\lim_{x 	o 0} \frac{e^x - 1}{x}$.

Accepted Answer

$1$

Question 5

Find $\lim_{x 	o 0} \frac{\sin 3x}{x}$.

Accepted Answer

$3$

Question 6

A cube's side is increasing at a constant rate of $0.2$ cm/s. What is the rate of change of the volume, in cubic centimeters per second, when the surface area is $6$ cm$^2$?

Accepted Answer

$0.15$

Question 7

Brent is riding his bicycle north away from an intersection at a rate of $14$ miles per hour. Sally is driving her car toward the intersection from the west at a rate of $36$ miles per hour. If Brent is $4$ miles from the intersection and Sally is $3$ miles from the intersection, what is the rate at which the distance between them is changing?

Accepted Answer

$22$ miles per hour

Question 8

Given $f(x) = (1 - \sin x)^3$, use the local linear approximation at $x = \frac{\pi}{2}$ to estimate $f(0.5)$. Is the estimate an overestimate or underestimate?

Accepted Answer

The estimate is an overestimate.

Question 9

If $y = x^2 e^x$, what is $\frac{dy}{dx}$?

Accepted Answer

$(2x + x^2) e^x$

Question 10

A medication is administered to a patient and the concentration $C(t)$ in the bloodstream decreases according to $C(t) = C_0 e^{-kt}$. What is the rate of change of concentration at time $t$?

Accepted Answer

$-k C_0 e^{-kt}$

Question 11

A car's position is given by $s(t) = t^3 - 6t^2 + 9t + 2$. At what time(s) is the car at rest?

Accepted Answer

$t = 1$ and $t = 3$

Question 12

A bacteria population grows according to $P(t) = 100e^{0.05t}$. What is the instantaneous rate of growth at $t = 10$?

Accepted Answer

$5e^{0.5}$

Question 13

If $y = \ln(x^2 + 1)$, what is $\frac{dy}{dx}$?

Accepted Answer

$\frac{2x}{x^2 + 1}$

Question 14

A ball is thrown upward with velocity $v(t) = 20 - 9.8t$. At what time does it reach its maximum height?

Accepted Answer

$t = 2.04$ s

Question 15

If $f(x)$ is differentiable and $f'(a) = 0$ and $f''(a) > 0$, what can be concluded about $f(x)$ at $x = a$?

Accepted Answer

$f(x)$ has a local minimum at $x = a$

Question 16

A tank is being filled with water at a rate of $5$ liters per minute, but is leaking at a rate proportional to the amount of water present, $L(t) = kW(t)$. Which differential equation models the amount of water $W(t)$ in the tank?

Accepted Answer

$\frac{dW}{dt} = 5 - kW$

Unit 4 Review: Contextual Applications of Differentiation

Study Guide - Practice Questions

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