AP Calculus AB Unit 4: Contextual Applications of Differentiation

Interpreting a derivative in context with units, straight-line motion (position, velocity, acceleration and speed), rates of change in other settings, related rates, local linearity and tangent-line approximation, and L'Hospital's rule for $\frac00$ and $\frac\infty\infty$ forms.

CED topics in this unit: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7.

Key formulas and facts

• $v(t)=x'(t)$, $a(t)=v'(t)=x''(t)$, speed $=|v(t)|$
• Speed increases when $v$ and $a$ have the same sign, and decreases when they have opposite signs
• Tangent-line approximation: $f(x)\approx f(a)+f'(a)(x-a)$ for $x$ near $a$
• L'Hospital's rule: if $\lim f=\lim g=0$ (or both are infinite), then $\lim\frac fg=\lim\frac{f'}{g'}$ when the right side exists
• Related rates: write an equation linking the quantities, differentiate with respect to $t$, then substitute the values at the instant
• Cone $V=\frac13\pi r^2h$; sphere $V=\frac43\pi r^3$, $S=4\pi r^2$

Worked examples

• Units: if $W(t)$ is the number of liters in a tank after $t$ minutes, $W'(5)=-3$ means that at $t=5$ the volume is falling at $3$ liters per minute
• Motion: $x(t)=t^3-6t^2+9t$ gives $v=3(t-1)(t-3)$ and $a=6t-12$; at $t=2$, $v=-3$ and $a=0$
• Related rates: a circle's radius grows at $2$ cm/s; when $r=5$, $\frac{dA}{dt}=2\pi r\frac{dr}{dt}=20\pi\text{ cm}^2/\text{s}$
• Linear approximation: $\sqrt{26}\approx5+\frac{1}{10}(1)=5.1$
• L'Hospital: $\lim_{x\to0}\frac{e^{2x}-1}{\sin x}=\lim_{x\to0}\frac{2e^{2x}}{\cos x}=2$

Common traps

• Substituting the values at the instant before differentiating in a related-rates problem.
• Saying speed increases because acceleration is positive: check that $v$ and $a$ have the same sign.
• Using L'Hospital's rule without first checking for $\frac00$ or $\frac\infty\infty$, or using the quotient rule instead of differentiating the top and bottom separately.
• Leaving out the units or the time in an interpretation sentence.
• Calling a tangent-line approximation an overestimate or an underestimate without using concavity.

Practice questions

42 multiple-choice and 4 free-response questions. Try each one before opening the answer. Each line of a free-response scoring guide is one point. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

The cost, in dollars, of producing $n$ items is $C(n)=600+5n-0.004n^2$ for $0\le n\le 500$. Which gives the value of $C'(250)$ and its meaning?

1. $3$; when $250$ items have been produced, the cost is increasing at about 3 dollars per additional item
2. $3$; the average cost of each of the first $250$ items is 3 dollars
3. $1600$; producing $250$ items costs 1600 dollars
4. $6.4$; when $250$ items have been produced, the cost is increasing at about 6.40 dollars per additional item

Question 2 · Multiple choice

A particle moves along the $x$-axis with position $x(t)=t^3-18t^2+81t$ for $t\ge0$. For which values of $t$ is the speed of the particle increasing?

1. $t>6$ only
2. $0<t<3$ and $t>9$
3. $3<t<6$ and $t>9$
4. $3<t<9$

Question 3 · Multiple choice

The population of a town is modeled by $P(t)=\dfrac{800}{1+3e^{-0.4t}}$, where $t$ is in years. At what rate, in people per year, is the population changing at $t=5$?

1. $73.798$
2. $568.988$
3. $60.000$
4. $65.722$

Question 4 · Multiple choice

The volume of a cube is decreasing at $30\text{ cm}^3$ per minute. How fast is the surface area of the cube changing when each edge is $5$ cm long?

1. $-12$
2. $- \frac{2}{5}$
3. $-24$
4. $-60$

Question 5 · Multiple choice

A point moves along the curve $y=x^2$ so that $\dfrac{dx}{dt}=3$. How fast is the distance from the point to the origin changing when $x=2$?

1. $\frac{9 \sqrt{5}}{5}$
2. $\frac{27 \sqrt{5}}{5}$
3. $3 \sqrt{5}$
4. $36$

Question 6 · Multiple choice

$f(2)=5$, $f'(2)=-3$ and $f''(x)>0$ for all $x$. Using the line tangent to the graph of $f$ at $x=2$, what is the approximation of $f(2.1)$, and is it an overestimate or an underestimate?

1. $4.7$, an underestimate
2. $4.7$, an overestimate
3. $5.3$, an underestimate
4. $5.3$, an overestimate

Question 7 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{\cos(4x)-1}{x^2}$?

1. $-2$
2. $-8$
3. $8$
4. $0$

Question 8 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{\ln(x^3+1)}{\ln x}$?

1. $1$
2. $0$
3. $3$
4. $\infty$

Question 9 · Free response (9 points)

A particle moves along the $x$-axis. Its position at time $t$ is $x(t)=2t^3-27t^2+84t-5$ for $0\le t\le8$.

1. (a) Find the velocity and the acceleration of the particle at $t=1$. Is the speed of the particle increasing or decreasing at $t=1$? Give a reason. [3 points]
2. (b) At what times does the particle change direction? Justify your answer. [2 points]
3. (c) Find the total distance traveled by the particle for $0\le t\le8$. [3 points]
4. (d) Find the minimum velocity of the particle for $0\le t\le8$. [1 point]

Question 10 · Free response (9 points)

The legs of a right triangle change with time. The horizontal leg $x$ is increasing at $3$ cm per second and the vertical leg $y$ is decreasing at $2$ cm per second. At time $t=0$, $x=8$ cm and $y=6$ cm. Let $\theta$ be the angle opposite the vertical leg, so that $\tan\theta=\frac{y}{x}$.

1. (a) Find the rate of change of the area of the triangle at $t=0$. Indicate units of measure. [2 points]
2. (b) Find the rate of change of the length of the hypotenuse at $t=0$. [3 points]
3. (c) Find the rate of change of $\theta$ at $t=0$. [3 points]
4. (d) Use the rate from part (a) to estimate the area of the triangle at $t=0.5$ seconds. [1 point]

Question 11 · Multiple choice

The cost, in dollars, of producing $x$ chairs is $C(x)$, and $C'(100)=12$. Which is the best estimate of the additional cost of producing the $101$st chair?

1. $12$
2. $1200$
3. $\frac{3}{25}$
4. $112$

Question 12 · Multiple choice

The temperature of a cooling drink is $H(t)=60+20e^{-0.1t}$ degrees Fahrenheit, $t$ minutes after it is poured. What is $H'(10)$, in degrees Fahrenheit per minute?

1. $-0.736$
2. $67.358$
3. $-0.271$
4. $0.736$

Question 13 · Multiple choice

The number of people who have heard a rumor is modeled by $P(t)=\dfrac{2000}{1+e^{-t}}$, where $t$ is measured in days. At what rate, in people per day, is $P$ increasing at $t=0$?

1. $1000$
2. $250$
3. $500$
4. $2000$

Question 14 · Multiple choice

The volume of a sphere of radius $r$ centimeters is $V(r)=\frac43\pi r^3$ cubic centimeters. What is $V'(3)$, and what are its units?

1. $36\pi$ cubic centimeters
2. $36\pi$ cubic centimeters per centimeter
3. $12\pi$ square centimeters
4. $108\pi$ cubic centimeters per centimeter

Question 15 · Multiple choice

A particle moves along the $x$-axis so that its position at time $t\ge0$ is $x(t)=t^3-6t^2+9t$. At what times is the particle at rest?

1. $t=2$ only
2. $t=0$ and $t=3$ only
3. $t=0$, $t=1$ and $t=3$
4. $t=1$ and $t=3$ only

Question 16 · Multiple choice

For $0<t<6$, an object's velocity along a straight track is $v(t)=t^2-7t+10$. During which times is the object speeding up?

1. $\left(\frac72,6\right)$
2. $(2,5)$
3. $(0,2)\cup\left(\frac72,5\right)$
4. $\left(2,\frac72\right)\cup(5,6)$

Question 17 · Multiple choice

A particle's position is $x(t)=te^{-t}$ for $t\ge0$. At what time does the particle change direction?

1. $0$
2. $2$
3. $e$
4. $1$

Question 18 · Multiple choice

A particle moves with position $x(t)=t^3-3t^2$ for $t\ge0$. What is the acceleration of the particle at the time $t>0$ when its velocity is $0$?

1. $6$
2. $0$
3. $-6$
4. $12$

Question 19 · Multiple choice

A particle moves along the $x$-axis with velocity $v(t)=\sin(t^2)$ for $0<t<3$. How many times does the particle change direction?

1. $1$
2. $3$
3. $2$
4. $0$

Question 20 · Multiple choice

The cost, in dollars, of making $x$ bicycles is $C(x)=0.02x^2+5x+300$. What is the marginal cost $C'(100)$, in dollars per bicycle?

1. $1000$
2. $5$
3. $7$
4. $9$

Question 21 · Multiple choice

The concentration of a drug in the blood is $C(t)=\dfrac{5t}{t^2+4}$ milligrams per liter, $t$ hours after an injection. What is the rate of change of the concentration at $t=1$?

1. $1$
2. $- \frac{3}{5}$
3. $\frac{3}{5}$
4. $\frac{1}{5}$

Question 22 · Multiple choice

The brightness of a light at distance $d$ meters is $I(d)=\dfrac{400}{d^2}$ units. What is the rate of change of $I$ with respect to $d$ at $d=5$?

1. $16$
2. $- \frac{32}{5}$
3. $-16$
4. $- \frac{8}{5}$

Question 23 · Multiple choice

The population of a town is $P(t)=500e^{0.03t}$ people, $t$ years after 2020. At what rate is the population growing at $t=10$, in people per year?

1. $674.929$
2. $15.000$
3. $20.248$
4. $0.405$

Question 24 · Multiple choice

The volume $V(t)$ of water in a tank, in liters, is measured at selected times $t$ in minutes. Using the data, what is the best estimate of $V'(4.5)$?

1. $-3$
2. $-5$
3. $-4$
4. $-9$

Question 25 · Multiple choice

A store sells $q(p)=1000-5p^2$ games per month when the price is $p$ dollars. What is $\dfrac{dq}{dp}$ when $p=4$?

1. $-40$
2. $920$
3. $-10$
4. $40$

Question 26 · Multiple choice

The variables $x$ and $y$ are differentiable functions of $t$ with $x^2+y^2=25$. When $x=3$, $y=4$ and $\dfrac{dx}{dt}=3$, what is $\dfrac{dy}{dt}$?

1. $\frac{9}{4}$
2. $- \frac{9}{4}$
3. $-4$
4. $- \frac{3}{4}$

Question 27 · Multiple choice

The radius of a circle is increasing at $2$ centimeters per second. How fast is the area increasing when the radius is $5$ centimeters?

1. $20 \pi$
2. $10 \pi$
3. $25 \pi$
4. $4 \pi$

Question 28 · Multiple choice

If $y=x^3$ and $\dfrac{dx}{dt}=4$, what is $\dfrac{dy}{dt}$ when $x=2$?

1. $12$
2. $32$
3. $8$
4. $48$

Question 29 · Multiple choice

The volume of a cube is increasing at $12$ cubic inches per minute. How fast is the edge length increasing when the edge is $2$ inches long?

1. $1$
2. $3$
3. $4$
4. $\frac{1}{4}$

Question 30 · Multiple choice

A $13$-foot ladder leans against a vertical wall. The bottom slides away from the wall at $2$ feet per second. How fast, in feet per second, is the top sliding down the wall when the bottom is $5$ feet from the wall?

1. $\frac{12}{5}$
2. $\frac{10}{13}$
3. $\frac{5}{6}$
4. $\frac{5}{12}$

Question 31 · Multiple choice

Water is poured at $4$ cubic feet per minute into a tank shaped like an inverted cone with radius $3$ feet and height $6$ feet. How fast is the water level rising when the water is $2$ feet deep?

1. $\frac{1}{\pi}$
2. $\frac{4}{\pi}$
3. $\frac{16}{\pi}$
4. $\frac{2}{\pi}$

Question 32 · Multiple choice

Air is pumped into a spherical balloon at $100$ cubic centimeters per second. How fast is the radius increasing when the radius is $5$ centimeters?

1. $\frac{4}{\pi}$
2. $\frac{1}{4 \pi}$
3. $\frac{1}{\pi}$
4. $\frac{5}{\pi}$

Question 33 · Multiple choice

Two cars leave the same intersection at the same time, one heading north at $30$ miles per hour and the other east at $40$ miles per hour. How fast is the distance between them increasing $2$ hours later?

1. $70$
2. $50$
3. $100$
4. $10$

Question 34 · Multiple choice

A $6$-foot-tall person walks away from a $15$-foot lamppost at $5$ feet per second. How fast is the length of the person's shadow increasing?

1. $\frac{5}{3}$
2. $\frac{25}{3}$
3. $2$
4. $\frac{10}{3}$

Question 35 · Multiple choice

Using the line tangent to $y=\sqrt x$ at $x=16$, what is the approximation of $\sqrt{16.2}$?

1. $\frac{81}{20}$
2. $\frac{21}{5}$
3. $\frac{161}{40}$
4. $\frac{321}{80}$

Question 36 · Multiple choice

A differentiable function $f$ has $f(2)=5$ and $f'(2)=-3$. Using the tangent line at $x=2$, what is the approximation of $f(2.1)$?

1. $\frac{53}{10}$
2. $2$
3. $\frac{47}{10}$
4. $- \frac{3}{10}$

Question 37 · Multiple choice

What is the approximation of $\ln(0.9)$ given by the line tangent to $y=\ln x$ at $x=1$?

1. $\frac{1}{10}$
2. $- \frac{9}{10}$
3. $- \frac{1}{10}$
4. $0$

Question 38 · Multiple choice

The line tangent to $y=e^x$ at $x=0$ is used to approximate $e^{0.1}$. Which gives the approximation and describes it correctly?

1. 1.1, an overestimate
2. 0.9, an underestimate
3. 1.1, an underestimate
4. 1.01, an overestimate

Question 39 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{e^{4x}-4x-1}{x^2}$?

1. $4$
2. $16$
3. $8$
4. $0$

Question 40 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{\ln x}{\sqrt{x}}$?

1. $1$
2. $0$
3. $2$
4. $\infty$

Question 41 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{\sin x-x}{x^3}$?

1. $\frac{1}{6}$
2. $0$
3. $- \frac{1}{3}$
4. $- \frac{1}{6}$

Question 42 · Multiple choice

What is $\displaystyle\lim_{x\to1}\frac{\ln x}{x^3-1}$?

1. $\frac{1}{3}$
2. $1$
3. $3$
4. $0$

Question 43 · Multiple choice

What is $\displaystyle\lim_{x\to0}\frac{1-\cos(3x)}{x\sin x}$?

1. $\frac{3}{2}$
2. $\frac{9}{2}$
3. $9$
4. $3$

Question 44 · Multiple choice

What is $\displaystyle\lim_{x\to\pi/2}\frac{1-\sin x}{\cos x}$?

1. $1$
2. $0$
3. $-1$
4. $\infty$

Question 45 · Free response (9 points)

A particle moves along the $x$-axis so that its position at time $t$, for $0\le t\le5$, is $x(t)=2t^3-15t^2+36t$.

1. (a) Find the velocity and the acceleration of the particle at $t=2.5$. [2 points]
2. (b) On what open interval is the particle moving to the left? Justify your answer. [2 points]
3. (c) Is the speed of the particle increasing or decreasing at $t=1$? Give a reason. [2 points]
4. (d) Find the total distance traveled by the particle over $0\le t\le5$. [3 points]

Question 46 · Free response (9 points)

The radius $r(t)$, in centimeters, of a circular puddle is a twice-differentiable, increasing function of time $t$, in minutes. Selected values are in the table. In addition, $r'(4)=1.2$ and $r''(t)<0$ for all $t$.

1. (a) Use the data in the table to estimate $r'(3)$. Show the computations, and indicate units of measure. [2 points]
2. (b) Use the line tangent to the graph of $r$ at $t=4$ to approximate $r(4.5)$. [2 points]
3. (c) Find the rate at which the area of the puddle is changing at $t=4$. Indicate units of measure. [3 points]
4. (d) Is the approximation in part (b) an overestimate or an underestimate? Give a reason. [2 points]