AP Calculus AB Rates of Change in Context and Straight-Line Motion Test Solutions and Scoring (PDF)

AP Calculus AB math test solutions: Rates of Change in Context and Straight-Line Motion – Maths PDF

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This rates motion test answers for AP Calculus AB covers rates of change in context and straight-line motion (Common Core AP CALC AB 4.1, AP CALC AB 4.2, AP CALC AB 4.3), with a free PDF to print.

Use this answer key to score the rates of change test out of 20 points. Each solution shows the work and a scoring guide that says how the points are awarded. Give partial credit for a correct setup, a correct derivative or integral, and a correct final statement with units where they apply.

AP Calculus AB · Common Core State Standards: AP CALC AB 4.1, AP CALC AB 4.2, AP CALC AB 4.3

Rates motion test answers: worked solutions

Question 1 – Interpreting derivatives [2 pts]

At day 8 the plant is growing at 1.5 cm per day.
At month 6 the share price is falling at \$2.30 per month.

Scoring: 1 point each for value, units, and direction in a full sentence.

Question 2 – Position, velocity, acceleration [3 pts]

$v=3t^{2}-12$ and $a=6t$.
$3t^{2}-12=0$ gives $t=2$ (since $t\ge 0$).
$a(2)=12\text{ m/s}^{2}$.

Scoring: 1 point per part.

Question 3 – Direction from a sign chart [3 pts]

At $t=1$ and $t=4$, where $v$ changes sign.
Left on $(1,4)$, where $v < 0$.
$v(2)=-2 < 0$ and $a(2)=-1 < 0$: same sign, so speeding up.

Sign chart for the velocity of a particle.

Scoring: 1 point per part.

Question 4 – Geometric rates [2 pts]

$A^{\prime}=2\pi r=16\pi\approx 50.3\text{ cm}^{2}\text{ per cm}$.
$V^{\prime}=4\pi r^{2}$, so $V^{\prime}(3)=36\pi\approx 113.1$ cubic units per unit of radius.

Scoring: 1 point per part.

Question 5 – Velocity from a table [3 pts]

$\frac{84-0}{6}=14$ m/s.
$\frac{84-12}{4}=18$ m/s.
The averages are $6$, $14$ and $22$ m/s, increasing, so the sled is speeding up.

Scoring: 1 point per part.

Question 6 – Reading a velocity graph [3 pts]

At $t=3$ (where $3-t=0$) and $t=4.33$ (where $3(t-4)-1=0$), about $t=\frac{13}{3}$.
Slowing down when $v$ and $a$ have opposite signs: on $(0,3)$ ( $v > 0$, $a=-1 < 0$) and on $\left(4,\frac{13}{3}\right)$ ( $v < 0$, $a=3 > 0$).
The slope is $3\text{ ft/s}^{2}$.

Velocity in feet per second.

Scoring: 1 point per part.

Question 7 – Exponential rate [2 pts]

$N^{\prime}(t)=20e^{0.4t}$, so $N^{\prime}(3)=20e^{1.2}\approx 66.4$.

At day 3 the algae are increasing by about 66.4 thousand cells per day.

Scoring: 1 point for the derivative and value; 1 point for the interpretation with units.

Question 8 – Total distance [2 pts]

$v=3(t-2)(t-4)$ changes sign at $t=2$ and $t=4$. Positions: $s(0)=0$, $s(2)=20$, $s(4)=16$, $s(5)=20$.

Distance: $20+4+4=28$ feet.

Scoring: 1 point for the turning points; 1 point for the total.

Total score

Total

20 points

Key facts behind this rates motion test answers

Key facts

A derivative has units of output per input, such as dollars per pencil or feet per second.
Velocity is s′(t), acceleration is v′(t), and speed is the absolute value of velocity.
Speed increases when velocity and acceleration have the same sign.
Total distance adds the distances between turning points, where velocity changes sign.

How to use this rates motion test answers

Score your test with the point values shown for each question. Then read the worked solution of every question where you lost points and note the method to review.

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