AP Calculus AB Unit 4 — Rates of Change and Related Rates Worksheet

All 8 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one.

1. Q1Interpreting the Meaning of the Derivative in Context

   A rooftop rain cistern is being drained through a filter. Let $W(t)$ be the volume of water in the cistern, in liters, where $t$ is measured in hours after the valve is opened.

   1. State the units of $W^{\prime }(t)$, and explain in one sentence where those units come from.
   2. Given that $W(3)=418$ and $W^{\prime }(3)=-26$, write one sentence interpreting $W^{\prime }(3)$ in the context of the cistern, using correct units.
   3. Explain what would be different about the cistern if $W^{\prime }(3)$ were $+26$ instead.
2. Q2Straight-Line Motion - Position, Velocity and Acceleration

   A maintenance cart runs along a straight rail. Its position, in meters from the depot, is $$s(t)=2t^3-15t^2+24t+5,0\le t\le 5,$$ where $t$ is in seconds.

   1. Find $v(t)$ and $a(t)$, with units.
   2. Find every time in $(0,5)$ at which the cart is at rest.
   3. At $t=2$ seconds, is the cart speeding up or slowing down? Justify your answer.
3. Q3Rates of Change in Applied Contexts Other Than Motion

   A greenhouse tank is filled and then drawn down over one working day. The volume of water in it is $$V(t)=240+36t-3t^2\text{liters},0\le t\le 10,$$ where $t$ is in hours after 7 a.m.

   1. Find $V^{\prime }(t)$ and state its units.
   2. Find $V^{\prime }(8)$ and interpret it in one sentence, with units.
   3. At what time is the volume of water greatest? Justify your answer using $V^{\prime }$.
4. Q4Introduction to Related Rates

   Two quantities $x$ and $y$ are both functions of time $t$ and satisfy $$xy+x^2=40.$$

   1. Differentiate both sides with respect to $t$. State which differentiation rule the term $xy$ requires, and why.
   2. Find the value of $y$ when $x=4$.
   3. Find ${\displaystyle \frac{dy}{dt}}$ at the instant when $x=4$ and ${\displaystyle \frac{dx}{dt}}=-2$.
   4. A student differentiates $x^2$ with respect to $t$ and writes $2x$. State the missing factor and explain in one sentence why it is there.
5. Q5Solving Related Rates Problems

   A weather balloon is inflated so that it stays spherical. Helium is pumped in at a constant rate of $36\pi$ cubic feet per minute. For a sphere of radius $r$, $V=\frac{4}{3}\pi r^3$ and $S=4\pi r^2$.

   1. How fast is the radius increasing at the instant the radius is $3$ feet? Include units.
   2. How fast is the surface area increasing at that same instant? Include units.
6. Q6Approximating Values Using Local Linearity and Linearization

   Let $f(x)=\sqrt[3]{x}$.

   1. Write the equation of the line tangent to the graph of $f$ at $x=8$.
   2. Use that tangent line to approximate $\sqrt[3]{8.06}$.
   3. Is your approximation an overestimate or an underestimate of the true value? Justify your answer using the concavity of $f$.
7. Q7Using L'Hospital's Rule for Indeterminate Forms

   Evaluate each limit. In each case, first show that the limit has an indeterminate form of type $\frac{0}{0}$ or $\frac{\infty }{\infty }$, so that L'Hospital's Rule may be applied.

   1. ${\displaystyle {lim}_{x\to 3}\frac{x^2-9}{\ln (x-2)}}$
   2. ${\displaystyle {lim}_{x\to \infty }\frac{4x^2-x}{3x^2+5}}$
8. Q8Synthesis — drawing on several topics in this unit

   A test sled is fired along a straight horizontal track. Let $s(t)$ be its position in meters from the launch point, with $t$ in seconds. Instruments record $s(4)=52$, $s^{\prime }(4)=9$ and $s^{″}(4)=-2$.

   1. Interpret $s^{\prime }(4)$ in one sentence in the context of the sled, with units.
   2. Use the line tangent to the graph of $s$ at $t=4$ to approximate $s(4.2)$, with units.
   3. Is the sled speeding up or slowing down at $t=4$? Justify your answer.
   4. Assuming $s^{″}(t)<0$ for all $t$ near $4$, is your answer to (b) an overestimate or an underestimate of the true position? Justify.

The 7 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

The interpretation sentence — this is where the free marks are

Q1 and Q8(a) both ask you to say what a derivative means. Readers of AP responses are not looking for the number, which you were given; they are looking for a sentence that names the quantity, the direction, the rate, the moment and the units. Miss one of the five and the sentence stops earning.

Reading a model for when a quantity peaks

Q3 gives the volume in a tank as a function of time and asks when it is greatest. The justification has to be about V', not about V:

Finding the t that makes V' zero is half a mark; saying what V' does on each side of it is the other half. "Volume rising before, falling after, so the turning point is a maximum" is the whole argument in one line. Plugging a couple of t values into V and comparing them is not — a maximum is a claim about every nearby point, and only the sign of the derivative can support it.

Straight-line motion: v, a, and what "speeding up" actually means

Position, velocity and acceleration are one chain of derivatives:

"At rest" means v(t) = 0 — solve it, keep only the times inside the given interval, and factor rather than expand, since the factored form is what you will need for signs a moment later. Q2 asks whether the cart is speeding up at a particular instant, and that is a question about speed, which is |v|, not about velocity.

Displacement and total distance are two different questions

Displacement over [a, b] is the net change in position, s(b) − s(a): one subtraction, sign included, direction included. Total distance is how far the object actually travelled, and it ignores direction entirely.

They agree only when the object never turns around. So the work is to find where it turns: solve v(t) = 0, keep the times strictly inside the interval, and check that v really changes sign there. Then split the interval at those times, take the change in position across each piece separately, and add the sizes of those changes.

Related rates: differentiate first, substitute last

Q4 is the mechanism stripped of any story, and Q5 is the same mechanism wearing one. Both come down to the fact that every variable is secretly a function of t, so differentiating the relation with respect to t turns a statement about quantities into a statement about their rates. The order below is not a style preference — swapping the last two steps is the single most common way these problems go wrong.

The chain rule factor that gets dropped

Q4(d) is built around it directly. Differentiating x² with respect to x gives 2x; differentiating it with respect to t gives

because x is itself a function of t and the chain rule asks for the derivative of the inside. The same reasoning puts a dy/dt on any y term and, in a product like xy, gives the product rule with a rate factor on each piece. A quick check on any related-rates line you have written: every variable term should carry exactly one d(something)/dt. A term with none of them is a term you differentiated as if t were not there.

Linearization, and which side of the curve the tangent line is on

The tangent line at x = a is the best straight-line stand-in for f near a:

Q6 builds it for the cube-root function and uses it near x = 8; Q8(b) does the same for a sled's position. Choose a to be the nearby point where f and f' are easy — 8 for a cube root, the instant the instruments reported for the sled — and then the arithmetic is one multiplication.

The follow-up, "is this an over- or an underestimate?", is always settled by the concavity, and the argument is one sentence:

L'Hospital's rule: check the form before you use it

Q7 asks you to first show that each limit is indeterminate, and that instruction is the question. The rule licenses differentiating the numerator and the denominator separately — not as a quotient — and it applies to exactly two forms:

So the first line of your solution substitutes the target value into the top and the bottom and reports what each one tends to. Only then does the rule come out. If one application leaves you with 0/0 again, apply it again, saying so each time.

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Getting the most out of it

Write the units before you write the number

You can fill in "liters per hour" or "meters per second squared" the moment you have read the question, long before you have differentiated anything. Doing it in that order means the units are never the thing you forget under time pressure, and a rate whose units you cannot name is usually a rate you have set up wrong.

Make a sign line for v before answering anything about motion

Factor v(t), mark its zeros on a number line, and note the sign of v on each stretch. That one picture answers "when is it at rest", "which way is it moving", "when does it turn around" and — once you add the sign of a — "is it speeding up", without you having to re-derive anything.

Keep everything a letter until the last possible line

In related rates, do the differentiation entirely in symbols and substitute the instant's values only into the finished derivative equation. It is slower to type and far faster to fix: if a number appears in your working before the d/dt line, it is worth checking whether the quantity it stands for was actually constant.

Say the answer back as a sentence

"The radius is growing by one foot every minute at the moment it reaches three feet." If the sentence sounds like nonsense, the arithmetic probably is. This is also exactly the sentence the interpretation parts want, so you have written the answer twice for the price of once.

Common questions