A B C A L C U L U S
Chapter 4
Contextual Applications of Differentiation
AP-Style Practice Questions
EASYMEDIUMHARD
Topics 4.1 – 4.7AB
Name:Period:
PART ITopics 4.1 – 4.7
Multiple Choice Questions
Show all supporting work on scratch paper. On the AP Exam, Section I is split into a no-calculator and a calculator-allowed part — each question below is labeled accordingly.
Q1EASY4.1 Interpreting DerivativeNo Calculator
$W(t)$ is the weight, in kg, of a calf at age $t$ weeks. The most appropriate units for $W'(t)$ are
- (A) kg
- (B) weeks
- (C) kg/week
- (D) weeks/kg
Q2EASY4.2 MotionNo Calculator
A particle's position is $s(t)=t^{2}-4t$. The particle is at rest when
- (A) $t=0$
- (B) $t=2$
- (C) $t=4$
- (D) $t=\pm 2$
Q3MEDIUM4.2 Speeding UpNo Calculator
A particle moves with velocity $v(t)=t^{2}-4t+3$. The particle is speeding up on
- (A) $(1,2)\cup(3,\infty)$
- (B) $(2,3)$ only
- (C) $(0,1)\cup(3,\infty)$
- (D) $(-\infty,2)$
Q4EASY4.3 Rates (Context)No Calculator
Oil is pumped into a tank so that $V(t)=10t-t^{2}/4$ gallons at time $t$ minutes. At $t=4$, oil enters the tank at
- (A) $6$ gal/min
- (B) $8$ gal/min
- (C) $10$ gal/min
- (D) $12$ gal/min
Q5MEDIUM4.4 Related RatesNo Calculator
A spherical balloon is inflated so that its volume increases at $36\pi$ cm³/s. At the instant $r=3$ cm, $\dfrac{dr}{dt}=$
- (A) $1$ cm/s
- (B) $2$ cm/s
- (C) $\dfrac{1}{3}$ cm/s
- (D) $3$ cm/s
Q6MEDIUM4.4 Related Rates (Shadow)Calculator
A $6$-ft-tall person walks away from a $15$-ft lamppost at $4$ ft/s. The tip of their shadow moves at
- (A) $\dfrac{20}{3}$ ft/s
- (B) $\dfrac{8}{3}$ ft/s
- (C) $6$ ft/s
- (D) $10$ ft/s
Q7MEDIUM4.5 Linear ApproximationNo Calculator
The linear approximation of $f(x)=\sqrt{x}$ at $x=9$ gives $\sqrt{9.4}\approx$
- (A) $3.033$
- (B) $3.067$
- (C) $3.100$
- (D) $3.200$
Q8HARD4.5 Error DirectionNo Calculator
Let $f$ be twice-differentiable with $f''<0$ on an open interval around $x=a$. The tangent-line approximation to $f$ near $x=a$ is
- (A) an overestimate
- (B) an underestimate
- (C) exact
- (D) not determinable from the given information
Q9EASY4.7 L'HôpitalNo Calculator
$\displaystyle\lim_{x\to 0}\dfrac{\sin x}{x}=$
- (A) $0$
- (B) $1$
- (C) $\infty$
- (D) does not exist
Q10MEDIUM4.7 L'HôpitalNo Calculator
$\displaystyle\lim_{x\to 0}\dfrac{e^{2x}-1-2x}{x^{2}}=$
- (A) $0$
- (B) $1$
- (C) $2$
- (D) $4$
Q11MEDIUM4.7 L'Hôpital (∞/∞)No Calculator
$\displaystyle\lim_{x\to\infty}\dfrac{\ln x}{\sqrt{x}}=$
- (A) $0$
- (B) $1$
- (C) $\infty$
- (D) does not exist
Q12HARD4.4 Related Rates (Angle)Calculator
A spotlight on the ground is $20$ m from a wall. A $2$-m-tall figure walks from the light toward the wall at $1$ m/s. When the figure is $4$ m from the wall, the length of the shadow on the wall changes at a rate closest to
- (A) $-0.31$ m/s
- (B) $-0.63$ m/s
- (C) $+0.63$ m/s
- (D) $+1.25$ m/s
Q13MEDIUM4.3 Rates (Economic)No Calculator
A company's cost in dollars for producing $x$ items is $C(x)=0.01x^{2}+20x+500$. The marginal cost at $x=100$ is
- (A) $20$
- (B) $22$
- (C) $40$
- (D) $120$
Q14EASY5.1 MVTPreview of Unit 5No Calculator
Note: The Mean Value Theorem is formally CED Topic 5.1 (Unit 5). Included here as a preview because tabular-rate FRQs in Unit 4 routinely cite it.
Which hypothesis is required for the Mean Value Theorem on $[a,b]$?
- (A) $f$ differentiable on $(a,b)$ and continuous on $[a,b]$.
- (B) $f$ continuous on $(a,b)$.
- (C) $f(a)=f(b)$.
- (D) $f$ is a polynomial.
Q15MEDIUM5.1 MVTPreview of Unit 5No Calculator
For $f(x)=x^{2}$ on $[1,4]$, the value $c$ guaranteed by the MVT is
- (A) $\dfrac{3}{2}$
- (B) $2$
- (C) $\dfrac{5}{2}$
- (D) $3$
Q16HARD4.2 Motion (Graph)No Calculator
A particle's velocity graph is shown on $[0,6]$. On what interval is the particle speeding up?
- (A) $(0,2)$
- (B) $(2,4)$
- (C) $(0,1)\cup(3,5)$
- (D) $(5,6)$
Q17MEDIUM4.5 Linearization ErrorNo Calculator
$L(x)$ is the tangent-line approximation to $f(x)=\sin x$ at $x=0$. Then $L(0.1)=$
- (A) $0$
- (B) $0.100$
- (C) $0.0998$
- (D) $0.050$
Q18HARD4.7 L'Hôpital (Twice)No Calculator
$\displaystyle\lim_{x\to 0}\dfrac{x-\sin x}{x^{3}}=$
- (A) $0$
- (B) $\dfrac{1}{6}$
- (C) $\dfrac{1}{2}$
- (D) $1$
PART IIShow All Work
Free-Response Questions
Contextual FRQs require labeled units, clear statement of what the derivative means in context, and justification using signs of $v$ and $a$ for motion problems. For related-rates, include a diagram, equation relating quantities, and differentiation with respect to time.
FRQ 1EASY4.2 MotionNo Calculator
A particle moves along the $x$-axis with velocity $v(t)=3t^{2}-12t+9$ for $t\ge 0$ (in m/s).
(a) Find all times $t\ge 0$ when the particle is at rest.
(b) On which interval(s) is the particle moving right?
(c) At $t=2$, is the particle speeding up or slowing down? Justify using the signs of $v$ and $a$.
FRQ 2MEDIUM4.4 Related RatesCalculator
A water trough is $10$ ft long with cross-section an isosceles triangle ($2$ ft wide at top, $2$ ft deep). Water fills the trough at $3$ ft³/min.
(a) Express $V$, the water volume, in terms of the depth $h$. Justify using similar triangles.
(b) At the instant $h=1$ ft, find $\dfrac{dh}{dt}$. Include units.
(c) Is the water surface width increasing faster or slower when $h=1$ ft vs $h=1.5$ ft? Justify.
FRQ 3MEDIUM4.5 Linearization & ErrorNo Calculator
Let $f(x)=\sqrt[3]{x}$.
(a) Find the linearization $L(x)$ of $f$ at $x=8$.
(b) Use $L$ to estimate $\sqrt[3]{8.6}$.
(c) Is the estimate an overestimate or underestimate? Justify using $f''$.
FRQ 4HARD4.3 / 4.4 Tabular RatesCalculator
A tank holds $G(t)$ gallons of water at time $t$ minutes. Selected values of $G$:
| $t$ (min) | $0$ | $2$ | $4$ | $6$ | $8$ |
|---|
| $G(t)$ (gal) | $120$ | $108$ | $88$ | $60$ | $24$ |
|---|
(a) Estimate $G'(4)$ using a symmetric difference quotient. Include units and interpret.
(b) Must there be a time $t\in(0,8)$ for which $G'(t)=-12$? Justify using the MVT. (MVT is formally Unit 5, Topic 5.1 — see Q14/Q15 preview above.)
(c) Use the linearization of $G$ at $t=4$ to estimate $G(4.5)$. Is the estimate likely an overestimate or underestimate, based on the concavity suggested by the table? Justify.
FRQ 5HARD4.7 L'Hôpital & ReasoningNo Calculator
Evaluate each limit. For each, state the indeterminate form before applying L'Hôpital's Rule, and justify each step.
(a) $\displaystyle\lim_{x\to 0}\dfrac{\tan x-x}{x^{3}}$
(b) $\displaystyle\lim_{x\to \infty}\dfrac{(\ln x)^{2}}{x}$
(c) $\displaystyle\lim_{x\to 0^{+}}x\ln x$
(d) Explain, with a short counterexample, why L'Hôpital's Rule does NOT apply to $\displaystyle\lim_{x\to 0}\dfrac{x+\sin x}{x}$ as an $\tfrac{0}{0}$ form — compute the limit directly.