A B C A L C U L U S
Chapter 3
Composite, Implicit & Inverse Functions
AP-Style Practice Questions
EASYMEDIUMHARD
Topics 3.1 – 3.6AB
Name:Period:
PART ITopics 3.1 – 3.6
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.
Q1EASY3.1 Chain RuleNo Calculator
If $f(x)=(3x^{2}+1)^{4}$, then $f'(x)=$
- (A) $4(3x^{2}+1)^{3}$
- (B) $24x(3x^{2}+1)^{3}$
- (C) $6x(3x^{2}+1)^{3}$
- (D) $4(6x)^{3}$
Q2EASY3.1 Chain (Trig)No Calculator
$\dfrac{d}{dx}\bigl[\,\sin(4x)\,\bigr]=$
- (A) $\cos(4x)$
- (B) $4\cos(4x)$
- (C) $-4\cos(4x)$
- (D) $4x\cos(4x)$
Q3EASY3.1 Chain (Exp)No Calculator
$\dfrac{d}{dx}\bigl[\,e^{\,x^{2}}\,\bigr]=$
- (A) $e^{\,x^{2}}$
- (B) $2x\,e^{\,x^{2}}$
- (C) $x^{2}e^{\,x^{2}-1}$
- (D) $2xe^{2x}$
Q4MEDIUM3.1 Chain (Log)No Calculator
$\dfrac{d}{dx}\bigl[\ln(x^{2}+3x)\bigr]=$
- (A) $\dfrac{1}{x^{2}+3x}$
- (B) $\dfrac{2x+3}{x^{2}+3x}$
- (C) $\dfrac{2x+3}{\ln(x^{2}+3x)}$
- (D) $2x+3$
Q5MEDIUM3.1 Nested ChainNo Calculator
If $y=\sin^{2}(3x)$, then $\dfrac{dy}{dx}=$
- (A) $2\sin(3x)$
- (B) $6\sin(3x)\cos(3x)$
- (C) $3\sin(3x)\cos(3x)$
- (D) $2\sin(3x)\cos(3x)$
Q6MEDIUM3.1 Chain (Table)No Calculator
Selected values of $f$ and $g$:
| $x$ | $f(x)$ | $f'(x)$ | $g(x)$ | $g'(x)$ |
|---|
| $1$ | $2$ | $5$ | $3$ | $-2$ |
| $3$ | $4$ | $-1$ | $6$ | $2$ |
If $h(x)=f(g(x))$, then $h'(1)=$
- (A) $-10$
- (B) $-2$
- (C) $2$
- (D) $5$
Q7MEDIUM3.2 ImplicitNo Calculator
If $x^{2}+y^{2}=25$, then $\dfrac{dy}{dx}=$
- (A) $-\dfrac{x}{y}$
- (B) $\dfrac{x}{y}$
- (C) $-\dfrac{y}{x}$
- (D) $\dfrac{y-x}{y}$
Q8MEDIUM3.2 Implicit (Mixed)No Calculator
If $xy+y^{3}=4$, then $\dfrac{dy}{dx}=$
- (A) $-\dfrac{y}{x+3y^{2}}$
- (B) $\dfrac{y}{x-3y^{2}}$
- (C) $-\dfrac{x}{y+3y^{2}}$
- (D) $\dfrac{1-y}{x+3y^{2}}$
Q9HARD3.2 Implicit TangentNo Calculator
For the curve $x^{2}+xy+y^{2}=7$, the slope of the tangent line at $(1,2)$ is
- (A) $-\dfrac{4}{5}$
- (B) $-\dfrac{1}{5}$
- (C) $\dfrac{4}{5}$
- (D) $-\dfrac{5}{4}$
Q10EASY3.3 Inverse DerivativeNo Calculator
If $f(x)=x^{3}+x+1$, then $(f^{-1})'(1)=$
- (A) $\dfrac{1}{4}$
- (B) $\dfrac{1}{2}$
- (C) $1$
- (D) $4$
Q11MEDIUM3.3 Inverse (Table)No Calculator
Let $f$ be differentiable and one-to-one. Selected values:
| $x$ | $1$ | $2$ | $3$ |
|---|
| $f(x)$ | $4$ | $7$ | $10$ |
|---|
| $f'(x)$ | $2$ | $5$ | $6$ |
|---|
$(f^{-1})'(7)=$
- (A) $\dfrac{1}{2}$
- (B) $\dfrac{1}{5}$
- (C) $\dfrac{1}{6}$
- (D) $5$
Q12EASY3.4 Inverse TrigNo Calculator
$\dfrac{d}{dx}\bigl[\arctan x\bigr]=$
- (A) $\dfrac{1}{1+x^{2}}$
- (B) $\dfrac{1}{\sqrt{1-x^{2}}}$
- (C) $\dfrac{-1}{1+x^{2}}$
- (D) $\sec^{2} x$
Q13MEDIUM3.4 Inverse Trig ChainNo Calculator
$\dfrac{d}{dx}\bigl[\arcsin(2x)\bigr]=$
- (A) $\dfrac{1}{\sqrt{1-4x^{2}}}$
- (B) $\dfrac{2}{\sqrt{1-4x^{2}}}$
- (C) $\dfrac{2}{\sqrt{1-x^{2}}}$
- (D) $\dfrac{1}{1-4x^{2}}$
Q14MEDIUM3.5 Higher-Order ImplicitNo Calculator
If $x^{2}+y^{2}=4$, then $\dfrac{d^{2}y}{dx^{2}}=$
- (A) $-\dfrac{1}{y}$
- (B) $-\dfrac{4}{y^{3}}$
- (C) $\dfrac{x^{2}}{y^{3}}$
- (D) $-\dfrac{x^{2}+y^{2}}{y^{3}}$
Q15MEDIUM4.4 Related RatesPreview of Unit 4No Calculator
Note: Related Rates is formally CED Topic 4.4 (Unit 4). Included here as a preview because related-rates problems synthesize the chain rule and implicit differentiation from this unit.
A circle's radius grows at $3$ cm/s. At $r=5$, the rate of change of its area is
- (A) $10\pi$ cm²/s
- (B) $15\pi$ cm²/s
- (C) $25\pi$ cm²/s
- (D) $30\pi$ cm²/s
Q16HARD4.4 Related Rates (Ladder)Preview of Unit 4Calculator
A $13$-ft ladder slides down a wall. When the base is $5$ ft from the wall and moving at $2$ ft/s away, the top is moving at a rate closest to
- (A) $-0.42$ ft/s
- (B) $-0.83$ ft/s
- (C) $-1.67$ ft/s
- (D) $-2.40$ ft/s
Q17HARD3.2 Horizontal TangentNo Calculator
The curve $x^{2}-xy+y^{2}=3$ has a horizontal tangent at points where
- (A) $x=0$
- (B) $y=2x$
- (C) $x=2y$
- (D) $x=y$
Q18HARD3.1 Chain + ProductNo Calculator
If $y=x^{2}\sin(\ln x)$, then $\dfrac{dy}{dx}=$
- (A) $2x\sin(\ln x)+x\cos(\ln x)$
- (B) $2x\sin(\ln x)+x^{2}\cos(\ln x)\cdot\dfrac{1}{x}$
- (C) $\sin(\ln x)+\cos(\ln x)$
- (D) Both (A) and (B) are correct.
PART IIShow All Work
Free-Response Questions
Free-response answers require explicit rule statements (chain, implicit, inverse), clear labeling of inner/outer functions, and unit-bearing answers for related-rates problems.
FRQ 1EASY3.1 Chain RuleNo Calculator
Differentiate each function. Label inner and outer functions.
(a) $y=(2x^{3}-5x+1)^{6}$
(b) $y=\sqrt{9-x^{2}}$
(c) $y=\cos\!\bigl(e^{2x}\bigr)$
FRQ 2MEDIUM3.2 Implicit DifferentiationNo Calculator
Consider the curve defined by $x^{2}+2xy+y^{3}=8$.
(a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.
(b) Verify that the point $(2,0)$ lies on the curve, and find the slope of the tangent line there.
(c) Write an equation of the tangent line to the curve at $(2,0)$.
FRQ 3MEDIUM3.3 / 3.4 Inverse FunctionsNo Calculator
Let $f(x)=x^{3}+2x-1$.
(a) Explain why $f$ is one-to-one on $\mathbb{R}$, using $f'$.
(b) Compute $f(1)$ and use the inverse-function formula to find $(f^{-1})'(2)$.
(c) Let $g(x)=\arctan(f(x))$. Find $g'(1)$.
FRQ 4HARD4.4 Related Rates (Cone)Preview of Unit 4Calculator
Water is being poured into a right-circular-cone tank (vertex down) of radius $4$ ft at the top and height $6$ ft, at a rate of $2$ ft³/min.
(a) Express the water volume $V$ in terms of the water's height $h$ alone. Justify using similar triangles.
(b) At the instant when $h=3$ ft, find $\dfrac{dh}{dt}$. Include units.
(c) Is the water height rising faster when $h=2$ ft or when $h=4$ ft? Justify with the relationship from part (b).
FRQ 5HARD3.1 / 3.2 / 3.5 Second DerivativeNo Calculator
Consider the curve $y^{2}=x^{3}+2x$.
(a) Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$.
(b) At the point $(1,\sqrt{3})$ on the curve, find the slope of the tangent line.
(c) Find $\dfrac{d^{2}y}{dx^{2}}$ at the point $(1,\sqrt{3})$.
(d) Is the curve concave up or concave down at $(1,\sqrt{3})$? Justify.