AP Calculus AB Unit 3 Differentiation Composite Implicit and Inverse Functions Questions

Explore AP Calculus Unit 3 questions on the chain rule, implicit differentiation, tangent lines, and related rates in context.

Covered syllabus points

3.1 The Chain Rule 3.2 Implicit Differentiation 3.3 Differentiating Inverse Functions 3.4 Differentiating Inverse Trigonometric Functions 3.5 Selecting Procedures for Calculating Derivatives 3.6 Calculating Higher-Order Derivatives

[Maximum number: 2]

Two particles, H and J, are moving along the x-axis. For $0 \leq t \leq 5$ , the position of particle H at time t is given by $x_{H}(t)=e^{t^{2}-4 t}$ and the velocity of particle J at time t is given by $v_{J}(t)=2 t\left(t^{2}-1\right)^{3}$ .

Find the velocity of particle H at time t=1. Show the work that leads to your answer.

A	Find the velocity of particle H at time t=1. Show the work that leads to your answer.
		$x_{H}{ }^{\prime}(t)=v_{H}(t)=(2 t-4) e^{t^{2}-4 t}$	Considers $x_{H}{ }^{\prime}$ Point 1 (P1)
		$x_{H}{ }^{\prime}(1)=v_{H}(1)=-2 e^{-3}$	Answer Point 2 (P2)
		Scoring Notes for Part A
		- P1 can be earned by presenting $x_{H}{ }^{\prime}(t), x_{H}{ }^{\prime}(1), x^{\prime}(t), x^{\prime}(1),(2 t-4) e^{t^{2}-4 t}$ , or $(2 \cdot 1-4) e^{1^{2}-4 \cdot 1}$ . - An unsupported answer of $-2 e^{-3}$ earns P2 but not P1.

[Maximum number: 2]

Consider the curve G defined by the equation $y^{3}-y^{2}-y+\frac{1}{4} x^{2}=0$ .

Show that $\frac{d y}{d x}=\frac{-x}{2\left(3 y^{2}-2 y-1\right)}$ .

Show that $\frac{d y}{d x}=\frac{-x}{2\left(3 y^{2}-2 y-1\right)}$ .
$\begin{aligned} & \frac{d}{d x}\left(y^{3}-y^{2}-y+\frac{1}{4} x^{2}\right)=0 \\ & \Rightarrow 3 y^{2} \frac{d y}{d x}-2 y \frac{d y}{d x}-\frac{d y}{d x}+\frac{x}{2}=0 \end{aligned}$	Implicit differentiation	Point 1 (P1)
$\begin{aligned} & \Rightarrow\left(3 y^{2}-2 y-1\right) \frac{d y}{d x}=-\frac{x}{2} \\ & \Rightarrow \frac{d y}{d x}=\frac{-x}{2\left(3 y^{2}-2 y-1\right)} \end{aligned}$	Verification	Point 2 (P2)
Scoring Notes for Part A
- P1 is earned only for the correct implicit differentiation of $y^{3}-y^{2}-y+\frac{1}{4}x^{2}=0$ . Alternative notation such as $y^{\prime}$ is acceptable. - It is sufficient to present $\left(3 y^{2}-2 y-1\right)\frac{d y}{d x}=-\frac{x}{2}$ to earn P2, provided there are no subsequent errors.

[Maximum number: 1]

Questions 9 and 10. Use the following table, which shows the values of differentiable functions $\boldsymbol{f}$ and $\boldsymbol{g}$ .

If $H(x)=f^{-1}(x)$ , then $H^{\prime}(3)$ equals

$-\frac{1}{16}$

$-\frac{1}{8}$

$\frac{1}{2}$

[Maximum number: 1]

$\frac{d}{d x}\left(\tan ^{-1} x+2 \sqrt{x}\right)=$

$-\frac{1}{\sin ^{2} x}+\frac{1}{\sqrt{x}}$

$\frac{1}{\sqrt{1-x^{2}}}-4 \sqrt[3]{x}$

$\frac{1}{\sqrt{1-x^{2}}}+\frac{1}{\sqrt{x}}$

$\frac{1}{1+x^{2}}-4 \sqrt[3]{x}$

$\frac{1}{1+x^{2}}+\frac{1}{\sqrt{x}}$

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