Unit 4 Overview (College Board AP® Calculus AB): Exam Questions

Functions $f$, $g$, and $h$ are twice-differentiable functions with $g\left(2\right)=h\left(2\right)=4$. The line $y=4+\frac{2}{3}\left(x-2\right)$ is tangent to both the graph of $g$ at $x=2$ and the graph of $h$ at $x=2$.

Find $h\text{'}\left(2\right)$.

Let $a$ be the function given by $a\left(x\right)=3x^{3}h\left(x\right)$. Write an expression for $a\text{'}\left(x\right)$. Find $a\text{'}\left(2\right)$.

The function $h$ satisfies $h\left(x\right)=\frac{x^2-4}{1-{\left(f\left(x\right)\right)}^3}$ for $x\ne 2$. It is known that $\underset{x\to 2}{\lim }h\left(x\right)$ can be evaluated using L'Hospital's Rule. Use $\underset{x\to 2}{\lim }h\left(x\right)$ to find $f\left(2\right)$ and $f\text{'}\left(2\right)$. Show the work that leads to your answers.

It is known that $g\left(x\right)\le h\left(x\right)$ for $1<x<3$. Let $k$ be a function satisfying $g\left(x\right)\le k\left(x\right)\le h\left(x\right)$ for $1<x<3$. Is $k$ continuous at $x=2$? Justify your answer.

Consider the curve given by the equation $6xy=2+y^3$.

Show that $\frac{\text{d}y}{\text{d}x}=\frac{2y}{y^2-2x}$.

Find the coordinates of a point on the curve at which the line tangent to the curve is horizontal, or explain why no such point exists.

Find the coordinates of a point on the curve at which the line tangent to the curve is vertical, or explain why no such point exists.

A particle is moving along the curve. At the instant when the particle is at the point $\left(\frac{1}{2},-2\right)$, its horizontal position is increasing at a rate of $\frac{\text{d}x}{\text{d}t}=\frac{2}{3}$ unit per second. What is the value of $\frac{\text{d}y}{\text{d}t}$, the rate of change of the particle's vertical position, at that instant?

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

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

There is a point $P$ on the curve $G$ near $\left(2,-1\right)$ with $x$-coordinate $1.6$. Use the line tangent to the curve at $\left(2,-1\right)$ to approximate the $y$-coordinate of point $P$.

For $x>0$ and $y>0$, there is a point $S$ on the curve $G$ at which the line tangent to the curve at that point is vertical. Find the $y$-coordinate of point $S$. Show the work that leads to your answer.

A particle moves along the curve $H$ defined by the equation $2xy+\text{ln}y=8$. At the instant when the particle is at the point $\left(4,1\right)$, $\frac{\text{d}x}{\text{d}t}=3$. Find $\frac{\text{d}y}{\text{d}t}$ at that instant. Show the work that leads to your answer.