Unit 1 test: AB Review: Limits and Derivatives

Evaluate $\displaystyle\lim_{x \to -\infty}\left(2x + \sqrt{4x^2 + 3x}\right)$.

Let $f(x) = \dfrac{2x^2 - 8}{x^2 - x - 2}$. Which describes the graph of $f$?

For a positive constant $k$, let $f(x) = \dfrac{1 - \cos(kx)}{x^2}$ for $x \ne 0$ and $f(0) = 8$. Find the value of $k$ that makes $f$ continuous at $x = 0$.

A function $f$ is continuous on $[1, 4]$ with $f(1) = 6$ and $f(4) = -2$. Which statement must be true?

Evaluate $\displaystyle\lim_{h \to 0}\frac{\ln(e + h) - 1}{h}$.

Suppose $f(3) = 2$, $f'(3) = -1$, $g(3) = 4$ and $g'(3) = 5$. Let $k(x) = \dfrac{f(x)g(x)}{x}$. Find $k'(3)$.

Which function is continuous at $x = 0$ but not differentiable there?

Find $\dfrac{d}{dx}\,e^{\sin(3x)}$.

The point $(1, 2)$ lies on the curve $x^2 + 3xy + y^2 = 11$. Find an equation of the tangent line to the curve at that point.

(AP-style) Let $f(x) = 2x + \cos x$, and let $g$ be the inverse function of $f$. What is $g'(1)$?

Let $h(x) = \arctan\left(e^x\right)$. Find $h'(0)$.

The volume of a spherical balloon increases at $100\pi$ cubic centimeters per second. How fast is its surface area increasing, in square centimeters per second, when the radius is 5 cm?

Evaluate $\displaystyle\lim_{x \to 0}\frac{x - \sin x}{x^3}$.

(AP-style) Let $f(x) = x^4 - 4x^3$. Which statement is true?

An open-top box with a square base is made from 108 square inches of material. What is the largest possible volume of the box, in cubic inches?