AP Calculus AB Practice Questions

What is $\lim_{x \to 3} \frac{x^2 - 9}{x - 3}$?

A function $f$ is defined as $f(x) = \begin{cases} x^2 + 1 & x < 2 \\ 3x - 1 & x \geq 2 \end{cases}$. Is $f$ continuous at $x = 2$?

What is $\lim_{x \to \infty} \frac{5x^3 + 2x}{3x^3 - 7}$?

What is $\lim_{x \to 0} \frac{\sin(5x)}{x}$?

If $f(x) = 4x^3 - 2x^2 + 7x - 5$, what is $f'(x)$?

If $h(x) = x^2 \sin(x)$, which expression gives $h'(x)$?

What is $\frac{d}{dx}[e^{3x}]$?

Using the limit definition, the derivative of $f(x)$ at $x=a$ is given by which expression?

If $g(x) = (3x^2 + 1)^4$, what is $g'(x)$?

For the curve $x^2 + y^2 = 25$, what is $\frac{dy}{dx}$?

If $f(x) = x^3 + x$ and $f(2) = 10$, what is $(f^{-1})'(10)$?

If $f(x) = \sin(2x)$, what is $f''(x)$?

A circular oil spill has radius increasing at $2$ m/s. How fast is the area increasing when $r = 5$ m?

The temperature of coffee $t$ minutes after being poured is $T(t) = 70 + 130e^{-0.04t}$ degrees Fahrenheit. What does $T'(10)$ represent?

Evaluate $\lim_{x \to 0} \frac{e^x - 1}{\sin x}$ using L'Hôpital's Rule.

The linearization of $f(x) = \sqrt{x}$ at $a = 25$ is used to approximate $\sqrt{26}$. What is the approximation?

If $f'(x) = x(x-2)(x+3)$, on which interval is $f$ increasing?

A farmer has $200$ meters of fencing to enclose a rectangular area against a barn wall (no fence needed on the barn side). What dimensions maximize the area?

If $f(x) = x^3 - 3x + 2$ on $[0, 2]$, the Mean Value Theorem guarantees a value $c$ in $(0,2)$ where $f'(c)$ equals:

If $f''(x) > 0$ for all $x$ in an interval, which statement is true about $f$ on that interval?