AP (Advanced Placement) · Calculus Ab

Basic results

25 practice questions, each with the answer and an explanation.

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Random questions from this topic, one at a time. You see the answer and the explanation straight after each one.

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What is the principal value of $cos^{- 1} (- \frac{1}{2})$ ?
What is $x \to 0 lim \frac{sin ( 3 x ) tan ( 2 x )}{1 - cos x}$ ?
What is $x \to 0 lim \frac{1 - cos ( 4 x )}{x ^{2}}$ ?
Let $f (x) = {a x^{2} + b, 4 x - 1, x \leq 1 x > 1$ . If $f$ is differentiable at $x = 1$ , what is the value of $a + 2 b$ ?
For $f (x) = x^{3} - 3 x$ on $[0, 2]$ , the Mean Value Theorem guarantees a value $c$ in $(0, 2)$ . What is $c$ ?
What is $x \to \infty lim (4 x^{2} + x - 2 x)$ ?
Let $f (x) = \frac{2 x ^{2} - x - 6}{x ^{2} - x - 2}$ . If the graph has vertical asymptotes at $x = p_{1}, \dots, p_{n}$ and a horizontal asymptote $y = H$ , what is $(p_{1} + \dots + p_{n}) + H$ ?
What is $h \to 0 lim \frac{( 2 + h ) ^{5} - 32}{h}$ ?
What is $x \to 0 lim \frac{e ^{3 x} - 1 - 3 x}{x ^{2}}$ ?
What is $x \to 2 lim \frac{x ^{3} - 8}{x + 7 - 3}$ ?
Let $f (x) = \frac{x ^{2} - k x + 6}{x - 2}$ for $x \neq = 2$ , and $f (2) = m$ . The constants $k$ and $m$ are chosen so that $f$ is continuous at $x = 2$ . What is $k + m$ ?
Evaluate the limit: $lim_{x \to 0} \frac{s i n ( 7 x )}{3 x}$ .
Evaluate the limit: $lim_{x \to 4} \frac{x ^{2} - 16}{x - 4}$ .
Evaluate the limit at infinity: $lim_{x \to \infty} \frac{5 x ^{3} - 2 x + 1}{2 x ^{3} + 7 x ^{2} - 4}$ .
Find the value of the constant $k$ that makes the function continuous at $x = 2$ :
$ $f (x) = {k x^{2} + 1, 3 x + 7, x \leq 2 x > 2$ $
Evaluate the limit: $lim_{x \to 0} \frac{x + 9 - 3}{x}$ .
Which of the following expressions represents the derivative of $f (x) = x^{3}$ at $x = 2$ using the limit definition of the derivative?
Evaluate the one-sided limit: $lim_{x \to 3^{-}} \frac{∣ x - 3∣}{x - 3}$ .
Evaluate the trigonometric limit: $lim_{x \to 0} \frac{1 - c o s ( x )}{x ^{2}}$ .
Let $f$ be a function such that $3 - x^{2} \leq f (x) \leq 3 + x^{2}$ for all $x$ . What is $lim_{x \to 0} f (x)$ ?
If $lim_{x \to c} f (x) = 5$ and $lim_{x \to c} g (x) = - 2$ , what is $lim_{x \to c} \frac{2 f ( x ) - [ g ( x ) ] ^{2}}{f ( x ) + g ( x )}$ ?
What is the principal value of $sin^{- 1} (\frac{1}{2})$ ?
The derivative of $sin x$ with respect to $x$ is
The derivative of $e^{x}$ with respect to $x$ is
The value of $\int \frac{1}{x} d x$ is

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