Q: A function $f(x)$ is continuous at $x = a$ if:

$\displaystyle \lim_{x \to a} f(x)$ exists, $f(a)$ exists, \text{ and } \displaystyle \lim_{x \to a} f(x) = f(a)$

Q: Evaluate $\displaystyle \int x^2 e^{x^3} dx$.

$\displaystyle \frac{1}{3} e^{x^3} + C$

Q: Which of the following is the correct application of the product rule for differentiation?

$\displaystyle (fg)' = f'g + fg'$

Q: Find the derivative of $f(x) = \sqrt{x^2 + 1}$.

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

Q: Which of the following limits does NOT exist?

$\displaystyle \lim_{x \to 0} \frac{1}{x}$

Q: Suppose $f(x)$ is not continuous at $x = a$ because $\displaystyle \lim_{x \to a} f(x)$ does not exist. Which type of discontinuity is this?

Essential (infinite or jump) discontinuity

Question 1

Which of the following statements best describes the meaning of $\displaystyle \lim_{x 	o a} f(x) = L$?

Accepted Answer

As $x$ approaches $a$, $f(x)$ becomes arbitrarily close to $L$.

Question 2

Evaluate $\displaystyle \lim_{x 	o 2} \frac{x^2 - 4}{x - 2}$.

Accepted Answer

$4$

Question 3

A function $f(x)$ is continuous at $x = a$ if:

Accepted Answer

$\displaystyle \lim_{x 	o a} f(x)$ exists, $f(a)$ exists, 	ext{ and } \displaystyle \lim_{x 	o a} f(x) = f(a)$

Question 4

Find the derivative of $f(x) = x^3 - 5x^2$.

Accepted Answer

$3x^2 - 10x$

Question 5

Evaluate $\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}$.

Accepted Answer

$1$

Question 6

Which of the following is the correct derivative of $f(x) = \ln(x^2 + 1)$?

Accepted Answer

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

Question 7

Suppose $f(x)$ is defined as $f(x) = \begin{cases} x^2 & x < 1 \ 2x + 1 & x \geq 1 \end{cases}$. For which value of $a$ is $f(x)$ continuous at $x = 1$?

Accepted Answer

$a = 3$

Question 8

Evaluate $\displaystyle \int x^2 e^{x^3} dx$.

Accepted Answer

$\displaystyle \frac{1}{3} e^{x^3} + C$

Question 9

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$\displaystyle (fg)' = f'g + fg'$

Question 10

A hospital monitors the rate at which a patient's medication is metabolized, modeled by $M(t) = 5e^{-0.2t}$. What is the rate of change of medication in the body at $t = 3$ hours?

Accepted Answer

$-1e^{-0.6}$

Question 11

Evaluate $\displaystyle \int_0^1 3x^2 dx$.

Accepted Answer

$1$

Question 12

Find the derivative of $f(x) = \sqrt{x^2 + 1}$.

Accepted Answer

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

Question 13

Evaluate $\displaystyle \lim_{x 	o 0} \frac{e^x - 1}{x}$.

Accepted Answer

$1$

Question 14

Which of the following is the correct antiderivative of $\displaystyle \frac{1}{1 + x^2}$?

Accepted Answer

$\arctan x + C$

Question 15

A chemical reaction's rate is given by $R(t) = 4t^3 - 2t$. What is the total change in concentration from $t = 0$ to $t = 2$?

Accepted Answer

$12$

Question 16

Which of the following limits does NOT exist?

Accepted Answer

$\displaystyle \lim_{x 	o 0} \frac{1}{x}$

Question 17

Find $\displaystyle \frac{d}{dx} \left( \frac{1}{x} \right)$.

Accepted Answer

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

Question 18

Evaluate $\displaystyle \int \cos(2x) dx$.

Accepted Answer

$\frac{1}{2} \sin(2x) + C$

Question 19

Suppose $f(x)$ is not continuous at $x = a$ because $\displaystyle \lim_{x 	o a} f(x)$ does not exist. Which type of discontinuity is this?

Accepted Answer

Essential (infinite or jump) discontinuity

Calculus I Final Exam Review: Limits, Derivatives, and Integrals

Study Guide - Practice Questions

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