Q: Which of the following statements about the continuity of a function $f(x)$ is TRUE?

A function $f(x)$ is continuous at $x = a$ if $\lim_{x \to a} f(x)$ exists, $f(a)$ exists, and $\lim_{x \to a} f(x) = f(a)$.

Q: Which of the following is the correct application of L'Hôpital's Rule to $\lim_{x \to 0} \frac{e^x - 1}{x}$?

$\lim_{x \to 0} \frac{e^x}{1} = 1$

Question 1

A car's velocity at time $t$ is given by $v(t) = 3t^2 + 2t$. What is the average velocity of the car between $t = 1$ and $t = 4$ seconds?

Accepted Answer

$\frac{1}{4-1} \int_{1}^{4} (3t^2 + 2t) \, dt$

Question 2

Which of the following statements about the continuity of a function $f(x)$ is TRUE?

Accepted Answer

A function $f(x)$ is continuous at $x = a$ if $\lim_{x 	o a} f(x)$ exists, $f(a)$ exists, and $\lim_{x 	o a} f(x) = f(a)$.

Question 3

Evaluate the limit: $\lim_{x 	o 0} \frac{\sin(5x)}{x}$.

Accepted Answer

$5$

Question 4

A pharmaceutical company models the concentration of a drug in the bloodstream as $C(t) = e^{-0.5t}$. What is the instantaneous rate of change of concentration at $t = 2$ hours?

Accepted Answer

$-0.5 e^{-1}$

Question 5

Which theorem guarantees that a continuous function $f(x)$ on $[a, b]$ takes on every value between $f(a)$ and $f(b)$?

Accepted Answer

Intermediate Value Theorem

Question 6

Find the derivative of $f(x) = x^2 \sin(x)$ using the product rule.

Accepted Answer

$2x \sin(x) + x^2 \cos(x)$

Question 7

A population of bacteria grows according to $P(t) = 1000 e^{0.2t}$. What is the rate of change of the population at $t = 5$ hours?

Accepted Answer

$200 e^{0.2 	imes 5}$

Question 8

Which of the following is the correct application of the chain rule to $f(x) = \cos(3x^2)$?

Accepted Answer

$-\sin(3x^2) \cdot 6x$

Question 9

A tank is being filled with water at a rate given by $R(t) = 4t$. How much water is added between $t = 0$ and $t = 3$ minutes?

Accepted Answer

$\int_{0}^{3} 4t \, dt$

Question 10

Which of the following is the correct statement of the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

If $F(x) = \int_{a}^{x} f(t) \, dt$, then $F'(x) = f(x)$

Question 11

A function $f(x)$ is defined as $f(x) = \frac{x^2 - 4}{x - 2}$. What is $\lim_{x 	o 2} f(x)$?

Accepted Answer

$4$

Question 12

A hospital monitors the temperature of a patient using $T(t) = 98.6 + 0.5 \sin(\pi t)$, where $t$ is in hours. What is the rate of change of temperature at $t = 2$ hours?

Accepted Answer

$0.5 \pi \cos(2\pi)$

Question 13

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

Accepted Answer

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

Question 14

A company wants to maximize profit given by $P(x) = -2x^2 + 12x - 20$. What value of $x$ gives the maximum profit?

Accepted Answer

$x = 3$

Question 15

Evaluate the definite integral $\int_{0}^{\pi} \sin(x) \, dx$.

Accepted Answer

$0$

Question 16

Which of the following is the correct application of L'Hôpital's Rule to $\lim_{x 	o 0} \frac{e^x - 1}{x}$?

Accepted Answer

$\lim_{x 	o 0} \frac{e^x}{1} = 1$

Question 17

Find the area between the curves $y = x^2$ and $y = x$ from $x = 0$ to $x = 1$.

Accepted Answer

$\int_{0}^{1} (x - x^2) \, dx$

Question 18

A ball is thrown upward and its height at time $t$ is $h(t) = 10t - 4.9t^2$. What is the velocity of the ball at $t = 2$ seconds?

Accepted Answer

$10 - 9.8 	imes 2 = -9.6$ m/s

Question 19

Which of the following is the correct linear approximation of $f(x) = \sqrt{x}$ at $x = 4$?

Accepted Answer

$f(x) \approx 2 + \frac{1}{4}(x - 4)$

Calculus I (MATH-2413) Syllabus and Core Concepts Study Guide

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