If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$, then $\lim_{x \to a} f(x) = L$.

Question 1

Which of the following best describes the definition of a limit in calculus?

Accepted Answer

The value that a function approaches as the input approaches some value.

Question 2

Suppose $f(x) = x^2$. What is the derivative of $f(x)$ with respect to $x$?

Accepted Answer

$2x$

Question 3

A car's position at time $t$ is given by $s(t) = 4t^3 - 2t + 5$. What is the car's velocity at $t = 2$ seconds?

Accepted Answer

$92$

Question 4

Which rule is used to differentiate the product of two functions?

Accepted Answer

Product Rule

Question 5

Given $f(x) = e^{2x}$, what is $f'(x)$?

Accepted Answer

$2e^{2x}$

Question 6

If $y = 	an(x)$, what is $\frac{dy}{dx}$?

Accepted Answer

$\sec^2(x)$

Question 7

Which of the following is the correct application of the Chain Rule for $f(x) = \sin(3x)$?

Accepted Answer

$3\cos(3x)$

Question 8

A population of bacteria grows according to $P(t) = P_0 e^{kt}$. What does the constant $k$ represent?

Accepted Answer

The growth rate of the population.

Question 9

Which of the following statements about the Fundamental Theorem of Calculus is correct?

Accepted Answer

It connects differentiation and integration.

Question 10

Evaluate the definite integral $\int_0^2 x dx$.

Accepted Answer

$2$

Question 11

Which of the following best describes a critical point of a function?

Accepted Answer

A point where the derivative is zero or undefined.

Question 12

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

Accepted Answer

$\frac{1}{x}$

Question 13

Which of the following is the correct derivative of $f(x) = \arcsin(x)$?

Accepted Answer

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

Question 14

A ball is thrown upward with velocity $v(t) = 20 - 9.8t$. What is the maximum height reached by the ball?

Accepted Answer

$20.41$ meters

Question 15

Which of the following is the correct formula for the area under the curve $y = f(x)$ from $x = a$ to $x = b$?

Accepted Answer

$\int_a^b f(x) dx$

Question 16

Which of the following is an example of implicit differentiation?

Accepted Answer

Differentiating $x^2 + y^2 = 1$ with respect to $x$

Question 17

Which rule is used to differentiate $f(x) = \ln(2x)$?

Accepted Answer

$\frac{1}{x}$

Question 18

If $f(x) = x^3 - 3x^2 + 2x$, at which $x$ does $f(x)$ have a local minimum?

Accepted Answer

$x = 2$

Question 19

Which of the following is the correct expression for the average value of a function $f(x)$ on $[a, b]$?

Accepted Answer

$\frac{1}{b-a} \int_a^b f(x) dx$

Question 20

Which of the following is a correct application of the squeeze theorem?

Accepted Answer

If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x 	o a} g(x) = \lim_{x 	o a} h(x) = L$, then $\lim_{x 	o a} f(x) = L$.

Introduction to Calculus: Course Outline and Key Topics

Study Guide - Practice Questions