Question 1

Which of the following best describes the domain of the function $$f(x) = \sqrt{1-x^2$}$$?

Accepted Answer

$$[-1, 1]$$

Question 2

A function $$f is $$defined as $$f(x) = x^3$ - 2x. $$Which of the following statements is true?

Accepted Answer

$$f is an $$odd function

Question 3

Given $$f(x) = 2x + 1$$ and $$g(x) = \frac{1}{x-3}$$, what is $$(f \circ g)(x)$$?

Accepted Answer

$$2\left(\frac{1}{x-3}\right) + 1$$

Question 4

Convert $$120^\circ to $$radians.

Accepted Answer

$$\frac{2\pi}{3}$$

Question 5

The area of a sector of a circle with radius $$r$$ and central angle $$	heta (in $$radians) is:

Accepted Answer

$$\frac{1}{2} r^2$ 	heta$$

Question 6

If $$y = \sin(x)$$ and $$x is $$measured in radians, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\cos(x)$$

Question 7

Which of the following is the correct derivative of $$y = x^2$ e^x$$?

Accepted Answer

$$x^2 e^x + 2x e^x$

Question 8

Evaluate $$\lim_{x 	o 1} \frac{x^2$ - 1}{x - 1}.$$

Accepted Answer

$$2$$

Question 9

Which of the following is the correct formula for the volume of a solid of revolution generated by rotating $$y = f(x)$$ about the $$x-$$axis from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b \pi [f(x)]^2 dx$$

Question 10

Suppose $$y = \ln(x^2$ + 1). $$What is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 11

A manufacturer wants to minimize the cost of a cylindrical tank with volume $$V$$ and surface area cost $$C = 80(\pi r^2$ + \frac{3}{r}). $$What is the first step in finding the optimal radius $$r$$?

Accepted Answer

Set $$\frac{dC}{dr} = 0$$ and solve for $$r$$

Question 12

If $$f(x) = x^4$ - 4x^2$, at which x$-values does f(x$$)$$ have local minima?

Accepted Answer

$$x = \pm 2$$

Question 13

Which of the following is the correct chain rule for $$y = f(g(x))$$?

Accepted Answer

$$\frac{dy}{dx} = f'(g(x))g'(x)$$

Question 14

The definite integral $$\$$int_0$$^1 (1 + x + x^2$ + x^3$) dx is $$equal to:

Accepted Answer

$$\frac{25}{12}$$

Question 15

If $$y = \arcsin(2x)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$\frac{2}{\sqrt{1-4x^2$}}$$

Question 16

Which of the following is the correct statement of the Fundamental Theorem of Calculus (Part 2)?

Accepted Answer

$$\int_a^b f(x) dx = F(b) - F(a)$$, where $$F'(x) = f(x)$$

Comprehensive Study Notes: Functions, Limits, Differentiation, and Integration for College Algebra & Calculus

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