Question 1

Given the function $$f(x) = \frac{1}{x^2$ - 4}$$, what is the domain of $$f(x)$$?

Accepted Answer

$$x \in \mathbb{R} \setminus \{2, -2\}$$

Question 2

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

Accepted Answer

$$1$$

Question 3

Which of the following is a correct Pythagorean identity?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 4

Simplify $$\frac{\sin(x)}{\cos(x)}$$ using a trigonometric identity.

Accepted Answer

$$	an(x)$$

Question 5

If $$	an(x) = \frac{\sin(x)}{\cos(x)}$$, what is $$\cot(x) in $$terms of $$\sin(x)$$ and $$\cos(x)$$?

Accepted Answer

$$\frac{\cos(x)}{\sin(x)}$$

Question 6

Which of the following is the double angle formula for $$\sin(2x)$$?

Accepted Answer

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

Question 7

What is the value of $$\lim_{x 	o 0} \frac{1 - \cos(x)}{x}$$?

Accepted Answer

$$0$$

Question 8

Given $$f(x) = \frac{1}{x}$$, which of the following statements about its domain is correct?

Accepted Answer

$$x \in \mathbb{R} \setminus \{0\}$$

Question 9

Which of the following is a reciprocal identity?

Accepted Answer

$$\csc(x) = \frac{1}{\sin(x)}$$

Question 10

If $$\lim_{x 	o a} f(x) = L$$ exists, which of the following must be true?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - a| \u003c \delta$$

Question 11

Using the unit circle, what is the value of $$\sin(\frac{\pi}{2})$$?

Accepted Answer

$$1$$

Question 12

Which of the following is the correct double angle formula for $$\cos(2x)$$?

Accepted Answer

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

Question 13

If $$	an(x) = 1$$, what is the value of $$x in $$the interval $$[0, 2\pi)$$?

Accepted Answer

$$\frac{\pi}{4}$$ and $$\frac{5\pi}{4}$$

Question 14

Which of the following is NOT a valid step in proving $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

Assume $$f(x) is $$differentiable at $$x = a$$

Question 15

A medical device uses the function $$g(x) = \frac{1}{x^2$} to $$model the intensity of a signal. For which values of $$x is $$the function defined?

Accepted Answer

$$x \in \mathbb{R} \setminus \{0\}$$

Question 16

Which trigonometric identity would you use to simplify $$\sin^2$(x) + \cos^2$(x) in a $$chemical reaction rate equation?

Accepted Answer

$$\sin^2$(x) + \cos^2$(x) = 1$$

Question 17

A sound wave is modeled by $$h(x) = \sin(2x). $$Which identity can be used to rewrite $$h(x) in $$terms of $$\sin(x)$$ and $$\cos(x)$$?

Accepted Answer

$$\sin(2x) = 2\sin(x)\cos(x)$$

Question 18

If $$\cos(x) = 0$$, what are the possible values of $$x in $$the interval $$[0, 2\pi)$$?

Accepted Answer

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

Question 19

Which of the following is the correct reciprocal identity for $$\sec(x)$$?

Accepted Answer

$$\sec(x) = \frac{1}{\cos(x)}$$

Calculus Study Notes: Limits, Trigonometric Identities, and the Unit Circle

Study Guide - Practice Questions

Study Guide - Flashcards