Question 1

Which of the following is the correct solution to the equation $$2^{x+1} = 8$?

Accepted Answer

$$x = 2$$

Question 2

Find the exact value of $$\sin$$^{-1}$$\left(\dfrac{\sqrt{3}}{2}\right).$$

Accepted Answer

$$\dfrac{\pi}{3}$$

Question 3

Which of the following intervals contains a solution to the equation $$2x^3$$ + $$3x^2 - 5x - 3 = 0$ according to the Intermediate Value Theorem?

Accepted Answer

$$(-3, -2)$$

Question 4

Evaluate the following limit: $$\displaystyle \lim_{x 	o 0} \dfrac{\sin(4x)}{x}.$$

Accepted Answer

$$4$$

Question 5

Suppose $$f(x) = x^2$ + 2x - 6. $$Which of the following is a vertical asymptote of $$g(x) = \dfrac{f(x)}{x-2}$$?

Accepted Answer

$$x = 2$$

Question 6

Let $$f(x) = \begin{cases} x^2$ + 2x - 3 & x \leq 1 \ ax + b & x \u003e 1 \end{cases}. $$For which values of $$a$$ and $$b is$$ $$f(x)$$ continuous on $$(-\infty, \infty)$$?

Accepted Answer

$$a = 4,\ b = -6$$

Question 7

Find the domain and range of the function $$f(x) = \ln(x-1).$$

Accepted Answer

Domain: $$(1, \infty)$$, Range: $$(-\infty, \infty)$$

Question 8

Suppose $$f(x) = x^2$ and g(x$$) = \sin(x)$$. What is the derivative of h(x) = f(x)g(x$$)$$ at x = 0$?

Accepted Answer

$$h'(0) = 0$$

Question 9

Which of the following statements about the function $$f(x) = \dfrac{x-2}{|x-2|} is $$true?

Accepted Answer

$$f(x) is $$not continuous at $$x = 2$$

Question 10

Evaluate $$	an$$^{-1}$$(	an(\dfrac{5\pi}{3})).$$

Accepted Answer

$$-\dfrac{\pi}{3}$$

Question 11

Let $$f(x) = \begin{cases} \sin\left(\dfrac{1}{x}ight) & x 
eq 0 \ 0 & x = 0 \end{cases}. $$Which of the following is true about $$f(x) at$$ $$x = 0$$?

Accepted Answer

$$f(x) is $$not continuous at $$x = 0$$

Question 12

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f(c) = 0$$

Question 13

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 14

Find the limit: $$\displaystyle \lim_{x 	o 2} \dfrac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 15

Which of the following is the correct domain for the function $$f(x) = \sqrt{x^2$ - 4}$$?

Accepted Answer

$$(-\infty, -2] \cup [2, \infty)$$

Question 16

Let $$f(x) = e$$^{2x}$$. What is the equation of the tangent line to f(x$$)$$ at x = 0$?

Accepted Answer

$$y = 2x + 1$$

Question 17

Suppose $$f(x) is $$differentiable and $$f'(2) = 5. $$What is $$\displaystyle \lim_{h 	o 0} \dfrac{f(2 + 4h) - f(2)}{h}$$?

Accepted Answer

$$20$$

Question 18

Which of the following is the correct expression for $$\sin(\arccos(x))$$?

Accepted Answer

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

Question 19

If $$f(x) = \dfrac{1}{x}$$ and $$f^{-1}(x$$)$$ is its inverse, what is f$$^{-1}$$(x)$$?

Accepted Answer

$$\dfrac{1}{x}$$

Calculus I Study Guide: Functions, Limits, Derivatives, and Applications

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