Q: Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} \sin(\frac{1}{x}) & x \neq 0 \\ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

No, because $$\lim_{x \to 0} \sin(\frac{1}{x})$$ does not exist

Question 1

Which of the following is the correct solution to the equation $$2x^2 + x - 21 = 0$?

Accepted Answer

$$x = 3 or$$ $$x = -3.5$$

Question 2

Given $$y = e$$^{2x}$$, what is x$ in terms of y$?

Accepted Answer

$$x = \frac{1}{2} \ln(y)$$

Question 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

$$(-2, -1)$$

Question 5

Evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 6

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 2x - 3 & x \u003e 1 \ 3x^2$ + 2x - 3 & x \leq 1 \end{cases}. $$For which value(s) of $$x is$$ $$f(x)$$ continuous on $$(-\infty, \infty)$$?

Accepted Answer

$$x = 1$$ and all other $$x$$

Question 7

Find the horizontal asymptote of $$f(x) = \frac{2x^2$ + 4x + 7}{x^2$ - 6}.$$

Accepted Answer

$$y = 2$$

Question 8

Given $$f(x) = x^2$ and g(x$$) = \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 is the domain of the inverse function of $$f(x) = \frac{2x + 5}{x - 2}$$?

Accepted Answer

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

Question 10

If $$f(x) is $$continuous everywhere except at $$x = 2$$, which of the following could be the definition of $$f(x)$$?

Accepted Answer

$$f(x) = \frac{1}{x - 2}$$

Question 11

Evaluate $$\lim_{h 	o 0} \frac{\sqrt{4 + h} - 2}{h}.$$

Accepted Answer

$$\frac{1}{2}$$

Question 12

Which of the following is the correct expression for $$	an$$^{-1}$$(	an(\frac{5\pi}{7}))$$?

Accepted Answer

$$\frac{5\pi}{7} - \pi$$

Question 13

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} \sin(\frac{1}{x}) & x 
eq 0 \ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

No, because $$\lim_{x 	o 0} \sin(\frac{1}{x})$$ does not exist

Question 14

Given $$f(x) = x^2$ and f$$^{-1}$$(x) = \sqrt{x}$$, what is $$\frac{d}{dx} f$$^{-1}$$(x) at$$ $$x = 4$$?

Accepted Answer

$$\frac{1}{2}$$

Question 15

Find the domain and range of the function $$f(x) = \ln(x - 3).$$

Accepted Answer

Domain: $$x \u003e 3$$, Range: $$(-\infty, \infty)$$

Question 16

If $$f(x) = \frac{1}{x}$$, what is the derivative $$f'(x)$$?

Accepted Answer

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

Question 17

Which of the following is the correct formula for the derivative of the inverse function $$f^{-1}(x$$)$$ at x = a$?

Accepted Answer

$$\frac{1}{f'(f$$^{-1}$$(a))}$$

Question 18

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

Accepted Answer

$$2x \ln(x) + x^2$ \frac{1}{x}$$

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

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