Q: Which of the following is the correct domain of the function $$f(x) = \ln\left(\dfrac{x^2$ + 4x + 7}{x - 6}\right)$$?

$$x \in \mathbb{R},\ x \neq 6$$

Q: Let $$f(x) = \begin{cases} \sin\left(\dfrac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

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

Q: Suppose $$f(x) = \begin{cases} x \sin\left(\dfrac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ differentiable at $$x = 0$$?

Yes, because $$\lim_{x \to 0} \dfrac{f(x) - f(0)}{x - 0}$$ exists and equals $$0$$

Question 1

Which of the following is the correct domain of the function $$f(x) = \ln\left(\dfrac{x^2$ + 4x + 7}{x - 6}\right)$$?

Accepted Answer

$$x \in \mathbb{R},\ x 
eq 6$$

Question 2

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

Accepted Answer

$$\dfrac{\pi}{6}$$

Question 3

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

Accepted Answer

$$a = 4$$

Question 4

Evaluate $$\lim_{x 	o 0} \dfrac{\sin(4x) + \sqrt{4 + x} - 6}{x}.$$

Accepted Answer

$$\dfrac{1}{2}$$

Question 5

Given $$f(x) = e$$^{2x}$$, what is the value of f$$^{-1}$$(x)$$?

Accepted Answer

$$\dfrac{1}{2} \ln(x)$$

Question 6

Find the horizontal asymptote of $$y = \dfrac{2x^2$ + 4x + 7}{x^2$ - 6}.$$

Accepted Answer

$$y = 2$$

Question 7

Suppose $$f(x) = \dfrac{x^2$ + 4x + 7}{x - 6}. $$What is the vertical asymptote?

Accepted Answer

$$x = 6$$

Question 8

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

Accepted Answer

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

Question 9

Let $$f(x) = \begin{cases} \sin\left(\dfrac{1}{x}ight) & x 
eq 0 \ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

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

Question 10

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

Accepted Answer

$$(0, 1)$$

Question 11

Suppose $$f(x) = \dfrac{x^2$ + 4x + 7}{x - 6}$$ and $$f'(x) is $$its derivative. What is $$f'(x)$$?

Accepted Answer

$$\dfrac{2x + 4}{x - 6} - \dfrac{x^2$ + 4x + 7}{(x - 6)^2$}$$

Question 12

If $$f(x) = x^2$ + 1$$ for $$x \leq 1$$ and $$f(x) = \dfrac{3x^2$ + 2x - 3}{x^2$ + 2x - 3}$$ for $$x \u003e 1$$, what value of $$a$$ makes $$f(x)$$ continuous at $$x = 1$$?

Accepted Answer

$$a = 4$$

Question 13

Evaluate $$\lim_{x 	o \infty} \dfrac{2x^2$ + 4x + 7}{x^2$ - 6}.$$

Accepted Answer

$$2$$

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

Find the exact value of $$	an$$^{-1}$$\left(\dfrac{\sqrt{3}}{3}ight).$$

Accepted Answer

$$\dfrac{\pi}{6}$$

Question 17

If $$f(x) = \dfrac{x^2$ + 4x + 7}{x - 6}$$, what is the value of $$f(7)$$?

Accepted Answer

$$46$$

Question 18

Suppose $$f(x) = \begin{cases} x \sin\left(\dfrac{1}{x}ight) & x 
eq 0 \ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ differentiable at $$x = 0$$?

Accepted Answer

Yes, because $$\lim_{x 	o 0} \dfrac{f(x) - f(0)}{x - 0}$$ exists and equals $$0$$

Calculus I Exam 1 Study Guide: Functions, Limits, Derivatives, and Applications

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