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 $$\lim_{x \to 0} \sin\left(\dfrac{1}{x}\right)$$ does not exist.

Q: Let $$f(x) = \begin{cases} x^2$ \sin\left(\dfrac{1}{x}\right) & x \neq 0 \\ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ 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 solution to the equation $$2^{x+1}$$ - $$2^{x-1} = 0$?

Accepted Answer

$$x = -2$$

Question 2

Given $$\log_2$ x = 2.4$$ and $$\log_3$ x = 3.6$$, what is the value of $$x$$?

Accepted Answer

$$x = 5.278$$

Question 3

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

Accepted Answer

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

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 $$\lim_{x 	o 0} \dfrac{\sin(4x)}{x}.$$

Accepted Answer

$$4$$

Question 6

Suppose $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ + 4x + 7 & x \u003c 2 \ 3 & x \geq 2 \end{cases}. $$Compute $$\lim_{h 	o 0} \dfrac{f(2+h) - f(2)}{h}.$$

Accepted Answer

$$8$$

Question 7

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

Accepted Answer

$$y = 1$$

Question 8

Given $$f(x) = x^2$ - 4x + 7$$ and $$g(x) = x^2$ - 6$$, what is $$\lim_{x 	o \infty} \dfrac{f(x)}{g(x)}$$?

Accepted Answer

$$1$$

Question 9

If $$f(x) is $$continuous on $$(-\infty, \infty)$$ and $$f(1) = 2$$, $$f(3) = 5$$, what must be true according to the Intermediate Value Theorem?

Accepted Answer

There exists $$c in$$ $$(1,3)$$ such that $$f(c) = 4$$

Question 10

Which of the following is the correct domain for the inverse function of $$f(x) = \dfrac{2x+5}{x-2}$$?

Accepted Answer

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

Question 11

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 $$\lim_{x 	o 0} \sin\left(\dfrac{1}{x}ight)$$ does not exist.

Question 12

Suppose $$f(x)$$ and $$f^{-1}(x$$)$$ are inverse functions and f(a) = b$. What is $$\$$left(f^{-1}$$\right)'(b)$$ in terms of f$$'(a)$$?

Accepted Answer

$$\dfrac{1}{f'(a)}$$

Question 13

Find the derivative of $$F(x) = \dfrac{f(x)}{g(x)} at$$ $$x = 0$$ given $$f(0) = 2$$, $$f'(0) = 3$$, $$g(0) = 1$$, $$g'(0) = 2.$$

Accepted Answer

$$4$$

Question 14

Which of the following is the correct range for the function $$f(x) = 	an$$^{-1}$$ x$$?

Accepted Answer

$$\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$$

Question 15

Evaluate $$\lim_{x 	o 2} \dfrac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$2$$

Question 16

If $$f(x) = x^2$ + 4x + 7$$, what is the slope of the tangent line to $$f(x) at$$ $$x = 1$$?

Accepted Answer

$$6$$

Question 17

Which of the following statements is true about the function $$f(x) = \dfrac{x^2$ + 4x + 7}{x^2$ - 6}$$?

Accepted Answer

It has a vertical asymptote at $$x = \sqrt{6}$$ and $$x = -\sqrt{6}.$$

Question 18

Let $$f(x) = \begin{cases} x^2$ \sin\left(\dfrac{1}{x}ight) & x 
eq 0 \ 0 & x = 0 \end{cases}. Is$$ $$f(x)$$ 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 Study Guide: Functions, Limits, Derivatives, and Trigonometry

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