Q: For the function $$f(x)$$ graphed below, which of the following is true at the point where the function jumps?

$$\lim_{x \to a^-} f(x) \neq \lim_{x \to a^+} f(x)$$

Question 1

A ship travels 20 miles on a bearing of 20 degrees north of west. At the end of this time, how much further west is the ship than it was originally?

Accepted Answer

$$20 \cos(20^\circ)$$

Question 2

Which of the following expressions is indeterminate when involved in the calculation of limits?

Accepted Answer

$$\infty - \infty$$

Question 3

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 4

At what point is the function $$f(x) = \frac{4}{x-3}$$ discontinuous?

Accepted Answer

$$x = 3$$

Question 5

Which of the following is the correct simplification of $$\log_3$ 81$$?

Accepted Answer

$$4$$

Question 6

If $$\sin(	heta) = \frac{\sqrt{3}}{2}$$ and $$	heta is in $$the first quadrant, what is $$	heta$$?

Accepted Answer

$$60^\circ$$

Question 7

Suppose $$f(x) is $$discontinuous at $$x = a. $$Why would the derivative $$f'(a)$$ fail to exist?

Accepted Answer

Because the function is not defined at $$x = a$$, so the limit defining the derivative does not exist.

Question 8

Which of the following real-world functions is most likely to be continuous over its natural domain?

Accepted Answer

The length of a tree's shadow as a function of its height.

Question 9

Find the derivative of $$f(x) = x^2$ - 3x$$ using the definition of derivative.

Accepted Answer

$$2x - 3$$

Question 10

If $$	an(\frac{7\pi}{2}) is $$evaluated, what is the result?

Accepted Answer

Undefined

Question 11

In which quadrant is $$\sin(x)$$ negative and $$\cos(x)$$ positive?

Accepted Answer

Quadrant IV

Question 12

Which of the following is the correct simplification of $$\ln(e$$^{4x}$$)$$?

Accepted Answer

$$4x$$

Question 13

Evaluate $$\lim_{x 	o 3} \frac{x^2$ - 9}{x - 3}.$$

Accepted Answer

$$6$$

Question 14

Which of the following is a correct interpretation of the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L.$$

Question 15

Which of the following is the correct value for $$\cos(60^\circ)$$?

Accepted Answer

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

Question 16

If $$f(x) is $$defined as $$f(x) = \begin{cases} x^2$ & x \u003c 2 \ 4x - 6 & x \geq 2 \end{cases}$$, at which value of $$x is$$ $$f(x)$$ potentially discontinuous?

Accepted Answer

$$x = 2$$

Question 17

Which of the following is the correct simplification of $$\ln((x + 4)/(x + 1))$$?

Accepted Answer

$$\ln(x + 4) - \ln(x + 1)$$

Question 18

Which of the following best explains why the tax as a function of income is continuous, even with progressive income tax rates?

Accepted Answer

Because the tax function does not have jumps or breaks as income increases.

Question 19

For the function $$f(x)$$ graphed below, which of the following is true at the point where the function jumps?

Accepted Answer

$$\lim_{x 	o a^-} f(x) 
eq \lim_{x 	o a^+} f(x)$$

Question 20

If $$\cos(x) = 0$$ and $$x is $$between $$0$$ and $$2\pi$$, what is $$x$$?

Accepted Answer

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

Calculus Study Notes: Trigonometry, Limits, Continuity, and Derivatives

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