Q: What is the U.S. copper consumption in 2005 according to $$p(t) = 277t^2$ + 6000$$?

$$p(15) = 277 \times 15^2$ + 6000$$

Q: If $$f(x)$$ has a vertical asymptote at $$x = 2$$, which of the following is true?

$$\lim_{x \to 2} f(x)$$ does not exist.

Q: Suppose $$P(x) = 10x - 0.03x^2$ - 500. $$What is the marginal profit when $$x = 20$$?

$$P'(20) = 10 - 0.06 \times 20$$

Q: If $$N(x) = \dfrac{10x + 100}{x^2$ + 2}$$, which of the following best describes the behavior of $$N(x) as$$ $$x$$ becomes very large?

$$N(x)$$ approaches $$0 as$$ $$x \to \infty.$$

Question 1

Which of the following statements best describes the behavior of a function $$f(x)$$ that is continuous everywhere except at $$x = 3$$, where it is not differentiable, and has a vertical asymptote at $$x = 2$$?

Accepted Answer

The function $$f(x) is $$continuous except at $$x = 2 ($$vertical asymptote), and not differentiable at $$x = 3.$$

Question 2

Given $$N(x) = \dfrac{10x + 100}{x^2$ + 2}$$, what is the average rate of change of $$N(x) on $$the interval $$[0, 1]$$?

Accepted Answer

$$\dfrac{N(1) - N(0)}{1 - 0} = \dfrac{\dfrac{110}{3} - 50}{1}$$

Question 3

The total profit from the sale of $$x$$ machines is given by $$P(x) = 10x - 0.03x^2$ - 500. $$What is the rate at which average profit per machine is changing when $$5$$ machines have been sold?

Accepted Answer

$$\dfrac{d}{dx}\left(\dfrac{P(x)}{x}\right) \Big|_{x=5}$$

Question 4

Given $$p(t) = 277t^2$ + 6000$$, where $$t is $$years since 1990, use the definition of the derivative to find $$p'(t).$$

Accepted Answer

$$p'(t) = 277 \cdot 2t$$

Question 5

What is the U.S. copper consumption in 2005 according to $$p(t) = 277t^2$ + 6000$$?

Accepted Answer

$$p(15) = 277 	imes 15^2$ + 6000$$

Question 6

Find the rate of copper consumption in 2005 using $$p(t) = 277t^2$ + 6000.$$

Accepted Answer

$$p'(15) = 554 	imes 15$$

Question 7

Jon throws a stone vertically from the edge of a cliff. The height of the stone above the water is $$h(t) = -4.9t^2$ + 34t + 118. $$What is the velocity of the stone after $$5$$ seconds?

Accepted Answer

$$h'(5) = -9.8 	imes 5 + 34$$

Question 8

For the stone thrown by Jon, what is the acceleration of the stone at any time $$t$$?

Accepted Answer

$$h''(t) = -9.8$$

Question 9

If $$f(x)$$ has a vertical asymptote at $$x = 2$$, which of the following is true?

Accepted Answer

$$\lim_{x 	o 2} f(x)$$ does not exist.

Question 10

Given $$f(x) is $$continuous but not differentiable at $$x = 3$$, which of the following could be true about $$f(x) at$$ $$x = 3$$?

Accepted Answer

$$f(x)$$ has a cusp or corner at $$x = 3.$$

Question 11

Suppose $$P(x) = 10x - 0.03x^2$ - 500. $$What is the marginal profit when $$x = 20$$?

Accepted Answer

$$P'(20) = 10 - 0.06 	imes 20$$

Question 12

If $$N(x) = \dfrac{10x + 100}{x^2$ + 2}$$, which of the following best describes the behavior of $$N(x) as$$ $$x$$ becomes very large?

Accepted Answer

$$N(x)$$ approaches $$0 as$$ $$x 	o \infty.$$

Question 13

Given $$h(t) = -4.9t^2$ + 34t + 118$$, how long does it take for the stone to hit the water (i.e., $$h(t) = 0$$)?

Accepted Answer

Solve $$-4.9t^2 + 34t + 118 = 0$ for t$.

Question 14

If $$p(t) = 277t^2$ + 6000$$, what is the instantaneous rate of change of copper consumption at $$t = 0$$?

Accepted Answer

$$p'(0) = 0$$

Question 15

Which of the following best describes the meaning of the average rate of change of $$N(x) on$$ $$[0, 1]$$?

Accepted Answer

It is the slope of the secant line connecting $$(0, N(0))$$ and $$(1, N(1)).$$

Question 16

If $$f(x) is $$continuous at $$x = a$$ but not differentiable at $$x = a$$, which of the following is most likely?

Accepted Answer

$$f(x)$$ has a sharp corner or cusp at $$x = a.$$

Question 17

Suppose $$P(x) = 10x - 0.03x^2$ - 500. $$What is the average profit per machine when $$x = 10$$?

Accepted Answer

$$\dfrac{P(10)}{10}$$

Question 18

If $$h(t) = -4.9t^2$ + 34t + 118$$, what is the velocity of the stone when it hits the water?

Accepted Answer

Find $$t$$ such that $$h(t) = 0$$, then compute $$h'(t).$$

Calculus Quiz: Functions, Rates of Change, and Applications

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