Q: Which of the following best describes the instantaneous velocity at time $$t = a$$?

$$\displaystyle \lim_{\Delta t \to 0} \frac{s(a + \Delta t) - s(a)}{\Delta t}$$

Q: Suppose the cost function for producing $$x$$ items is $$C(x) = 1000 + 50x - 0.04x^2$. What is the marginal cost when x = 100$?

$$C'(100) = 50 - 0.08 \times 100 = 42$$

Question 1

Given the position function $$s(t) of an $$object moving along a line, which of the following expressions correctly represents the average velocity of the object over the interval $$[a, a + \Delta t]$$?

Accepted Answer

$$\displaystyle \frac{s(a + \Delta t) - s(a)}{\Delta t}$$

Question 2

If the position of a rock thrown upward is given by $$s(t) = -16t^2$ + 96t$$, at what time does the rock reach its maximum height?

Accepted Answer

$$t = 3$$ seconds

Question 3

For the same rock in the previous question, what is the velocity of the rock when it hits the ground?

Accepted Answer

$$-96 ft/sec$$

Question 4

Which of the following best describes the instantaneous velocity at time $$t = a$$?

Accepted Answer

$$\displaystyle \lim_{\Delta t 	o 0} \frac{s(a + \Delta t) - s(a)}{\Delta t}$$

Question 5

A company’s cost function is $$C(x). $$What does the marginal cost $$C'(x)$$ represent?

Accepted Answer

The approximate cost to produce one additional item after producing $$x$$ items

Question 6

Suppose the number of worldwide Internet users is modeled by $$p(t) = 6t^2$ + 98t + 431.2$$, where $$t is $$years since 2000. What is the instantaneous growth rate of Internet users in 2011?

Accepted Answer

$$230$$ million users per year

Question 7

Given the cost function $$C(x)$$, which formula gives the average cost per item for $$x$$ items?

Accepted Answer

$$\displaystyle \frac{C(x)}{x}$$

Question 8

If the position function $$s(t) of an $$airliner is as shown in the graph, what does a negative slope of the tangent line at $$t = 6$$ hours indicate?

Accepted Answer

The airliner is returning to Seattle (moving in the negative direction)

Question 9

For a cost function $$C(x)$$, what does $$C(x+1) - C(x)$$ represent?

Accepted Answer

The cost of producing one additional item after $$x$$ items

Question 10

If the velocity of an object is given by $$v(t) = s'(t)$$, what is the acceleration at time $$t$$?

Accepted Answer

$$a(t) = v'(t) = s''(t)$$

Question 11

A plane travels from Seattle to Minneapolis and back, as shown in the position-time graph. At what time(s) is the velocity of the plane zero?

Accepted Answer

When the plane is at its maximum distance from Seattle (the top of the curve)

Question 12

Suppose the cost function for producing $$x$$ items is $$C(x) = 1000 + 50x - 0.04x^2$. What is the marginal cost when x = 100$?

Accepted Answer

$$C'(100) = 50 - 0.08 	imes 100 = 42$$

Question 13

If the speed of an object is $$|v(t)|$$, which of the following statements is true?

Accepted Answer

Speed is always non-negative, regardless of the direction of motion

Question 14

A company finds that the average cost of producing 900 items is $$32 per item, and the marginal cost at 900 items is 14. $$What does this imply?

Accepted Answer

Producing one more item (the 901st) will cost less than the average cost per item so far

Question 15

If $$p(t) = 6t^2$ + 98t + 431.2$$ models the number of Internet users (in millions) since 2000, what is the predicted number of users in 2020?

Accepted Answer

$$4791$$ million

Question 16

Which of the following best describes the relationship between the average cost and marginal cost as production increases?

Accepted Answer

The average cost typically decreases and approaches the marginal cost as production increases

Derivatives as Rates of Change: Velocity, Acceleration, and Marginal Cost

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