Question 1

Which of the following best describes the meaning of the derivative $$f'(x) at a $$point $$x$$?

Accepted Answer

It gives the slope of the graph of $$f(x) at $$the point $$(x, f(x)).$$

Question 2

Given the function $$f(x) = 3x^2$ - 2x + 5$$, what is the instantaneous rate of change of $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$f'(2) = 7$$

Question 3

The average rate of change of $$f(x) on $$the interval $$[a, b] is $$given by which formula?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 4

If $$s(t) = -16t^2$ + 32t + 128$$ gives the height (in feet) of an object thrown from a window after $$t$$ seconds, what is the velocity of the object when it hits the ground?

Accepted Answer

$$-64 ft/s$$

Question 5

For the same function $$s(t) = -16t^2$ + 32t + 128$$, what is the acceleration of the object at any time $$t$$?

Accepted Answer

$$-16 ft/s$^2$$

Question 6

Suppose a business’s profit $$P(x) is $$modeled by $$P(x) = 100x - 2x^2$, where x$ is the number of units sold. What is the marginal profit when x = 10$?

Accepted Answer

$$P'(10) = 80$$

Question 7

Which of the following statements is true about the relationship between velocity and acceleration?

Accepted Answer

Acceleration is the derivative of velocity.

Question 8

A company’s revenue $$R(t) is $$increasing most rapidly at the time when:

Accepted Answer

$$R'(t) is at $$its maximum.

Question 9

If the graph of $$f(t) is $$flat (horizontal) at a certain time, what does this indicate about the rate of change at that time?

Accepted Answer

The rate of change is zero.

Question 10

Given the function $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at what value of $$x is$$ $$f(x)$$ increasing most rapidly?

Accepted Answer

$$x = 2$$

Question 11

A car’s position is given by $$s(t) = 5t^2$ + 2t. $$What is the car’s velocity at $$t = 3$$ seconds?

Accepted Answer

$$32$$

Question 12

Which of the following best describes the average rate of change of a function over an interval?

Accepted Answer

The slope of the secant line connecting two points on the graph.

Question 13

If $$f(x) is $$decreasing most rapidly at $$x = a$$, what can you say about $$f''(a)$$?

Accepted Answer

$$f''(a) is $$negative and at its maximum (most negative).

Question 14

A company’s cost function is $$C(x) = 50 + 10x + 0.5x^2$. What is the marginal cost when x = 20$?

Accepted Answer

$$C'(20) = 30$$

Question 15

If the position of a particle is given by $$s(t) = t^3$ - 6t^2$ + 9t$$, at what time is the velocity zero?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 16

Which of the following is the correct interpretation of $$\frac{ds}{dt} in a $$real-world context?

Accepted Answer

It is the velocity, or rate of change of position with respect to time.

Question 17

A ball is thrown upward from a window, and its height is given by $$s(t) = -16t^2$ + 32t + 128. $$How long does it take for the ball to hit the ground?

Accepted Answer

$$t = 4$$

Question 18

If a function $$f(x)$$ has a constant rate of change, which of the following must be true?

Accepted Answer

$$f(x) is a $$linear function.

Question 19

Suppose the graph of $$f(t) is $$increasing most rapidly at $$t = 5. $$What does this mean in terms of the second derivative?

Accepted Answer

$$f''(5) is at $$its maximum.

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