Q: A function $$f(x) is $$said to be nondifferentiable at $$x = a if$$:

The limit $$\lim_{h \to 0} \frac{f(a + h) - f(a)}{h}$$ does not exist

Question 1

A company manufactures plastic planter boxes. The revenue (in dollars) from selling $$x$$ boxes is given by $$R(x) = 20x - 0.02x^2$ for 0$$ \u003c x \u003c 1,000$$. What is the average rate of change in revenue as production increases from 100 to 400 boxes?

Accepted Answer

$$10$$

Question 2

Given the position function for a falling object $$y = 16x^2$, what is the average velocity of the object between x = 2$ seconds and x = 3$ seconds?

Accepted Answer

$$80 ft/s$$

Question 3

Which of the following best describes the derivative $$f'(x) of a $$function $$f(x)$$?

Accepted Answer

The instantaneous rate of change of $$f(x)$$ with respect to $$x$$

Question 4

If $$f(x) = -16x^2$ + 80x + 6$$, what is the first step in the four-step process to find $$f'(x)$$?

Accepted Answer

Find $$f(x + h)$$

Question 5

A ball is dropped from a tower and its position after $$x$$ seconds is given by $$y = 16x^2$. What is the instantaneous velocity at x = 2$ seconds?

Accepted Answer

$$64 ft/s$$

Question 6

Which of the following statements is TRUE regarding the secant and tangent lines to a curve?

Accepted Answer

A tangent line touches the curve at exactly one point, while a secant line passes through two points on the curve.

Question 7

The derivative of a function at a point $$x = a$$ does NOT exist if:

Accepted Answer

The limit defining the derivative does not exist at $$x = a$$

Question 8

Suppose the revenue function is $$R(x) = 20x - 0.02x^2$. What is the instantaneous rate of change of revenue when x = 400$?

Accepted Answer

$$4$$

Question 9

Which of the following is the correct difference quotient for the average rate of change of $$f(x)$$ from $$x = a to$$ $$x = a + h$$?

Accepted Answer

$$\frac{f(a + h) - f(a)}{h}$$

Question 10

A car travels from mile marker 120 to mile marker 250 in 2 hours. What is the average velocity?

Accepted Answer

$$65$$ mph

Question 11

If $$f(x) = 16x^2$, what is the general expression for the average velocity from x = 2$ to x = 2 + h$?

Accepted Answer

$$\frac{16(2 + h)^2$ - 16(2)^2$}{h}$$

Question 12

Which of the following is NOT an interpretation of the derivative?

Accepted Answer

The total accumulated value of $$f(x)$$ over an interval

Question 13

If the average rate of change of a function over an interval is positive, what does this indicate about the function on that interval?

Accepted Answer

The function is increasing on that interval

Question 14

Which step in the four-step process for finding the derivative involves dividing by $$h$$?

Accepted Answer

Step 3

Question 15

If $$f(x) = -16x^2$ + 80x + 6$$, what is $$f'(x)$$ using the power rule?

Accepted Answer

$$-32x + 80$$

Question 16

A function $$f(x) is $$said to be nondifferentiable at $$x = a if$$:

Accepted Answer

The limit $$\lim_{h 	o 0} \frac{f(a + h) - f(a)}{h}$$ does not exist

Limits and the Derivative: Average and Instantaneous Rates of Change 2.4

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