Q: Which formula represents the instantaneous rate of change of $$f(x) at$$ $$x = a$$?

$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: What is the slope of the tangent line to the graph of $$y = f(x) at $$the point $$(a, f(a))$$?

$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Question 1

Which of the following best describes the average rate of change of a function $$f(x)$$ from $$x = a to$$ $$x = a + h$$?

Accepted Answer

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

Question 2

A company’s revenue from selling $$x$$ planter boxes is given by $$R(x) = 20x - 0.02x^2$. What is the average rate of change in revenue when production increases from 100 to 400 planters?

Accepted Answer

$$\10$ per planter

Question 3

Given $$y = f(x) = 16x^2$, what is the average velocity of a falling object from x = 2$ seconds to x = 3$ seconds?

Accepted Answer

$$80$$ feet per second

Question 4

Which of the following is NOT a correct interpretation of the derivative $$f'(x)$$?

Accepted Answer

Average rate of change between two points

Question 5

For the function $$f(x) = -16x^2$ + 80x + 6$$, what is the derivative $$f'(x)$$ using the four-step process?

Accepted Answer

$$-32x + 80$$

Question 6

If the limit $$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$ does not exist, what can be concluded about $$f at$$ $$x = a$$?

Accepted Answer

The derivative does not exist at $$x = a$$

Question 7

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

Accepted Answer

$$64$$ feet per second

Question 8

Which step in the four-step process for finding the derivative involves calculating $$\lim_{h 	o 0} \frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

Step 4

Question 9

The line connecting two points on the graph of a function is called a:

Accepted Answer

Secant line

Question 10

If $$f(x) is $$differentiable on $$(a, b)$$, what does this imply?

Accepted Answer

The derivative $$f'(x)$$ exists for every $$x in$$ $$(a, b)$$

Question 11

A company increases production from 100 to 400 units. The revenue function is $$R(x) = 20x - 0.02x^2$. What is the change in revenue?

Accepted Answer

$$\3$$,000$

Question 12

Which formula represents the instantaneous rate of change of $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 13

If $$f(x) is $$the position of a moving object at time $$x$$, what does $$f'(x)$$ represent?

Accepted Answer

Velocity

Question 14

Which of the following is true about the domain of the derivative $$f'(x)$$?

Accepted Answer

It is a subset of the domain of $$f(x)$$

Question 15

Which real-world scenario best illustrates the concept of instantaneous rate of change?

Accepted Answer

Reading the speedometer of a car at a specific moment

Question 16

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

Accepted Answer

$$64 + 16h$$

Question 17

What is the slope of the tangent line to the graph of $$y = f(x) at $$the point $$(a, f(a))$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 18

Which process is used to find the derivative of a function?

Accepted Answer

Differentiation

Question 19

If a function is nondifferentiable at $$x = a$$, what does this mean?

Accepted Answer

The derivative does not exist at $$x = a$$

Limits and the Derivative: Average and Instantaneous Rates of Change

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