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

Study Guide - Practice Questions

Study Guide - Flashcards