Question 1

Given the function $$f(x) = x^2$ - 3$$, use the 3-step method to find and simplify $$\frac{f(x+h) - f(x)}{h}. $$What is the result?

Accepted Answer

$$2x + h$$

Question 2

What is the first step in the 3-step method for finding the difference quotient of a function $$f(x)$$?

Accepted Answer

Find $$f(x+h)$$

Question 3

If $$f(x) = x^2$ - 3$$, what is $$f(x+h)$$?

Accepted Answer

$$(x+h)^2 - 3$

Question 4

After expanding $$f(x+h)$$ for $$f(x) = x^2$ - 3$$, what is the result?

Accepted Answer

$$x^2 + 2xh$$ + $$h^2 - 3$

Question 5

What is $$f(x+h) - f(x)$$ for $$f(x) = x^2$ - 3$$?

Accepted Answer

$$2xh + h^2$

Question 6

Which expression represents the difference quotient for a general function $$f(x)$$?

Accepted Answer

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

Question 7

For $$f(x) = x^2$ - 3$$, what is the simplified form of $$\frac{f(x+h) - f(x)}{h}$$?

Accepted Answer

$$2x + h$$

Question 8

If $$f(x) = x^2$ - 3$$, what is the value of the difference quotient $$\frac{f(x+h) - f(x)}{h}$$ when $$h = 0$$?

Accepted Answer

$$2x$$

Question 9

Which of the following best describes the purpose of the 3-step method shown in the study material?

Accepted Answer

To find and simplify the difference quotient for a function

Question 10

In the context of the difference quotient, what does the variable $$h$$ represent?

Accepted Answer

A small increment added to $$x$$

Question 11

Suppose a caterer uses the difference quotient to estimate the rate of change in cost as the number of guests increases. If the cost function is $$C(x) = x^2$ - 3$$, what does $$\frac{C(x+h) - C(x)}{h}$$ represent in this scenario?

Accepted Answer

The average rate of change in cost per additional guest over an interval of $$h$$ guests

Question 12

If $$f(x) = x^2$ - 3$$, what is the value of $$\frac{f(x+h) - f(x)}{h}$$ when $$x = 2$$ and $$h = 1$$?

Accepted Answer

$$5$$

Question 13

Which step in the 3-step method involves expanding $$(x+h)^2$$?

Accepted Answer

Step 1: Find $$f(x+h)$$

Question 14

If $$f(x) = x^2$ - 3$$, what is the result of $$f(x+h) - f(x)$$ after simplifying?

Accepted Answer

$$2xh + h^2$

Question 15

Why is it important to simplify $$\frac{f(x+h) - f(x)}{h} in $$calculus and precalculus?

Accepted Answer

It helps to find the instantaneous rate of change or the derivative

Difference Quotient and the 3-Step Method in Precalculus

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