Question 1

Given the function $$g(a) = a^2$ - 2a + 1$$, what is the simplified form of $$\frac{g(a+h) - g(a)}{h}$$?

Accepted Answer

$$2a + h - 2$$

Question 2

What is the first step in evaluating $$\frac{g(a+h) - g(a)}{h}$$ for $$g(a) = a^2$ - 2a + 1$$?

Accepted Answer

Substitute $$a+h$$ into $$g(a) to $$get $$g(a+h) = (a+h)^2$ - 2(a+h) + 1$$

Question 3

After expanding $$g(a+h)$$, what is the correct expression for $$g(a+h)$$?

Accepted Answer

$$a^2 + 2ah$$ + $$h^2 - 2a - 2h + 1$

Question 4

Which algebraic property is used to simplify $$g(a+h) - g(a)$$?

Accepted Answer

Combining like terms

Question 5

What is the result of $$g(a+h) - g(a)$$ for $$g(a) = a^2$ - 2a + 1$$?

Accepted Answer

$$2ah + h^2$ - 2h$$

Question 6

Which of the following best describes the process used to simplify $$\frac{g(a+h) - g(a)}{h}$$?

Accepted Answer

Difference quotient

Question 7

If $$g(a) = a^2$ - 2a + 1$$, what is $$g'(a)$$ using the definition of the derivative?

Accepted Answer

$$2a - 2$$

Question 8

In the context of the difference quotient, what does the $$h in$$ $$\frac{g(a+h) - g(a)}{h}$$ represent?

Accepted Answer

A small change in $$a$$

Question 9

Which step involves factoring $$h$$ from the numerator in the simplification process?

Accepted Answer

Step 4

Question 10

Why is it important to simplify $$\frac{g(a+h) - g(a)}{h}$$ before substituting $$h = 0$$?

Accepted Answer

To avoid division by zero

Question 11

Suppose $$g(a)$$ models the cost (in dollars) to produce $$a$$ units of a product. What does $$\frac{g(a+h) - g(a)}{h}$$ represent in this context?

Accepted Answer

The average cost per unit for producing $$h$$ additional units

Question 12

If $$g(a) = a^2$ - 2a + 1$$, what is $$g(3)$$?

Accepted Answer

$$4$$

Question 13

Which of the following is a common mistake when simplifying $$\frac{g(a+h) - g(a)}{h}$$?

Accepted Answer

Not expanding $$(a+h)^2$$ correctly

Question 14

If $$g(a) = a^2$ - 2a + 1$$, what is the value of $$\frac{g(5) - g(3)}{5-3}$$?

Accepted Answer

$$6$$

Question 15

Which real-world scenario could be modeled by the difference quotient $$\frac{g(a+h) - g(a)}{h}$$?

Accepted Answer

The average speed of a car over a time interval

Difference Quotient and Function Evaluation: Step-by-Step Solution

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