*Homework 1.3 - Average rates of change of functions
Score: Answered:
Question 10
Find the average rate of change of on the interval . Your answer will be an
expression involving
Submit Question

Ask by Valdez Chambers. in the United States

Mar 21,2025

Question

*Homework 1.3 - Average rates of change of functions
Score: Answered:
Question 10
Find the average rate of change of on the interval . Your answer will be an
expression involving
Submit Question

Ask by Valdez Chambers. in the United States

Mar 21,2025

UpStudy AI · Accepted Answer

To find the average rate of change of the function $$ f(x) = 2x^2 - 8 $$ on the interval $$[1, b]$$, we can use the formula for the average rate of change:

$$
\text{Average Rate of Change} = \frac{f(b) - f(1)}{b - 1}
$$

Now, let's calculate $$ f(1) $$ and $$ f(b) $$:

1. Calculate $$ f(1) $$:
   $$
   f(1) = 2(1)^2 - 8
   $$

2. Calculate $$ f(b) $$:
   $$
   f(b) = 2(b)^2 - 8
   $$

Now, we can substitute these values into the average rate of change formula. Let's perform the calculations step by step.

First, we will calculate $$ f(1) $$ and $$ f(b) $$.
Simplify the expression by following steps:
- step0: Solution:
  $$2b^{2}-8$$
Calculate or simplify the expression $$ 2*(1)^2 - 8 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2\times 1^{2}-8$$
- step1: Evaluate the power:
  $$2\times 1-8$$
- step2: Multiply:
  $$2-8$$
- step3: Subtract the numbers:
  $$-6$$
We have calculated the following:

1. $$ f(1) = -6 $$
2. $$ f(b) = 2b^2 - 8 $$

Now, we can substitute these values into the average rate of change formula:

$$
\text{Average Rate of Change} = \frac{(2b^2 - 8) - (-6)}{b - 1}
$$

This simplifies to:

$$
\text{Average Rate of Change} = \frac{2b^2 - 8 + 6}{b - 1} = \frac{2b^2 - 2}{b - 1}
$$

Next, we can factor the numerator:

$$
\text{Average Rate of Change} = \frac{2(b^2 - 1)}{b - 1}
$$

Recognizing that $$ b^2 - 1 $$ can be factored as $$ (b - 1)(b + 1) $$:

$$
\text{Average Rate of Change} = \frac{2(b - 1)(b + 1)}{b - 1}
$$

Assuming $$ b \neq 1 $$, we can cancel $$ b - 1 $$:

$$
\text{Average Rate of Change} = 2(b + 1)
$$

Thus, the average rate of change of $$ f(x) = 2x^2 - 8 $$ on the interval $$[1, b]$$ is:

$$
2(b + 1)
$$
The average rate of change of $$ f(x) = 2x^2 - 8 $$ on the interval $$[1, b]$$ is $$ 2(b + 1) $$.

*Homework 1.3 - Average rates of change of functions
Score: Answered:
Question 10
Find the average rate of change of on the interval . Your answer will be an
expression involving
Submit Question

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*Homework 1.3 - Average rates of change of functions Score: Answered: Question 10 Find the average rate of change of on the interval . Your answer will be an expression involving Submit Question

Solución de inteligencia artificial de Upstudy

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Solución

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Latest Calculus Questions

*Homework 1.3 - Average rates of change of functions
Score: Answered:
Question 10
Find the average rate of change of on the interval . Your answer will be an
expression involving
Submit Question