/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 8 Use a graphing utility to graph ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 7: Problem 8

Use a graphing utility to graph the function and find its average rate of change on the interval. Compare this rate with the instantaneous rates of change at the endpoints of the interval. $$ f(x)=x^{3 / 2} ;[1,4] $$

Short Answer

Expert verified
The average rate of change of the function \(f(x)=x^{3 / 2}\) on the interval [1,4] is computed using the formula \((f(b) - f(a)) / (b - a)\). The instantaneous rates of change at the interval's endpoints are given by the derivative of the function, evaluated at these points which is \((3/2) \sqrt{x}\). Comparing these values provides insight into the function's behavior over this interval.

Step by step solution

01

Compute Average Rate of Change

The average rate of change of a function on an interval [a, b] is given by the formula \((f(b) - f(a))/(b - a)\). Substitute \(a = 1\) and \(b = 4\) into the formula. Thus, the average rate of change of \(f\) on [1,4] is \((f(4) - f(1))/(4 - 1) = ((4^{3/2} - 1^{3/2})/ (4 - 1))\). Simplify this to get the answer.
02

Compute Instantaneous Rate of Change at the Endpoints

The instantaneous rate of change at a point is the derivative of the function evaluated at that point. The derivative of \(f(x)\) is given by \(f'(x) = (3/2) \sqrt{x}\). Compute \(f'(1)\) and \(f'(4)\) to find the instantaneous rates of change at the endpoints of the interval [1, 4].
03

Compare the Instantaneous Rates of Change with the Average Rate of Change

Compare the result from Step 1 with the results from Step 2. This will provide insight into how the function behaves on the interval [1,4], specifically how the average rate of change relates to the instantaneous rates of change at the endpoints of the interval.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(x^{2}-9\right)^{2 / 3} $$

Use the General Power Rule to find the derivative of the function. $$ f(x)=\left(4 x-x^{2}\right)^{3} $$

Find the derivative of the function. State which differentiation rule(s) you used to find the derivative. $$ y=\left(\frac{4 x^{2}}{3-x}\right)^{3} $$

Use a symbolic differentiation utility to find the derivative of the function. Graph the function and its derivative in the same viewing window. Describe the behavior of the function when the derivative is zero. $$ f(x)=\sqrt{x}\left(2-x^{2}\right) $$

Find an equation of the tangent line to the graph of the function at the given point. Then use a graphing utility to graph the function and the tangent line in the same viewing window. $$ f(t)=\left(t^{2}-9\right) \sqrt{t+2} \quad(-1,-8) $$
See all solutions

Recommended explanations on Math Textbooks

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.