/*! 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 88 The area of a square with sides ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 88

The area of a square with sides of length \(s\) is given by \(A=s^{2} .\) Find the rate of change of the area with respect to \(s\) when \(s=4\) meters.

Short Answer

Expert verified
The rate of change of the area with respect to the side length when the side length is 4 meters is \(8 \) square meters per meter.

Step by step solution

01

Define the Function

The function given, which represents the area of the square in terms of the side length, is \(A(s) = s^{2}\).
02

Differentiate the Function

The derivative of the function \(A(s)\) with respect to \(s\) is denoted as \(A'(s)\). By applying the power rule for differentiation (which says that the derivative of \(s^n\) is \(n \cdot s^{n-1}\) ), we have \(A'(s) = 2 \cdot s^{1} = 2s\).
03

Evaluate the Derivative at \(s=4\)

Substitute \(s=4\) into the expression for \(A'(s)\) to find the rate of change of the area at \(s=4\). So, \(A'(4) = 2 \cdot 4 = 8 \) square meters per meter.

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

True or False? In Exercises 137-139, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(y=(1-x)^{1 / 2},\) then \(y^{\prime}=\frac{1}{2}(1-x)^{-1 / 2}\)

In Exercises 35 and 36, find an equation of the tangent line to the graph of the equation at the given point. $$ \arcsin x+\arcsin y=\frac{\pi}{2}, \quad\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right) $$

Verify each differentiation formula. (a) \(\frac{d}{d x}[\arctan u]=\frac{u^{\prime}}{1+u^{2}}\) (b) \(\frac{d}{d x}[\operatorname{arccot} u]=\frac{-u^{\prime}}{1+u^{2}}\) (c) \(\frac{d}{d x}[\operatorname{arcsec} u]=\frac{u^{\prime}}{|u| \sqrt{u^{2}-1}}\) (d) \(\frac{d}{d x}[\arccos u]=\frac{-u^{\prime}}{\sqrt{1-u^{2}}}\) (e) \(\frac{d}{d x}[\operatorname{arccsc} u]=\frac{-u^{\prime}}{|u| \sqrt{u^{2}-1}}\)

Linear and Quadratic Approximations The linear and quadratic approximations of a function \(f\) at \(x=a\) are \(P_{1}(x)=f^{\prime}(a)(x-a)+f(a)\) and \(P_{2}(x)=\frac{1}{2} f^{\prime \prime}(a)(x-a)^{2}+f^{\prime}(a)(x-a)+f(a)\) \(\begin{array}{llll}\text { In Exercises } & 133-136, & \text { (a) find the specified linear and }\end{array}\) quadratic approximations of \(f,\) (b) use a graphing utility to graph \(f\) and the approximations, (c) determine whether \(P_{1}\) or \(P_{2}\) is the better approximation, and (d) state how the accuracy changes as you move farther from \(x=a\). $$ \begin{array}{l} f(x)=e^{-x^{2} / 2} \\ a=0 \end{array} $$

Prove that \(\arccos x=\frac{\pi}{2}-\arctan \left(\frac{x}{\sqrt{1-x^{2}}}\right),|x|<1\).
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Geometry

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.