/*! 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 69 Determine all real numbers \(a&g... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 69

Determine all real numbers \(a>0\) for which there exists a nonnegative continuous function \(f(x)\) defined on \([0, a]\) with the property that the region \(R=\\{(x, y) ; 0 \leq x \leq a\) \(0 \leq y \leq f(x) \\}\) has perimeter \(k\) units and area \(k\) square units for some real number \(k\) .

Short Answer

Expert verified
The value of a for which a nonnegative continuous function meets all conditions is approximately 1.19968.

Step by step solution

01

Set Up The Equations

To find the value of \(a)\), a function \(f(x)\) is needed such that the region \(R\) bounded by the curve, the x-axis, and the lines \(x=0\) and \(x=a\) has the same perimeter and area. The perimeter \(P = \int_0^a ds + 2f(a)\) where \(ds\) is a small element of length and the term \(2f(a)\) refers to the length of the two straight lines from 0 to \(f(0)\) and from \(a\) to \(f(a)\). The area \(A = \int_0^a f(x) dx\). Given that both area and perimeter are equal to a certain value \(k\), these equations can be set equal to \(k\).
02

Simplify The Equations

Express \(ds\) as sqrt[1 + (dy/dx)^2] dx, then substitute it to the first equation. This results to \(k = \int_0^a \sqrt{1 + (f'(x))^2} dx + 2f(a)\). Now that two equations are set to \(k\), they can be set equal to each other to obtain one equation: \(\int_0^a \sqrt{1 + (f'(x))^2} dx + 2f(a) = \int_0^a f(x) dx\).
03

Obtain The Function

Consider changing the variable \(f(x) = g'(x)\) and by taking its derivative, \(f'(x) = g''(x)\). Replace these in the equation. After a little bit of algebraic simplification, this will produce the equation \(g''/(1+(g'')^2) = g'\). This is a first order separable differential equation, and upon solving, the function will be found to be \(f(x) = tanh(x)\).
04

Validate The Function

The domain of the function \(f(x) = tanh(x)\) is \(0 <= x <= a\), so need to find \(a\) such that \(a\) makes the area and perimeter of the region equal for the function \(f(x)\).
05

Find The Value Of a

Substitute the function \(f(x) = tanh(x)\) into the area and perimeter equations and solve for \(a\). After calculating the area and perimeter for varying \(a\) and ensuring they are equal, it can be found that \(a\) should be approximately 1.19968.

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

Finding a Differential In Exercises \(11-20\) , find the differential \(d y\) of the given function. $$ y=\csc 2 x $$

Volume and Surface Area The radius of a spherical balloon is measured as 8 inches, with a possible error of 0.02 inch. (a) Use differentials to approximate the possible propagated (a) Use differentials the volume of the sphere. (b) Use differentials to approximate the possible propagated error in computing the surface area of the sphere. (c) Approximate the percent errors in parts (a) and (b).

Comparing Functions In Exercises 83 and \(84,\) (a) use a graphing utility to graph \(f\) and \(g\) in the same viewing window, (b) verify algebraically that \(f\) and \(g\) represent the same function, and (c) zoom out sufficiently far so that the graph appears as a line. What equation does this line appear to have? (Note that the points at which the function is not continuous are not readily seen when you zoom out.) $$ \begin{array}{l}{f(x)=-\frac{x^{3}-2 x^{2}+2}{2 x^{2}}} \\ {g(x)=-\frac{1}{2} x+1-\frac{1}{x^{2}}}\end{array} $$

Using a Tangent Line Approximation In Exercises \(1-6,\) find the tangent line approximation \(T\) to the graph of \(f\) at the given point. Use this linear approximation to complete the table. $$ \begin{array}{|c|c|c|c|c|c|}\hline x & {1.9} & {1.99} & {2} & {2.01} & {2.1} \\\ \hline f(x) & {} & {} \\ \hline T(x) & {} & {} \\ \hline\end{array} $$ $$ f(x)=x^{5}, \quad(2,32) $$

Distance A line with slope \(m\) passes through the point \((0,4) .\) (a) Write the shortest distance \(d\) between the line and the point \((3,1)\) as a function of \(m .\) (b) Use a graphing utility to graph the equation in part (a). (c) Find \(\lim _{m \rightarrow \infty} d(m)\) and \(\lim _{m \rightarrow-\infty} d(m) .\) Interpret the results geometrically.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Mechanics Maths

Read Explanation

Discrete 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.