Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

4th Edition

ISBN: 9780133178579

Author: Ross L. Finney

Publisher: PEARSON

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Question

Chapter 5.4, Problem 42E

To determine

To find the minimum distance of semicircle.

Expert Solution & Answer

Answer to Problem 42E

The minimum distance is d=2 .

Explanation of Solution

Given information: Given equation y=16−x2 ,the point is (1,3) .

Calculation:

The equation of the semicircle is given by y=16−x2.

Distance of the point (1,3) from the point (x,16−x2) is d=(x−1)2+(16−x2−3)2.

Square of the distance is ,

D=(x−1)2+(16−x2−3)2 =x2+1−2x+(16−x2)+3−23(16−x2) =−2x+20−248−3x2

For any function f(x) the condition of minimum and maximum is obtained by find f′(x)=0 ,

Critical points and end points

D=−2x+20−248−3x2dDdx=−2−2(−6x)248−3x2 =−2+(6x)48−3x2

Critical points occur at

D′(x)=0,−4<x<4−2+(6x)48−3x2=0(6x)48−3x2=2 36x2=4(48−3x2) 12x2=48 x=±2

Consider x=2

Here dDdx<0 for−4<x<2anddDdx>0 for 2<x<4. this corresponds to minimum distance.

For x=2,

D=−2x+20−248−3x2 =−4+20−248−12 =4d=4=2

Thus, the minimum distance is d=2.

Chapter 5.4, Problem 41E

Chapter 5.4, Problem 43E

Chapter 5 Solutions

Calculus 2012 Student Edition (by Finney/Demana/Waits/Kennedy)

Ch. 5.1 - Prob. 1QR Ch. 5.1 - Prob. 2QR Ch. 5.1 - Prob. 3QR Ch. 5.1 - Prob. 4QR Ch. 5.1 - Prob. 5QR Ch. 5.1 - Prob. 6QR Ch. 5.1 - Prob. 7QR Ch. 5.1 - Prob. 8QR Ch. 5.1 - Prob. 9QR Ch. 5.1 - Prob. 10QR

Ch. 5.1 - Prob. 11QR Ch. 5.1 - Prob. 12QR Ch. 5.1 - Prob. 1E Ch. 5.1 - Prob. 2E Ch. 5.1 - Prob. 3E Ch. 5.1 - Prob. 4E Ch. 5.1 - Prob. 5E Ch. 5.1 - Prob. 6E Ch. 5.1 - Prob. 7E Ch. 5.1 - Prob. 8E Ch. 5.1 - Prob. 9E Ch. 5.1 - Prob. 10E Ch. 5.1 - Prob. 11E Ch. 5.1 - Prob. 12E Ch. 5.1 - Prob. 13E Ch. 5.1 - Prob. 14E Ch. 5.1 - Prob. 15E Ch. 5.1 - Prob. 16E Ch. 5.1 - Prob. 17E Ch. 5.1 - Prob. 18E Ch. 5.1 - Prob. 19E Ch. 5.1 - Prob. 20E Ch. 5.1 - Prob. 21E Ch. 5.1 - Prob. 22E Ch. 5.1 - Prob. 23E Ch. 5.1 - Prob. 24E Ch. 5.1 - Prob. 25E Ch. 5.1 - Prob. 26E Ch. 5.1 - Prob. 27E Ch. 5.1 - Prob. 28E Ch. 5.1 - Prob. 29E Ch. 5.1 - Prob. 30E Ch. 5.1 - Prob. 31E Ch. 5.1 - Prob. 32E Ch. 5.1 - Prob. 33E Ch. 5.1 - Prob. 34E Ch. 5.1 - Prob. 35E Ch. 5.1 - Prob. 36E Ch. 5.1 - Prob. 37E Ch. 5.1 - Prob. 38E Ch. 5.1 - Prob. 39E Ch. 5.1 - Prob. 40E Ch. 5.1 - Prob. 41E Ch. 5.1 - Prob. 42E Ch. 5.1 - Prob. 43E Ch. 5.1 - Prob. 44E Ch. 5.1 - Prob. 45E Ch. 5.1 - Prob. 46E Ch. 5.1 - Prob. 47E Ch. 5.1 - Prob. 48E Ch. 5.1 - 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