   Chapter 8.4, Problem 64E

Chapter
Section
Textbook Problem

# Conjecture(a) Find formulas for the distances between (0, 0) and (a, a2) along the line between these points and along the parabola y = x 2 .(b) Use the formulas from part (a) to find the distances for a = 1 and a = 10.(c) Make a conjecture about the difference between the two distances as a increases.

(a)

To determine

The formula for the distances between the points (0,0) and (a,a2) along the line between them and along the parabola y=x2.

Explanation

Consider the figure shown below,

The distance between the points (0,0) and (a,a2) along the line between them is given by,

d1=(a0)2+(a20)2=a2+a4=a2(1+a2)=a1+a2

The distance between the points (0,0) and (a,a2) along the along the parabola y=x2 is given by,

d2=0a1+(dydx)2

As,

y=x2

Thus,

dydx=2x

Therefore,

d2=0a1+(2x)2dx=20a(12)2+(x)2dx

As,

a2+x2dx=12xa2+x2+a22ln|x+a2+x2a|+C

Thus,

d2=20a(12)2+(x)2dx=2[12x14

(b)

To determine

To calculate: The distance at a=1 and a=10 by the use of the formulas obtained in part (a).

(c)

To determine

What happens with the distance when a increases.

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