YA y =r B L A

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Dancing on a parabola Two people, A and B, walk along the
parabola y = x2 in such a way that the line segment L between
them is always perpendicular to the line tangent to the parabola at
A’s position. The goal of this exercise is to determine the positions
of A and B when L has minimum length. Assume the coordinates
of A are (a, a2).

a. Find the slope of the line tangent to the parabola at A, and find
the slope of the line that is perpendicular to the tangent line at A.
b. Find the equation of the line joining A and B.
c. Find the position of B on the parabola.
d. Write the function F(a) that gives the square of the distance
between A and B as it varies with a. (The square of the distance
is minimized at the same point that the distance is minimized;
it is easier to work with the square of the distance.)
e. Find the critical point of F on the interval a > 0.
f. Evaluate F at the critical point and verify that it corresponds to
an absolute minimum. What are the positions of A and B that
minimize the length of L? What is the minimum length?
g. Graph the function F to check your work.

YA
y =r
B
L
A
Transcribed Image Text:YA y =r B L A
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