To find: The closest or minimum distance such that the semicircle
Answer to Problem 42E
The minimum distance is
Explanation of Solution
Given information: The equation of the semicircle
Formula used:
Distance d between two points
The minimum value of the function or the closest distance will be at the critical point of that function.
Calculation:
Let the point on the given semicircle be
Consider the distance between the points
Determine the distance d between the points
Square on both sides of the obtained equation and simplify the resulting equation.
Differentiate the obtained equation on both sides with respect to x.
Equate the obtained derivative to zero to get the critical point.
This implies that
Since the distance along with the point
Now, determine the second derivative of the obtained distance by differentiating the obtained first derivative.
Determine the value of the second derivative at x=2.
Since the second derivative at x=2 is positive, the distance will be closest or minimum at x=2.
Determine the minimum distance by substituting 2 for x into the obtained distance function.
Substitute
As a result, the curve would come to close the point
Chapter 4 Solutions
Advanced Placement Calculus Graphical Numerical Algebraic Sixth Edition High School Binding Copyright 2020
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