   Chapter 4.7, Problem 50E Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Solutions

Chapter
Section Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

Area Find the area of the largest rectangle that can be inscribed under the curve y = e − x 2 in the first and second quadrants.

To determine

To calculate: The largest area of a rectangle that can fit inside the provided curve y=ex2 and the x-axis.

Explanation

Formula used:

For a function f that is twice differentiable on an open interval I, if f'(c)=0 for some c, then,

If f''(c)>0 the function f has relative minima at c if f''(c)<0 the function f has relative maxima at c.

Calculation:

The provided curve is a bell curve. So the rectangle would be of the form:

Let the point (x,f(x)) lie on the intersection of the curve and the rectangle. From the symmetry of the bell curve, the other intersection point in the second quadrant would be (x,f(x)). This implies the length of the rectangle would be 2x and the breadth would be f(x).

Hence, the area can be written as:

A(x)=(2x)(f(x))=2x(ex2)

Differentiate the provided function with respect to x and equate that to 0 to get the critical point.

A'(x)=4x2ex2+2ex24x2ex2+2ex2=02e

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