To determine

To find: The area of the largest rectangle in the first quadrant of the given expression.

Answer

e−1.

Explanation

Given:

y=e−x

Formulae used:

The area of the rectangle A=l×w

Consider y=e−x

The expression of the area is

A=x×e−x.

Given the two sides of the rectangle are on axes and one vertex on the curve which probably constitutes the length. Hence the rectangle will be driven by the vertex of curve.

Hence, use the formulae of area of the rectangle A=l×w

Now according to the given expression, the value of the largest area is

To find the largest area maximize the expression A=xe−x

dAdx=e−x−xe−x

To find the value of x put the derivative of area dAdx=0

e−x−xe−x=0e−x(1−x)=0x=1

Put the value of x=1 in the expression A=xe−x and get the value of the largest area

A=xe−x=(1)(e−1)=e−1

Conclusion:

The largest area of the function bounded by the first quadrant axes and one vertex on the curve y=e−x is e−1.