SOLUTION: Find the area of the largest rectangle that has two sides on the positive x-axis and the positive y-axis one vertex at the origin and one vertex on the curve y = e^(-x) Explain

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Question 999522: Find the area of the largest rectangle that has two sides on the positive x-axis and the positive y-axis one vertex at the origin and one vertex on the curve y = e^(-x)
Explain how you know the value you found is the maximum area.
Thank you

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
area of the rectangle is x * e^-x

x is the length of the rectangle.
e^-x is the width of the rectangle.

the area will be maximum when x = 1.

you can solve this graphically, or you can use calculus to find the max/min point.

the graphical solution is shown below:

the area is equal to x * e^(-x).

you graph that function and find the maximum point on the curve.

the maximum point on the curve is when x = 1 and y = e^-1 = .3678794412.

this is shown on the graph as (1,.3679).

at that point, the value of the x-coordinate is 1 and the value of the y-coordinate is .3679.

the graphing software i used automatically finds the maximum point.

here's the graph:

$$$

the red line is y = e^(-x).

the blue line is y = x * e^(-x).

the blue line is the graph of the area of the rectangle.

the area of a rectangle is length * width.
in the formula of y = x * e^(-x), the length is x and the width is e^(-x).

you could also have used calculus.
the max/min point of the equation is when the derivative of the equation is equal to 0.
the derivative of x * e^(-x) is equal to -(x-1)*e^(-x)

set that equal to 0 to get -(x-1)*e^(-x) = 0
remove the parentheses to get -x * e^(-x) + e^(-x) = 0
add x*(e^(-x) to both sides of this equation to get e^(-x) = x*e^(-x)
divide both sides of this equation by e^(-x) to get 1 = x
solve for x to get x = 1.

note that, in the equation of -(x-1)*e^(-x) = 0, i could also have divided both sides of the equation by -(x-1) to get e^(-x) = 0.

i would have then solved that by taking the natural log of both sides of the equation to get:

ln(e^(-x)) = ln(0).

since ln(0) is invalid, that would not have yielded a good result.

i then went to the alternate method of removing the parentheses, which led to the correct result.

even though dividing both sides of the equation by -(x-1) was a valid algebraic operation, it did not provide a satisfactory solution to this problem because of the restrictions on the value that you can take the natural log of. you can only take the natural log of a positive number.

i used the following online derivative calculator to find the derivative of x * e^(-x).

link to online derivative calculator











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