<p>This problem is from CalcAB textbook.
(But it's Chapter 4, it can't be HW)</p>
<p>f(x) = xe^(-x), x>0</p>
<p>Rectangle is drawn using the segment of x-axis(segment ab) as the base
and two points on the graph.</p>
<p>How do you get the maximum area and the coordinate of a?</p>
<p>The problem originally was pure calculator problem. So I did...
(Making table of a, b, and Area, doing scatter plot, regression eq, estimate.)</p>
<p>But it's been bugging me b/c I feel like this can be done w/ better way.
(The way I couldn't find it... maybe it requires more than calculus?)</p>
<p>The only way I could think of was using parametric equation...</p>
<p>Appreciate any help ...</p>
<p>
[quote]
f(x) = xe^(-x), x>0</p>
<p>Rectangle is drawn using the segment of x-axis(segment ab) as the base
and two points on the graph.</p>
<p>How do you get the maximum area and the coordinate of a?
[/quote]
</p>
<p>To find the coordinate of a, just take the derivative of f(x), set them equal zero, and do the second derivative test to see if it's really where the maximum is located. The coordinate of a would be (a,f(a)), with a as the location of the maximum value [whatever x value you get after setting up the derivative equal zero), and f(a) its value. </p>
<p>To find the area, it depends on the value of delta x, or to how many rectangles you divide the area under the graph -since we know that f(x) will never be below zero when x>0-. delta x would be (b-a) / n, with n the number of rectangles to which the area is divided. And the area is f(a) * delta x. Or if you're taking the integral, the area of greatest differential element would be f(a) * dx.</p>