Find the area of the largest rectangle cut from the first quadrant by a line tangent to the curve y=e^(-x^2)
The curve is "bell-shaped"
http://www.wolframalpha.com/input/?i=plot+y+%3De%5E(-x%5E2)
Let the point of contact in quad I be P(x,y)
Did you mean "largest triangle" ??
If you want the "largest rectangle" then the tangent line is not needed.
As Reiny says, a rectangle whose base goes from 0 to x has area
a = xy = xe^(-x^2)
da/dx = (1-2x^2) e^(-x^2)
so there is a max at x = 1/√2
That is assuming, of course that the rectangle has its base on the x-axis. It might be possible that a rotated rectangle could be bigger, but that gets too messy.
Oh, wow. Maybe that's what you meant. A rotated rectangle could have its top side tangent to the curve!
But that still gets messy, as issues of concavity and boundaries of the rectangle come into play.
The answer is 2/e and I don't even have a clue on how to solve it.
To find the area of the largest rectangle cut from the first quadrant by a line tangent to the curve y=e^(-x^2), we can follow these steps:
Step 1: Find the derivative of the curve y=e^(-x^2) with respect to x to get the slope of the tangent line.
The derivative of y=e^(-x^2) can be found using the chain rule as follows:
dy/dx = -2x * e^(-x^2)
Step 2: Find the equation of the tangent line by using the point-slope form.
The equation of a line with slope m through the point (a, b) is given by:
y - b = m(x - a)
Since the tangent line passes through the first quadrant, we know that the point (a, b) has positive x and y coordinates.
Step 3: Substitute the derivative of the curve into the point-slope form to get the equation of the tangent line.
Using the derivative we found in Step 1, we can substitute it into the point-slope form, assuming that x and y are the coordinates of the point of tangency:
y - e^(-x^2) = -2x * e^(-x^2)(x - x-coordinate of tangency)
Step 4: Find the x-coordinate of the point of tangency by equating the derivative of the curve with the slope of the tangent line.
Set the derivative -2x * e^(-x^2) equal to the slope -2 of the tangent line, and solve for x:
-2x * e^(-x^2) = -2
Step 5: Once you find the x-coordinate of the point of tangency, substitute it back into the original curve equation y=e^(-x^2) to find the corresponding y-coordinate.
Step 6: The width of the rectangle will be twice the x-coordinate of the point of tangency (since it lies in the first quadrant), and the height will be the y-coordinate.
Step 7: Multiply the width and height together to find the area of the largest rectangle cut from the first quadrant by the tangent line.
Follow these steps to find the area of the largest rectangle cut from the first quadrant by the tangent line to the curve y=e^(-x^2).