I have to find the area of the largest possible rectangle that can be inscribed under the curve y=e^(-x^2) in the first and second quadrants.
How do I do this?
It is symmetrical, so one could ignore the second quadrant, and work on the largest rectangle in the first quadrant.
Area= xy= x e^(-x^2)
dArea/dx=0 = ....you do it, solve for x.
To find the area of the largest possible rectangle that can be inscribed under the curve y = e^(-x^2) in the first and second quadrants, we can start by focusing on the largest rectangle in the first quadrant. Since the curve is symmetrical, we can then double the area to account for the second quadrant.
The area of a rectangle is given by the formula A = x * y, where x and y represent the dimensions of the rectangle. In this case, x represents the width of the rectangle, which is a segment along the x-axis, and y represents the height of the rectangle, which is the corresponding value of y on the curve.
So, the area of the rectangle is A = x * e^(-x^2).
To find the maximum area, we need to find the value of x that maximizes the area. We can do this by taking the derivative of the area function with respect to x (dA/dx) and setting it equal to zero. When the derivative is zero, it means we have reached a local maximum or minimum.
So, let's find the derivative of the area function:
dA/dx = e^(-x^2) - 2x^2 * e^(-x^2)
Now, set the derivative equal to zero and solve for x:
0 = e^(-x^2) - 2x^2 * e^(-x^2)
Simplifying the expression, we get:
e^(-x^2) = 2x^2 * e^(-x^2)
Dividing both sides by e^(-x^2), we have:
1 = 2x^2
Now, divide both sides by 2:
x^2 = 1/2
Taking the square root of both sides:
x = ±√(1/2)
So, we have two possible values for x: x = √(1/2) and x = -√(1/2).
Now that we have the possible values for x, we can find the corresponding value of y using the equation y = e^(-x^2):
For x = √(1/2):
y = e^(-√(1/2)^2) = e^(-1/2)
For x = -√(1/2):
y = e^(-(-√(1/2))^2) = e^(-1/2)
Since the exponential function is always positive, we can ignore the negative value of y and focus on the positive value.
So, the largest possible rectangle in the first quadrant has dimensions √(1/2) for the width (x) and e^(-1/2) for the height (y).
To find the area, we multiply the width and height:
Area = √(1/2) * e^(-1/2)
Finally, since we need to account for the second quadrant as well, we double the area:
Largest possible area = 2 * √(1/2) * e^(-1/2)
And that is how you can find the area of the largest possible rectangle inscribed under the curve y = e^(-x^2) in the first and second quadrants.