The figure shows the graph of f(x)=xe^x, x greater than or equal to 0.
figure: f(x) curve is drawn and under the max. point/ concave down curve there is an inscribed rectangle. with width from a(left) to b(right) and height up to a certain point of f(x).
Let a > 0 and b > 0 be given as shown in the figure. Complete the following table where A is the area of the rectangle in the figure.
a b A
0.1
0.2
0.3
.
.
.
1
I have no idea how to fill in b and A columns. The answer starts with b as 3.71 and A as 0.33, which I don't understand because a*b in this case does not give 0.33.
x e^x is concave up for x > -2
I don't see how your description can be right.
my function is xe^(-x).
If a = 0.1, f(a) = 0.09048
Draw that horizontal line, and it intersects the curve at b=3.71495
Thus, A = 0.09038(3.71495-0.1) = 0.32672
Do the others the same way:
find f(a)
see where else f(b) = f(a)
calculate A = f(a)*(b-a)
To determine the values of b and A in the table, we will need to find the coordinates of the points that define the rectangle on the graph of f(x). This can be done by finding the x-coordinate of the maximum point of f(x) and then finding the corresponding y-coordinate.
Given that f(x) = xe^x, we can differentiate f(x) with respect to x to find the critical points and determine the maximum value of f(x). Taking the derivative of f(x) yields:
f'(x) = e^x + xe^x.
To find the critical points, we set f'(x) equal to zero and solve for x:
e^x + xe^x = 0.
This equation does not have an algebraic solution, so we will need to solve it numerically. Using a numerical solver or a graphing calculator, we find that the critical point is approximately x = -1. In the given information, it is stated that x ≥ 0, so we disregard the negative value for x.
To find the maximum value of f(x), we substitute x = 0 into the original function:
f(0) = 0e^0 = 0.
Therefore, the maximum point is at (0, 0).
Now, we can proceed to find the rectangle's dimensions and the corresponding area:
1. Finding the value of b:
To determine the right side of the rectangle, we need to find the x-coordinate at which f(x) intersects the x-axis (i.e., where f(x) = 0). Solving f(x) = 0:
xe^x = 0.
Since the exponential term e^x is always positive, the only way for the product to be zero is if x = 0. Therefore, the right side of the rectangle is at x = 0.
2. Finding the value of A:
The height of the rectangle will be the y-coordinate corresponding to the maximum point (0, 0) on the graph of f(x).
Substituting x = 0 into f(x):
f(0) = 0e^0 = 0.
Therefore, the height of the rectangle is 0.
The area of the rectangle (A) is given by the product of the width (b - a) and the height (0):
A = (b - a) * 0 = 0.
Based on these calculations, it seems that there may be an error in the table or the given solution. The area (A) is expected to be 0, rather than 0.33.
Please double-check the information provided or the given answer, as it appears to be incorrect.