Question

If m ≤ f(x) ≤ M for a ≤ x ≤ b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then m(b − a) ≤ b f(x) dx a≤ M(b − a).
Use this property to estimate the value of the integral.

The integral from 0 to 2 of 5xe^−x .

Answer 1

To estimate the integral, we first need to find the absolute minimum (m) and absolute maximum (M) of the function f(x) = 5xe^(-x) on the interval [0, 2].

To find critical points of the function, we need to find its first derivative and then set it equal to 0.

f'(x) = 5(e^(-x) - x*e^(-x))

Now, we set f'(x) = 0 and solve for x:

5(e^(-x) - x*e^(-x)) = 0
e^(-x) (1 - x) = 0

Since e^(-x) is never 0, we must look for when (1-x) is 0:

1 - x = 0
x = 1

Thus, we have one critical point at x = 1.

Now, we check the boundary points and the critical point to find the minimum and maximum of the function:

f(0) = 5(0)*e^(0) = 0
f(1) = 5(1)*e^(-1) = 5/e
f(2) = 5(2)*e^(-2) = 10/e^2

We can see that f(0) is the absolute minimum (m = 0), and f(1) is the absolute maximum (M = 5/e).

Now, using the inequality from the problem, we have:

m(b - a) ≤ ∫[a, b] f(x) dx ≤ M(b - a)
0(2 - 0) ≤ ∫[0, 2] 5xe^(-x) dx ≤ (5/e)(2 - 0)

0 ≤ ∫[0, 2] 5xe^(-x) dx ≤ 10/e

Thus, the value of the integral is between 0 and 10/e ≈ 3.68. This gives us a good estimate for the integral but does not provide an exact value. To find the exact value, we would need to evaluate the integral directly.