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 .
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.
To estimate the value of the integral ∫(0 to 2) 5xe^(-x), we can use the property mentioned in the question.
According to the property, 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) ≤ ∫(a to b) f(x) dx ≤ M(b - a).
In our case, we need to find the absolute minimum and maximum of the function f(x) = 5xe^(-x) on the interval [0, 2].
First, let's find the derivative of f(x) to locate any critical points:
f'(x) = 5e^(-x) - 5xe^(-x)
Setting f'(x) = 0, we get:
5e^(-x) - 5xe^(-x) = 0
e^(-x)(5 - 5x) = 0
This equation is true when either e^(-x) = 0 (which is not possible) or 5 - 5x = 0.
Solving 5 - 5x = 0, we find x = 1.
So, the critical point is x = 1.
Next, we need to check the values of f(x) at the endpoints and the critical point:
f(0) = 0
f(1) = 5e^(-1) ≈ 1.8393
f(2) = 10e^(-2) ≈ 0.7358
Therefore, the absolute minimum (m) of f(x) on the interval [0, 2] is 0, and the absolute maximum (M) is approximately 1.8393.
Now we can use the property mentioned earlier to estimate the value of the integral:
m(b - a) ≤ ∫(a to b) f(x) dx ≤ M(b - a)
For our function f(x) = 5xe^(-x) and the interval [0, 2]:
0(2 - 0) ≤ ∫(0 to 2) 5xe^(-x) dx ≤ 1.8393(2 - 0)
Simplifying:
0 ≤ ∫(0 to 2) 5xe^(-x) dx ≤ 3.6786
Therefore, we can estimate that the value of the integral ∫(0 to 2) 5xe^(-x) is between 0 and 3.6786.
To estimate the value of the integral from 0 to 2 of 5xe^(-x), we can use the property mentioned: m(b - a) ≤ ∫[a, b] f(x)dx ≤ M(b - a).
First, we need to find the absolute minimum and maximum values of f(x) = 5xe^(-x) on the interval [0, 2].
To find the minimum, we can take the derivative of f(x) with respect to x and set it equal to zero to find the critical points:
f'(x) = 5e^(-x) - 5xe^(-x) = 0
Factor out e^(-x):
5e^(-x)(1 - x) = 0
Set each factor equal to zero:
e^(-x) = 0 (not possible)
1 - x = 0
Solving for x, we get x = 1.
To determine if this critical point is a minimum, we can check the second derivative:
f''(x) = 5e^(-x) + 5xe^(-x)
At x = 1:
f''(1) = 5e^(-1) + 5e^(-1) > 0
Since the second derivative is positive, x = 1 is a local minimum.
Next, we need to find the maximum. Since f(x) is continuous on the closed interval [0, 2], we can evaluate f(x) at the endpoints and the critical point:
f(0) = 5(0)e^(-0) = 0
f(1) = 5(1)e^(-1) = 5e^(-1)
f(2) = 5(2)e^(-2) = 10e^(-2)
Comparing these values, we see that the maximum occurs at x = 0 (in this case, the absolute maximum is also the endpoint maximum).
Therefore, m = 0 and M = 5e^(-1).
Now, let's estimate the value of the integral using the property mentioned:
m(b - a) ≤ ∫[a, b] f(x)dx ≤ M(b - a)
0(2 - 0) ≤ ∫[0, 2] 5xe^(-x)dx ≤ 5e^(-1)(2 - 0)
0 ≤ ∫[0, 2] 5xe^(-x)dx ≤ 10e^(-1)
Simplifying:
0 ≤ ∫[0, 2] 5xe^(-x)dx ≤ 10e^(-1)
Now, we can evaluate the definite integral:
∫[0, 2] 5xe^(-x)dx = [-5xe^(-x) - 5e^(-x)] evaluated from 0 to 2
= (-10e^(-2) - 10e^(-2)) - (0 - 5e^(-0) - 5e^(-0))
= -20e^(-2) + 10
Since we know that 0 ≤ ∫[0, 2] 5xe^(-x)dx ≤ 10e^(-1), we can estimate the value of the integral to be between 0 and 10e^(-1).
Note: The exact value of the integral can be found using calculus techniques, such as integration by parts.