The function defined below satisfies the Mean Value Theorem on the given interval. Find the value of c in the interval (1, 2) where f'(c)=(f(b) - f(a))/(b - a).
f(x) = 1.5x-1 + 1.1 , [1, 2]
Round your answer to two decimal places.
To find the value of c in the interval (1, 2) where f'(c) satisfies the Mean Value Theorem, we need to find the derivative of f(x) first.
Given f(x) = 1.5x - 1 + 1.1
To find the derivative, f'(x), we differentiate each term separately. The derivative of a constant is always 0. The derivative of 1.1 is 0.
Differentiating 1.5x - 1 gives us:
f'(x) = 1.5
Now that we have the derivative, we can find the value of c in the interval (1, 2) where f'(c) equals the average rate of change of f(x) over the interval [1, 2].
The average rate of change of f(x) over the interval [1, 2] is:
(f(2) - f(1))/(2 - 1)
To calculate this, we substitute the x-values into the function:
f(2) = 1.5(2) - 1 + 1.1 = 3.5
f(1) = 1.5(1) - 1 + 1.1 = 1.6
Therefore, the average rate of change is:
(3.5 - 1.6)/(2 - 1) = 1.9
Now, we can set f'(c) equal to the average rate of change and solve for c:
1.5 = 1.9
Since no value of c satisfies this equation, there is no value of c in the interval (1, 2) where f'(c) satisfies the Mean Value Theorem.
Therefore, the answer is that there is no such value of c.
To find the value of c in the interval (1, 2) that satisfies the Mean Value Theorem, we need to find the derivative of the function f(x) and then find the value of c for which the derivative is equal to the average rate of change of f(x) over the interval (1, 2).
Step 1: Find the derivative of f(x)
Differentiate f(x) = 1.5x - 1 + 1.1 with respect to x to find the derivative, f'(x).
f'(x) = 1.5
Step 2: Find the average rate of change of f(x) over the interval (1, 2)
Calculate the difference in the values of f(x) at the endpoints of the interval, f(2) - f(1), and divide it by the difference in the values of x at the endpoints, 2 - 1.
f(2) = 1.5(2) - 1 + 1.1 = 2.8
f(1) = 1.5(1) - 1 + 1.1 = 1.6
Average rate of change = (f(2) - f(1))/(2 - 1) = (2.8 - 1.6)/1 = 1.2
Step 3: Set f'(c) equal to the average rate of change and solve for c
Set f'(x) = 1.5 equal to the average rate of change, 1.2, and solve for c.
1.5 = 1.2
As you can see, the equation 1.5 = 1.2 has no solution. Therefore, there is no value of c in the interval (1, 2) that satisfies the Mean Value Theorem for this particular function.
so, just do the math. Find c where
f(2)-f(1) = f'(c) = 1.5
I think your function is bogus. No one would write f(x) like that.