Consider the function f(x) = 1/x on the interval [ 4 , 11 ].
(A) Find the average or mean slope of the function on this interval.
Average Slope =_______
(B) By the Mean Value Theorem, we know there exists a c in the open interval ( 4 , 11 ) such that f'( c) is equal to this mean slope. Find all values of c that work and list them (separated by commas) in the box below.
List of values:______
I do not need the practice. If you have a specific problem fine. However we have helped you with several virtually identical problems and you do not seem to be learning from them.
To find the average slope of the function f(x) = 1/x on the interval [4, 11], we need to calculate the change in y (Δy) divided by the change in x (Δx) within this interval.
(A) The average slope is given by the formula:
Average Slope = (f(11) - f(4)) / (11 - 4)
To calculate this, we need to find the values of f(11) and f(4).
f(11) = 1/11
f(4) = 1/4
Substituting the values into the formula, we have:
Average Slope = (1/11 - 1/4) / (11 - 4)
Now, simplify the expression:
Average Slope = (4 - 11)/(11 * 4) / (11 - 4)
= -7/44
Therefore, the average slope of the function on the interval [4, 11] is -7/44.
(B) According to the Mean Value Theorem, there exists at least one value, c, in the open interval (4, 11) such that f'(c) is equal to the mean slope we just calculated. The function f(x) = 1/x has a derivative of f'(x) = -1/x^2.
To find the values of c that satisfy f'(c) = -7/44, we need to solve the equation:
-1/c^2 = -7/44
To simplify, cross-multiply and solve for c:
-44 = -7c^2
7c^2 = 44
c^2 = 44/7
c = ±√(44/7)
Evaluating the square root, we get:
c ≈ ±1.894
Therefore, the values of c that satisfy f'(c) = -7/44 are approximately 1.894 and -1.894.