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:
the mean slope is just (1/11 - 1/4)/(11-4) = -1/44
so, where is f'(x) = -1/44
To find the average or mean slope of the function f(x) = 1/x on the interval [4, 11], we need to calculate the slope of the function between these two values.
(A) Average Slope:
The average slope can be calculated using the formula: (f(b) - f(a)) / (b - a), where (a, f(a)) and (b, f(b)) are two points on the function.
In this case, our interval is [4, 11]. Let's choose two points within this interval, such as (4, 1/4) and (11, 1/11).
Using the slope formula, we have:
Average slope = (1/11 - 1/4) / (11 - 4)
= (4/44 - 11/44) / 7
= (-7/44) / 7
= -1/44
Therefore, the average slope of the function on the interval [4, 11] is -1/44.
(B) Using the Mean Value Theorem, we know that there exists a point c in the open interval (4, 11) such that f'(c) is equal to the mean slope we calculated in part (A).
To find the value of c, we need to find the derivative of the function f(x) = 1/x and set it equal to the mean slope.
The derivative of f(x) = 1/x is given by:
f'(x) = -1/x^2
Setting f'(x) equal to -1/44, we have:
-1/x^2 = -1/44
Cross-multiplying, we get:
44 = x^2
Taking the square root of both sides, we have:
x = ±√44 = ±2√11
Since c must lie in the open interval (4, 11), only the positive square root is within that interval.
Therefore, the value of c that satisfies f'(c) = -1/44 is c = 2√11.
The list of values for c is: 2√11.