Question

Given the region bounded by the function (y = ln(x)/sinleft(x/5right)) on the interval ([11,14]):

a) Estimate the arc length of (y=f(x)) on the given interval.
b) Determine the exact arc length.
c) The average value must be between ___ₐnd ____ (min and max).
d) What is the average value?
e) The mean value theorem states that there must be at least one (x)-value (c) in ([11,14]) such that (fₐᵥₑ = f(c)). Use technology to find the values. (x = ?)

Answer 1

To estimate the arc length, divide the interval into subintervals. To find the exact arc length, use the formula for arc length with the derivative of the function. The average value must be between the function's minimum and maximum values. The mean value theorem states there is an x-value where the average value of the function is equal to its value.

Step-by-step explanation:

To estimate the arc length of the function y = ln(x)/sin(x/5) on the interval [11, 14], we can divide the interval into small subintervals and calculate the lengths of the corresponding line segments. The more subintervals we use, the more accurate the estimate will be.

To determine the exact arc length, we need to use calculus. We can use the formula for arc length: L = ∫ sqrt(1 + (dy/dx)^2) dx, where dy/dx represents the derivative of the function.

The average value of the function must be between its minimum and maximum values on the interval. To find the average value, we need to calculate the definite integral of the function over the interval and divide it by the length of the interval.

The mean value theorem states that there must be at least one x-value in the interval [11, 14] where the average value of the function is equal to the function's value at that x-value.