Question

Recall the mean value theorem (MVT), which states: If f is continuous on closed interval [a, b] and differentiable on (a, b), then there is at least one point c (a, b) such that

(f(b) - f(a)) / (b - a) = f'(c)
Use this theorem to answer the following questions.
(a) What is the slope of the secant line passing through (a, f(a)) and (b, f(b))?

Answer 1

The slope of the secant line through the points (a, f(a)) and (b, f(b)) is calculated by the ratio (f(b) - f(a)) / (b - a), which is equal to the slope of the tangent line at some point c within that interval as per the Mean Value Theorem.

Step-by-step explanation:

The slope of the secant line that passes through the points (a, f(a)) and (b, f(b)) is found by taking the ratio of the difference in y-values to the difference in x-values.

This is expressed as (f(b) - f(a)) / (b - a), which is essentially the average rate of change of the function f on the interval [a, b].

According to the Mean Value Theorem (MVT), this slope is also equal to the slope of the tangent line to the curve at some point c within the interval (a, b), where the tangent line is represented by the derivative f'(c).