A function is defined and a closed interval is given. Decide...

Name: Problem 2, Chapter 3: Calculus
Uploaded: 2021-06-14T20:50:42-08:00
Duration: 24 s
Channel: Donald Albin
Description: A function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of c ; if not, state the reason. In each problem, sketch the graph of the given function on the given interval. g(x)=|x| ;[-2,2]

A function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of $c ;$ if not, state the reason. In each problem, sketch the graph of the given function on the given interval. $$g(x)=|x| ;[-2,2]$$

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Key Concepts

Differentiability

Differentiability refers to the existence of a derivative at every point within an interval. A function must be smooth, without sharp corners or cusps, on an open interval in order to meet the conditions of the Mean Value Theorem.

Absolute Value Function

The absolute value function is a common example in calculus that is continuous everywhere but has a sharp corner at the origin. This corner point makes the function non-differentiable there, illustrating how even continuous functions might fail to meet the differentiability criterion required for the Mean Value Theorem.

Mean Value Theorem

The Mean Value Theorem (MVT) is a fundamental result in calculus which asserts that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within that interval where the function's instantaneous rate of change (its derivative) equals its average rate of change over the entire interval.

Continuity

Continuity is a key property in calculus that ensures a function has no breaks, jumps, or holes in its graph over a specified interval. It is a necessary condition for many theorems, including the Mean Value Theorem, to be applicable.

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