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Chapter 4: Problem 34

Interpret the Mean Value Theorem when it is applied to any linear function.

Short Answer

Expert verified
Answer: When the Mean Value Theorem is applied to a linear function, it confirms that the slope of the linear function is constant throughout its domain.

Step by step solution

01

Understanding the Mean Value Theorem

The Mean Value Theorem states that if a continuous function f(x) is differentiable on the open interval (a, b) and continuous on the closed interval [a, b], then there exists at least one point c in the open interval (a, b) such that: f'(c) = (f(b) - f(a)) / (b - a)
02

Finding a linear function example

Consider a linear function, f(x) = mx + c, where m and c are constants. This function is continuous and differentiable for all real x-values.
03

Applying the Mean Value Theorem to the linear function

As the function is continuous and differentiable, we can apply the Mean Value Theorem: f'(c) = (f(b) - f(a)) / (b - a) First, we need to find the derivative f'(x) of f(x) = mx + c, which is: f'(x) = m Now let's find f(b) - f(a): f(b) - f(a) = m*b + c - (m*a + c) = m*(b-a) Thus, we can now write the equation as: m = m*(b-a) / (b - a)
04

Analyzing the result

Since the function is linear, the slope m is constant throughout the entire function. Hence, the Mean Value Theorem holds true for all values of c in the open interval (a, b), as the derivative (slope) is the same everywhere. In conclusion, the Mean Value Theorem, when applied to a linear function, confirms that the slope of the linear function is constant throughout its domain.

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