Problem 36 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Answer: When applied to a linear function, the Mean Value Theorem implies that the average rate of change is constant and equal to the slope throughout the entire interval. This means that the instantaneous rate of change (derivative) and the average rate of change will always be equal for linear functions.

Step 1: Understanding the Mean Value Theorem

The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in the interval (a, b) such that the derivative of the function at c (f'(c)) is equal to the average rate of change of the function over the interval [a, b]. Mathematically, it can be written as: f'(c) = \frac{f(b) - f(a)}{b - a}

Step 2: Form of a Linear Function

A linear function can generally be written in the form: f(x) = mx + b where m is the slope and b is the y-intercept.

Step 3: Derivative of a Linear Function

Now we will find the derivative of the linear function. Since the linear function is a polynomial of degree 1, it is differentiable everywhere. Thus, its derivative is: f'(x) = \frac{d}{dx}(mx + b) = m

Step 4: Applying the Mean Value Theorem to the Linear Function

Applying the Mean Value Theorem to the linear function, we have: f'(c) = \frac{f(b) - f(a)}{b - a} Substituting the linear function and its derivative into the equation, we get: m = \frac{(mb + b) - (ma + b)}{b - a}

Step 5: Interpreting the Result

Simplifying the equation, we get m = m, which holds for any value of m. This implies that for any linear function, the Mean Value Theorem holds for every point c in the interval (a, b). In other words, for linear functions, the average rate of change is constant and equal to the slope throughout the entire interval. This result makes intuitive sense since, in a straight line, the slope is constant, and thus, the instantaneous rate of change (derivative) and the average rate of change will always be equal.