Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.4.56

Chapter 5, Problem 5.4.56

Average value of the derivative Suppose ƒ ' is a continuous function for all real numbers. Show that the average value of the derivative on an interval [a, b] is ƒ⁻' = (ƒ(b) ―ƒ(a))/ (b―a) . Interpret this result in terms of secant lines.

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Step 1: Recall the Mean Value Theorem (MVT) for derivatives, which states that if a function ƒ is continuous on [a, b] and differentiable on (a, b), then there exists a point c in (a, b) such that ƒ'(c) = (ƒ(b) - ƒ(a)) / (b - a). This theorem provides the foundation for the average value of the derivative.

Step 2: Define the average value of the derivative over the interval [a, b]. The average value of ƒ'(x) on [a, b] is given by the integral formula: (1 / (b - a)) ∫[a, b] ƒ'(x) dx. This formula calculates the mean rate of change of the function over the interval.

Step 3: Apply the Fundamental Theorem of Calculus to the integral ∫[a, b] ƒ'(x) dx. According to this theorem, the integral of the derivative of a function over an interval is equal to the net change in the function over that interval: ∫[a, b] ƒ'(x) dx = ƒ(b) - ƒ(a).

Step 4: Substitute the result from Step 3 into the average value formula. Replace ∫[a, b] ƒ'(x) dx with ƒ(b) - ƒ(a) to get: (1 / (b - a)) ∫[a, b] ƒ'(x) dx = (ƒ(b) - ƒ(a)) / (b - a). This shows that the average value of the derivative is equal to the slope of the secant line connecting the points (a, ƒ(a)) and (b, ƒ(b)).

Step 5: Interpret the result in terms of secant lines. The average value of the derivative on [a, b] represents the slope of the secant line between the endpoints of the interval. This means that the average rate of change of the function over the interval is equivalent to the slope of the straight line connecting the points (a, ƒ(a)) and (b, ƒ(b)) on the graph of ƒ.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Average Value of a Function

The average value of a function over an interval [a, b] is defined as the integral of the function over that interval divided by the length of the interval. For a continuous function f, this is expressed mathematically as (1/(b-a)) * ∫[a to b] f(x) dx. This concept is crucial for understanding how the behavior of a function can be summarized over a specific range.

Derivative and Its Interpretation

The derivative of a function at a point measures the rate of change of the function with respect to its variable. It is represented as f'(x) and can be interpreted as the slope of the tangent line to the curve at that point. Understanding derivatives is essential for analyzing how functions behave locally and for connecting to the concept of average rates of change over intervals.

Secant Lines

A secant line is a straight line that connects two points on a curve, representing the average rate of change of the function between those two points. The slope of the secant line between points (a, f(a)) and (b, f(b)) is given by (f(b) - f(a)) / (b - a). This concept helps visualize the average value of the derivative, as the slope of the secant line approximates the derivative as the interval narrows.

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