Average Value of a Function

AP Calculus · Axiom Academy

LESSON Average Value of a Function Finding the mean height of a curve over an interval Step 1: From Discrete to Continuous You already know how to average a finite set of numbers: add them up and divide by how many there are. But what does it mean to find the average value of a continuous function over an interval? Start by sampling equally spaced values of on . The average of those samples is: As , the spacing shrinks to zero and the sum becomes a Riemann integral. The factor turns into , giving us the continuous average. Step 2: The Average Value Formula The average (mean) value of a continuous function on the interval is: Geometric interpretation: is the height of a rectangle with base whose area equals the area under from to . Step 3: Mean Value Theorem for Integrals The Mean Value Theorem for Integrals guarantees that a continuous function actually attains its average value somewhere on the interval. Why must such a exist? By the Intermediate Value Theorem, a continuous function takes on every value between its minimum and maximum. Since the average lies between the min and max, the function must hit that average value at some point. Problem: Find the average value of on the interval . Then find the value guaranteed by the MVT for Integrals. Step 2: Evaluate the integral. Step 5: AP Exam Tips & Common Mistakes Average value questions appear frequently on both the multiple-choice and free-response sections. Here are the patterns you need to know:

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