Average Value of Functions

Real Analysis · Axiom Academy

LESSON Average Value of Functions The average of a function is the height of the equal-area rectangle — and the Mean Value Theorem guarantees the function actually reaches it. The average of n samples is their sum over n . Slice [a,b] into many pieces, sample f on each, and average: as the pieces shrink, the sum of values times widths becomes an integral, and the count n becomes the length b-a . The picture. is the height of the rectangle over [a,b] holding the same area as the region under y=f(x) . The animation flattens the area under the curve into that equal-area bar. Let's compute a real one. Integrate from 0 to , then divide by the interval length . 3. The Mean Value Theorem for Integrals That guarantee isn't luck — it's a theorem. A continuous function is squeezed between its minimum m and maximum M , its average must land in that band, and continuity then forces the function to hit the average somewhere. By the Extreme Value Theorem, for all x in [a,b] . Dividing by b-a traps the average: . The IVT: a continuous f hits every value between m and M — so some c gives . Average velocity is total displacement over time elapsed; expected value of a continuous random variable is a density-weighted average; RMS / average power integrates the instantaneous signal over a period. All are this one idea: integrate, then divide by the span. You've turned an average into an integral, computed it on a real function, and proved the function must reach it.

This is the written version of the interactive lesson above. See the full Real Analysis course.