Unit 12: Averages, area and slope
About this unit
The average of a function over [a, b] is its integral divided by b − a, which is also the slope of the straight line joining two points of its antiderivative.
By the end, you can: Find the average value of a function over an interval with an integral, and see why it equals the slope of the antiderivative's secant line.
Modules
5 modulesLearn
- Averaging a whole curveArticle · Average infinitely many values by sampling evenly, which is a Riemann sum divided by the length; the limit is (1/(b − a))∫f, the height of the rectangle with the same area.Not started
- The average is a slopeArticle · With F′ = f, (1/(b − a))∫f = (F(b) − F(a))/(b − a), the slope of F's secant: average speed is total distance over time. So f hits its average somewhere.Not started
- Find the intervalExperiment · Slide the secant of F(x) = x³/3 until its slope, the average of x² over [0, h], is 4/3.Not started
- Averages in one placeReview · Average = integral over length = secant slope of the antiderivative; the function equals its average somewhere; don't forget to divide.Not started