Average vs Instantaneous Rate of Change: 3 Ways to Estimate | MHF4U

300 km in 3 hours is 100 km/h on average, but your speedometer says something different every second. That's the difference between the average and the instantaneous rate of change, and here are three ways to find the second one. In this lecture: • Average rate of change from x = a to x = b: (f(b) − f(a)) / (b − a), the slope of the secant line → y = x² from 0 to 3: 9 / 3 = 3 • Instantaneous rate of change: the slope of the tangent line. At one point we get 0/0, so we estimate • Method 1, preceding and following intervals at x = 2: m₁ = (4 − 2.25)/(2 − 1.5) = 3.5, m₂ = (6.25 − 4)/(2.5 − 2) = 4.5, average (3.5 + 4.5)/2 = 4 • Method 2, the centred interval [1.5, 2.5]: (6.25 − 2.25)/1 = 4 • Method 3, shrinking intervals: 3, 3.5, 3.9, 3.99, 3.999 from the left and 5, 4.5, 4.1, 4.01, 4.001 from the right, both heading to 4. So the instantaneous rate of change at x = 2 is 4 (and at a sharp corner, where the two sides disagree, there's no single answer) Try this one: estimate the instantaneous rate of change of y = x³ at x = 1, using the centred interval from 0.9 to 1.1. (Answer below, after the spoiler gap) . . . . . Answer: (1.331 − 0.729)/(1.1 − 0.9) = 0.602/0.2 = 3.01. The exact answer is 3: very close, but not exact. Ontario: MHF4U (Grade 12 Advanced Functions), rates of change TIMELINE 0:00 Intro 0:06 The Trip and the Speedometer 0:46 The Average Rate of Change 1:42 The Instantaneous Rate of Change 2:45 Method 1: Preceding and Following 4:26 Method 2: The Centred Interval 5:20 Method 3: Shrink the Intervals 7:07 Recap and Your Turn ━━━━━━━━━━━━━━━━━━━━ Dr Song Square Academy in Toronto, Ontario Website: www.drsongsquare.com Contact / enquiries: contactus@drsongsquare.com Music by Bensound.com/royalty-free-music License code: RDWONSRMYBO8RU89 Artist: : Benjamin Tissot #math #MHF4U #rateofchange