Average Rate Finder: Average Speed Done Right, Secant Slopes and Average-Speed Cameras

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Opens on Chapter 5’s example: 30 miles at 30 mph and 30 at 60; a camera pair 3 miles apart, 2 min 40 s; a ball falling d = 16t².

1. Between two points

First time

Value then

Second time

Value then

Load another example from the book

2. A trip in legs

Legs: miles and mph, one a line

Or: a route of (miles)

at an average of (mph)

3. An average-speed camera

Miles between cameras

Minutes

and seconds

4. Secant slopes on a curve

Formula feet, t in seconds

From t =

to t = one or more

Enter your numbers, then press Work it out.

How it works

Average speed is total distance ÷ total time, never the average of the speeds: slow legs take longer, so they count for more. On a graph the average rate between two points is the slope of the secant, the straight line through them. The mean value theorem says, in plain words, that on a smooth trip you were going at exactly your average speed at least once; that's why an average-speed camera's reading is a fair one. Shrink the interval toward one point and the secant slopes settle on the slope at that point: Chapter 6's idea.

From the book, Chapter 5. The ball falls 16 feet by 1 second and 144 by 3: 64 ft/s on average. $4,000 growing to $4,860 in 3 years is $286.67 a year on average. Route 9 at 15 mph would take 56 minutes. 30 miles at 30 and 30 at 60 take 1.5 hours: 40 mph, not 45. Three miles in 160 seconds is 67.5 mph. The ball averages 64 ft/s from 1 to 3 seconds; from 2 to 3, 2.5 and 2.1 the secant slopes are 80, 72 and 65.6. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.