Straight-Line Estimate: Tangent-Line Guesses for Square Roots, Squares and Cubes, the Percent Rule and Newton's Method

Home › Calculators › Straight-Line Estimate

Opens on Chapter 16’s example: √50 from √49 = 7; a 12-inch pizza against a 14-inch; √10 by Newton’s method from a guess of 3.

1. A tangent-line guess

Function

Near a point you know

Guess the value at

Load another example from the book

2. The percent rule

Old length inches

New length

Power 2 = an area, 3 = a volume

3. Two percent changes at once

A bill now

Price change

Amount change

4. Newton’s method for a square root

Square root of

First guess

Steps

Enter your numbers, then press Work it out.

How it works

The tangent line at a is f(a) + f′(a) × (x − a): the value you know, plus the slope there times how far you step. For √x the slope is 1 ÷ (2√a), so √50 ≈ 7 + 1/14. The same idea in percent form: a small percent change in a length makes about twice that percent change in an area, and three times in a volume (the exact change is a little more, because the curve bends up). Newton's method uses the tangent over and over: for √N, the next guess is x − (x² − N) ÷ 2x, and the number of correct digits roughly doubles each step.

From the book, Chapter 16. √50 ≈ 7.0714 (true 7.0711); √66 ≈ 8.125; but √80 from 49 gives 9.21, off by 0.27. The 14-inch pizza is 16.7% wider: the straight-line rule says 33.3% more pizza, the truth is 36.1%. Newton from 3: 3.1667, then 3.16228. 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.