FEED THE TIGER

We illustrate the power of linear approximation by returning to the story of the Tiger Who Eats Square Roots.

• We are stranded on a desert island with no calculator.
• We must provide the Tiger with decimal approximations of square roots of numbers near 100.

We use the best linear approximation to the square root function near 100 to get out of this mess. For values of x near 100,

sqrt(x) 10 + (x - 100)/20

This approximation of sqrt(x) is just the one provided by the linear function L₁₀₀(x) = 10+(x-100)/20. Now here's the point. Notice that if you are given a value of x, it's really EASY to compute 10+(x-100)/20: you just take your number, subtract 100, divide by 20, and then add 10. End of story. So quick, what's a good approximation for sqrt(99)?

Now quick, use L₁₀₀ to complete the next three approximations in under a minute. (Don't use your calculator!)

sqrt(102) ____ sqrt(101) ____ sqrt(97.8) ____

Using L₁₀₀(x) to approximate sqrt(x), you will find approximations are well within 0.001 of the desired square root. Now go ahead and verify the accuracy of the linear approximations with your calculator.

And here's one more just for fun...or is it?

sqrt(901) ____

To augment this discussion of linear approximations, we will briefly introduce the concept of quadratice approximations.