Linear functions can be used to approximate more

Chapter 4, Problem 36

(choose chapter or problem)

Linear functions can be used to approximate more complicated functions. This is one of the meanings or implications of the quotation by Professor Gleason on page 213. This exercise illustrates that idea. (a) Using calculus, it can be shown that the equation of the line that is tangent to the curve y x2 at the point (1, 1) is y 2x 1. Verify this visually by graphing the two functions y x2 and y 2x 1 on the same set of axes. (Suggestion: Use a viewing rectangle that extends from 2 to 3 in the x-direction and from 3 to 4 in the y-direction.) Note that the tangent line is virtually indistinguishable from the curve in the immediate vicinity of the point (1, 1). (b) For numerical rather than visual evidence of how well the linear function y 2x 1 approximates the function y x2 in the immediate vicinity of (1, 1), complete the following tables. x 0.9 0.99 0.999 x2 2x 1 x 1.1 1.01 1.001 x2 2x 1