Calculus: An Integrated Approach to Functions and their Rates of Change, Preliminary Edition Part 95. A major complaint of professors teaching calculus is that students don't have the appropriate background to work through the calculus course successfully. This text is targeted directly at this underprepared audience. This is a single-variable (2-semester) calculus text that incorporates a conceptual re-introduction to key precalculus ideas throughout the exposition as appropriate. This is the ideal resource for those schools dealing with poorly prepared students or for schools introducing a slower paced, integrated precalculus/calculus course | Approximating a Function by a Polynomial 921 Figure Refining the tangent line approximation Any polynomial approximation Pk x of sin x about x 0 certainly ought to be as good a local approximation as is the tangent line approximation P1 x x. Therefore like sin x the graph of Pk x must pass through 0 0 and must have a slope of 1 at x 0. This means that Pk 0 0 Pk 0 1. and Pk x a0 a1x a2x2 H-----F akxk so Pk 0 a0 0. Pk x a1 2a2x 3a3x2 kakxk 1 so Pk 0 a1 1. Therefore Pk x is of the form x a2x2 a3x3 akxk. Note that sin x lies below the tangent line for x 0 and above the tangent line for x 0. Therefore the approximation sin x x must be decreased for x 0 and increased for x 0 in order to improve upon it. Second Degree Approximation P2 x must be of the form x a2x2 where a2 is a constant. But a2x2 cannot be negative for x 0 and positive for x 0 as required we cannot improve upon the tangent line approximation by using a second degree polynomial. We need at least a third degree polynomial to improve upon the tangent line approximation. 922 CHAPTER 30 Series Before moving on let s look at the second degree polynomial approximation from a geometric viewpoint. If P2 x 0 fix a2x2 is to be the best parabolic approximation to x sin x about x 0 then it must satisfy the following three conditions. It has the same value as sin x at x 0. P2 0 0 It has the same slope as sin x at x 0. P2 0 0 It has the same concavity as sin x at x 0. 2 0 0 Each one of these conditions determines the value of one coefficient of P2 x . The first two result in a0 0 and 1 respectively. The second derivative of sin x at x 0 is zero. d 2 7 sin x dx2 cos x sin x 0 x 0 dx x 0 x 0 The parabola must have a second derivative of zero consequently it is not a parabola at all. Third Degree Approximation To determine the coefficients of the third degree polynomial of best fit we require that the polynomial P3 x a0 a1x a2x2 a3x3 and x sin x agree at x 0 and that each nonzero derivative of the polynomial is .
