(a) Calculate the equation of the tangent line to the function f(x) = x2 at x = 2. Do

Chapter 0, Problem 62

(choose chapter or problem)

(a) Calculate the equation of the tangent line to the function f(x) = x2 at x = 2. Do the same calculation for g(x) = x3 4x2 + 8x 7 at x = 1 and for h(x)=2x3 + 4x2 3x + 7 at x = 1. (b) Use a computer algebra system to divide f(x) by (x 2)2, giving your result in the form f(x) (x 2)2 = q(x) + r(x) (x 2)2 , where q(x) is the quotient and r(x) is the remainder. In addition, divide g(x) by (x 1)2 and h(x) by (x + 1)2. (c) For each of the functions, f, g, h, compare your answers to part (a) with the remainder, r(x). What do you notice? Make a conjecture about the tangent line to a polynomial p(x) at the point x = a and the remainder, r(x), obtained from dividing p(x) by (x a) 2. (d) Use the Taylor expansion of p(x) about x = a to prove your conjecture.10