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

Chapter 10, Problem 62

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(a) Calculate the equation of the tangent line to the functionf(x) = x2 at x = 2. Do the same calculationfor g(x) = x3 4x2 + 8x 7 at x = 1 and forh(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 formf(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 youranswers to part (a) with the remainder, r(x). Whatdo you notice? Make a conjecture about the tangentline to a polynomial p(x) at the point x = a andthe remainder, r(x), obtained from dividing p(x) by(x a)2.(d) Use the Taylor expansion of p(x) about x = a toprove your conjecture.10