Math with Mrs. Sarah

(Finney, AP Calculus BC)

1. Find a formula for the truncation error if we use P₆(x) to approximate 1/(1−2x) on the interval (−½, ½).
2. Use the Remainder Bounding Theorem to prove that the Maclaurin series for f(x) = sin5x converges to f(x) for all ℝ.
3. Suppose you use P₂(x) = 1 − x²/2 to approximate cos(0.5). Give a bound on the error based on:
   a) the alternating series bound
   b) the Lagrange Error Formula
4. The hyperbolic sine and hyperbolic cosine functions are defined as:
   sinh x = (eˣ − e⁻ˣ)/2      cosh x = (eˣ + e⁻ˣ)/2
   Find the Maclaurin series generated by sinh x and cosh x.
5. Use the Remainder Bounding Theorem to prove that cosh x equals its Maclaurin series for all real numbers x.
6. Find the linearization and the quadratic approximation of f(x) = sec x at x = 0.
7. Given f(x) = sin(5x + π/4), let P₃(x) be the third degree Taylor polynomial for f about x = 0.
   a) Find P(x).
   b) Use the Lagrange error bound to show that |f(1/10) − P₃(1/10)| < 1/100
   c) Let G(x) = ∫₀ˣ f(t) dt. Write the third-degree Taylor polynomial for G about x = 0.

1. Find a formula for the truncation error if we use P6(x) to approximate 1/(1-2x) on the interval (-1/2, 1/2).
2. Use the Remainder Bounding Theorem to prove that the Maclaurin series for f(x) = sin(5x) converges to f(x) for all real numbers.
3. Suppose you use P2(x) = 1 – x^2/2 to approximate cos(0.5). Give a bound on the error based on:
   a) the alternating series bound
   b) the Lagrange Error Formula
4. The hyperbolic sine and hyperbolic cosine functions are defined as:
   sinh(x) = (e^x – e^(-x))/2 cosh(x) = (e^x + e^(-x))/2
   Find the Maclaurin series generated by sinh(x) and cosh(x).
5. Use the Remainder Bounding Theorem to prove that cosh(x) equals its Maclaurin series for all real numbers x.
6. Find the linearization and the quadratic approximation of f(x) = sec(x) at x = 0.
7. Given f(x) = sin(5x + pi/4), let P3(x) be the third degree Taylor polynomial for f about x = 0.
   a) Find P(x).
   b) Use the Lagrange error bound to show that |f(1/10) – P3(1/10)| < 1/100
   c) Let G(x) = integral from 0 to x of f(t) dt. Write the third-degree Taylor polynomial for G about x = 0.