Taylor Polynomials

  1. Find the degree 3 Taylor polynomial centered at \(a=1\) for \(f(x)=x^3-4x-7\). Then multiply out and simplify your answer. (Did the result surprise you?) Solution ↗
  2. Find the degree 2 Taylor polynomial for \(f(x) = x^4-x+1\) centered at \(a=-1\). Solution ↗
  3. Consider the function \(f(x)=\sqrt[3]{x}\).
    (a) Find the degree 2 Taylor polynomial, \(T_2(x)\), for \(f\) centered at \(a=8\).
    (b) Use the polynomial you found in part (a) to approximate the value of \(\sqrt[3]{7}\). Solution ↗
  4. Find the degree 0, degree 1, degree 2, and degree 3 Taylor polynomials centered at \(a=4\) for the function given by \(f(x)=\sqrt{x}\). Use each of these to get successively better approximations for \(\sqrt{5}\). Solution ↗
  5. Find the degree 3 Taylor polynomial, \(T_3(x)\), for \(f(x)=e^{-x}\, \sin(x)\) centered at \(a=0\). Solution (Method 1) ↗ Solution (Method 2) ↗
  6. Find the degree 3 Taylor polynomial, \(T_3(x)\), for \(f(x)=xe^{-4x}\) centered at \(a=0\). Solution ↗