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 1  solution 2

6. Find the degree 3 Taylor polynomial, \(T_3(x)\), for \(f(x)=xe^{-4x}\) centered at \(a=0\).   solution