Discovering Taylor Polynomials
This exploration is meant to help make connections between Taylor polynomials and four different functions.
- Use the sliders to change the Degree of the polynomial, the center of approximation, as well one of the four different polynomials that you’ll need to find.
- Set n=1 and change the degree of the Taylor polynomial. Find the a general term for the Taylor polynomial that is generated when centered at x=0.
- ]Which of the four functions (f(x)=sin(x), f(x)=cos(x), f(x)=e^x, f(x)=ln(x+1), and f(x)=1/(x+1)) best matches the Taylor polynomial? Confirm by finding the 4th degree “tangent function” by hand.
- What appears to be the values of x where there is no error as the degree goes to infinity (use "show error")? Does the center of approximation have an effect on the values of x? If yes, what is the effect?
- Repeat for the functions for n=2, 3, and 4.