Taylor Series Assignment.sagews

Taylor Series Assignment

Question 0

Watch the lecture video here.

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Question 1

Use Taylor polynomials to approximate $\pi$ using the following steps:

• $A=\displaystyle\int_0^1 \frac{1}{1+x^2}\, dx=\arctan(1)-\arctan(0)=\frac{\pi}{4}$

• $T(x)=$ Taylor polynomial of degree 100 of $\displaystyle\frac{1}{1+x^2}$ centered at $x=0$

• $B=\displaystyle\int_0^1 T(x)\, dx$

• Since $A$ and $B$ are approximately equal, $\pi\approx 4B$. So calculate $4B$ and convert to a decimal.

Question 2

Estimate the value of $\displaystyle\int_0^1 e^{-x^2}\, dx$ as follows:

• Define $T20(x)=$ the Taylor polynomial of degree 20 of $e^{-x^2}$ centered at $x=0$.

• Calculate $\displaystyle\int_0^1 T20(x)\, dx$.

• Define $T50(x)=$ the Taylor polynomial of degree 50 of $e^{-x^2}$ centered at $x=0$.

• Calculate $\displaystyle\int_0^1 T50(x)\, dx$.

• Compare your results with the output from Sage's numerical_integral command: $0.746824132812427$. [Use the N() command to convert to decimals.]

Question 3

Let $\displaystyle f(x)=e^{\sin(x)}$, $T5(x)=$ the 5th-degree Taylor polynomial of $f$ centered at $x=\pi$, and $T10(x)=$ the 10th-degree Taylor polynomial of $f$ centered at $x=\pi$.

Graph all three on the window $0\le x \le 2\pi$, $0\le y \le 3$. Use black for $f$, blue for $T5$, and red for $T10$.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.