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Aaron Tresham Calculus Materials - Feb 2018 snapshot

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Taylor Series Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

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

  • A=0111+x2dx=arctan(1)arctan(0)=π4A=\displaystyle\int_0^1 \frac{1}{1+x^2}\, dx=\arctan(1)-\arctan(0)=\frac{\pi}{4}

  • T(x)=T(x)= Taylor polynomial of degree 100 of 11+x2\displaystyle\frac{1}{1+x^2} centered at x=0x=0

  • B=01T(x)dxB=\displaystyle\int_0^1 T(x)\, dx

  • Since AA and BB are approximately equal, π4B\pi\approx 4B. So calculate 4B4B and convert to a decimal.

Question 2

Estimate the value of 01ex2dx\displaystyle\int_0^1 e^{-x^2}\, dx as follows:

  • Define T20(x)=T20(x)= the Taylor polynomial of degree 20 of ex2e^{-x^2} centered at x=0x=0.

  • Calculate 01T20(x)dx\displaystyle\int_0^1 T20(x)\, dx.

  • Define T50(x)=T50(x)= the Taylor polynomial of degree 50 of ex2e^{-x^2} centered at x=0x=0.

  • Calculate 01T50(x)dx\displaystyle\int_0^1 T50(x)\, dx.

  • Compare your results with the output from Sage's numerical_integral command: 0.7468241328124270.746824132812427. [Use the N() command to convert to decimals.]

Question 3

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

Graph all three on the window 0x2π0\le x \le 2\pi, 0y30\le y \le 3. Use black for ff, blue for T5T5, and red for T10T10.

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