Taylor Polynomials with Remainder

In all the following exercises, $c$ is taken as $0.$

  1. Let MATH The 5th Tayor polynomila for $f\left( x\right) $ is
    Using these formulas, or otherwise, find find MATH and for MATH for the functions

    1. $e^{-x}$

    2. $e^{x^{2}}$

  2. Use MATH for the function MATH to approximate MATH and MATH Determine the accuracy of this approximation knowing that MATH.

  3. Determine MATH for MATH and for MATH

  4. For MATH find MATH and MATH Use your results to approximate MATH and give the error exacly and using the formula for MATH

  5. Find MATH for MATH and use your results to approximate $\sqrt{1.5}$ and $\sqrt{0.5}.$ Which result is a better approximation of the corresponding true value? Suggest a way to obtain better accuracy for the less accurate result.

  6. Find MATH for MATH and then use your results to estimate the accuracy acheived when using MATH to approximate $f\left( x\right) $ on the interval MATH

  7. In the previous exercise, aproximate MATH and MATH by replacing $f\left( x\right) $ by MATH

  8. For the function MATH find MATH and MATH Find the Taylor polynomial of least degree to approximate $f\left( x\right) $ to $10^{-4}$ accuracy on the interval MATH and use it to approximate MATH

  9. On an Excel sheet draw the graphs of the functions MATH and its Taylor polynomials MATH over the interval MATH

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