Taylor and Maclaurin Series · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Building functions from their derivatives

Mathematics · Calculus & Analysis · ages 19-20
Name ______________________   Date ____________

  1. Let T2 be the degree 2 Maclaurin polynomial for e^x. What is T2(2)?

    Answer: ______________

  2. What is the Maclaurin polynomial of degree 2 for e^x?

    • 1 + x + x^2/2
    • 1 + x + x^2
    • x + x^2/2 + x^3/6
    • 1 + 2x + x^2

  3. What is the Maclaurin polynomial of degree 2 for e to the x?

    • 1 + x + x^2
    • 1 + x + x^2/2
    • x + x^2/2 + x^3/6

  4. Which series is the Maclaurin series for 1 over (1 minus x)?

    • 1 + x + x^2 + x^3 + ...
    • 1 - x + x^2 - x^3 + ...
    • x + x^2 + x^3 + ...

  5. Lena says the Lagrange remainder bound uses the largest value of the next derivative on the interval. Is Lena right?

    Circle one:   True   False

  6. Let T1 be the degree 1 Taylor polynomial for f(x) = 1/(1 - x) centered at 0. What is T1(0.2)?

    Answer: ______________

  7. Starting from e^x = 1 + x + x^2/2 + ..., what are the first three terms of e^(-x)?

    • 1 - x + x^2/2
    • 1 + x + x^2/2
    • x - x^2/2 + x^3/6
    • 1 - 2x + x^2

  8. Lena says the Lagrange remainder bound uses the largest value of the next derivative on the interval. Is Lena right?

    Circle one:   True   False

  9. Starting from the series for 1 over (1 minus x), what is the series for 1 over (1 + x)?

    • 1 - x + x^2 - x^3 + ...
    • 1 + x + x^2 + x^3 + ...
    • x - x^2 + x^3 - ...

  10. Noah says you can get the series for x times e to the x by multiplying each term of the e to the x series by x. Is Noah right?

    Circle one:   True   False

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Answer key

For grown-ups. Fold this page away before handing over the rest.

Building functions from their derivatives W1-mt_SgI9Pn9RIO-s1

  1. 5 · T2(x) = 1 + x + x^2/2, so T2(2) = 1 + 2 + 2 = 5.
  2. 1 + x + x^2/2 · Every derivative of e^x at 0 equals 1, so the terms are 1, x, and x^2 over 2 factorial.
  3. 1 + x + x^2/2 · Every derivative of e to the x at 0 equals 1, so the terms are 1, x, and x squared over 2 factorial.
  4. 1 + x + x^2 + x^3 + ... · Every derivative at 0 equals n factorial, and dividing by n factorial leaves coefficient 1 on each power.
  5. True · Lena is right: the bound multiplies the max of the next derivative by the power term over (n+1) factorial.
  6. 1.2 · T1(x) = 1 + x, so T1(0.2) = 1.2.
  7. 1 - x + x^2/2 · Substituting minus x for x flips the sign of each odd power and leaves even powers unchanged.
  8. True · That is exactly the bound: the max of the next derivative times the power term over (n + 1) factorial.
  9. 1 - x + x^2 - x^3 + ... · Replace x with minus x: odd powers flip sign while even powers stay put.
  10. True · Term by term multiplication by x is valid inside the interval, so each term simply gains one power.
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