← Calculus II

Unit 19: Taylor and Maclaurin Series

An infinite polynomial that reproduces a function exactly, built one matched derivative at a time.

8 topics · 35 questions

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Taylor Polynomials

  1. How it worksHow does each added term in a Taylor polynomial fix one more derivative to agree with the function at the center?
  2. Worked exampleFind the third-degree Taylor polynomial for f(x) = ln(x) centered at x = 1.
  3. Worked exampleFind the second-degree Taylor polynomial for f(x) = √x centered at x = 4.
  4. Common mistakeWhy isn't a Taylor polynomial a good approximation for values of x far away from its center?
Your turn1 of 2
Find the third-degree Taylor polynomial for f(x) = 4ln(x) centered at x = 1.

Maclaurin Polynomials of Increasing Degree

  1. How it worksHow do the degree 1, 3, 5, and 7 Maclaurin polynomials of sin(x) close in on the sine wave as the degree climbs?
  2. Worked exampleFind the fourth-degree Maclaurin polynomial for f(x) = e2x.
  3. Worked exampleFind the sixth-degree Maclaurin polynomial for f(x) = cos(x) and evaluate it at x = 1.
  4. Going furtherWhat happens to a Maclaurin polynomial's fit once x moves past the function's radius of convergence?
  5. Common mistakeWhy doesn't a higher-degree Taylor polynomial automatically fit a function better at every single point?
Your turn1 of 2
Find the fourth-degree Maclaurin polynomial for f(x) = e3x.

The Lagrange Error Bound

  1. How it worksHow does the size of the next unused derivative bound the leftover remainder left out of a Taylor polynomial?
  2. Worked exampleFind the Lagrange error bound for the third-degree Maclaurin polynomial of ex at x = 0.5.
  3. Worked exampleFind the smallest degree Maclaurin polynomial of sin(x) guaranteed to have error under 0.001 at x = 1.
  4. Common mistakeWhy isn't the Lagrange error bound the same thing as the actual error of the approximation?
Your turn1 of 2
Find the Lagrange error bound for the third-degree Maclaurin polynomial of 2ex at x = 0.624.

Finding a Taylor or Maclaurin Series

  1. How it worksHow does the general term built from the nth derivative divided by n factorial generate every coefficient?
  2. Worked exampleFind the Maclaurin series for f(x) = 1/(1 + x2) in general term form.
  3. Worked exampleFind the Taylor series for f(x) = ex centered at x = 2.
  4. Common mistakeWhy doesn't a function that's infinitely differentiable have to equal its own Taylor series everywhere?
  5. Going furtherWhat happens to a Taylor series' accuracy far from its center even when the series still converges there?
Your turn1 of 2
Find the coefficient of x6 in the Maclaurin series for f(x) = 1/(1 + 4x2).

The Standard Maclaurin Series Worth Knowing

  1. How it worksHow are the Maclaurin series for ex, sin(x), cos(x), and 1/(1 - x) all built from the same derivative pattern?
  2. Worked exampleFind the Maclaurin series for cos(2x) by substituting into the known cosine series.
  3. Worked exampleFind the first four nonzero terms of the Maclaurin series for ex times cos(x).
  4. Common mistakeWhy isn't the series for sin(x) just the series for cos(x) with every sign flipped?
Your turn1 of 2
Find the fourth-degree Maclaurin polynomial for f(x) = cos(3x) by substituting into the known cosine series.

Building New Series from Known Ones

  1. How it worksHow does multiplying a polynomial factor into a known series, term by term, build the series for a related function?
  2. Worked exampleFind the first four nonzero terms of the Maclaurin series for x2 times e−x.
  3. Worked exampleFind the Maclaurin series for sin(x2) by substituting x2 into the known series for sin(x).
  4. Common mistakeWhy doesn't substituting 2x for x in a series with radius 1 leave the radius of convergence unchanged?
Your turn1 of 2
Find the first four nonzero terms of the Maclaurin series for f(x) = 4x2 e−x.

Using Series to Evaluate Limits and Integrals

  1. How it worksHow does replacing an integrand with its series turn an integral with no elementary antiderivative into a truncatable sum?
  2. Worked exampleFind the limit as x approaches 0 of (sin(x) - x)/x3 using the Maclaurin series for sin(x).
  3. Worked exampleFind the integral from 0 to 1 of sin(x2) dx accurate to three terms using a series.
  4. Common mistakeWhy doesn't truncating a series after a few terms ever give an exact answer instead of an approximation?
  5. Ties togetherHow does using a series to evaluate a limit connect back to L'Hopital's rule from Calculus I on the same limit?
Your turn1 of 2
Find the limit as x approaches 0 of (sin(2x) - 2x)/x3 using the Maclaurin series for sin(x).

Applications of Taylor Series

  1. How it worksHow does approximating sin(theta) by theta alone simplify the equation governing a swinging pendulum?
  2. Worked exampleFind the second-degree Taylor approximation used to estimate √1.02 without a calculator.
  3. Common mistakeWhy doesn't a linear or quadratic Taylor approximation stay accurate no matter how far x moves from the center?
  4. Going furtherWhat happens to the accuracy of a linear approximation as x moves farther from the center point?
Your turn1 of 2
Find the second-degree Taylor approximation used to estimate √(7 + 0.036), centered at x = 7.