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Texas Instruments
9-12
45 Minutes
TI-Nspire™ CAS
3.2
Students will explore Taylor polynomials graphically and analytically, as well as graphically determine the interval where the Taylor polynomial approximates the function it models.
In this problem, students investigate a Taylor polynomial centered at zero. Students will recall that this Taylor polynomial is also known as a Maclaurin polynomial.Students find the 4th degree Taylor polynomial that approximates f(x) = ln(x + 5) at x = 0. They will notice that the values are closest at the center and become farther apart the further the x-values are from the center.
This problem also gives students the opportunity to explore different degrees of a Taylor polynomial. They use a slider to change the degree of the polynomial, which changes the graph.
This will demonstrate that a Taylor polynomial will only approximate the function over a given interval, no matter how large the Taylor Polynomial is.
In Problem 2, students work a problem where the Taylor polynomial is not centered at zero. They see that the polynomial follows the original function for a much smaller interval and they should use (x – a)n instead of xn. At the end of this activity, students will be able to find the first few terms of a Taylor series approximation to a function for any given value of x. They will also be able to graph a function and its Taylor polynomials of various degrees to show their convergence to a function.
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