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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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