Then use Wolfram Alpha to compute the Taylor series for sin 2 x up to the x 4 term. The starting terms of the Taylor series for f ( x ) = sin x are sin x = x − 3 ! x 3 + 5 ! x 5 − … Square the first two terms of this series to get the start of the power series for sin 2 x. Then use Wolfram Alpha to find the Taylor series of sec 2 x and compare your answers. You can transform trigonometric functions to their exponential forms, calculate symbolic antiderivatives, and derive closed-form series expansions. Differentiate both sides of the series you obtained in the previous problem to get the first terms of the power series for sec 2 x. (Also, if you are curious, there is a formula for this series with summation notation, but, unlike the Taylor series for e x and sin x, it is not a "nice" formula.) 4. Hint: you can type stuff into Wolfram Alpha in plain English, and it will figure out what you are asking most of the time. training a food wolfram we pet baby merseyside louis dog dogs you puppy. For trigonometric angles we have to convert angle in degree to radians. We will quantify this on the next homework). Taylor series expansion to derive cos angle values Deriving values of trigonometric angles by Taylor series expansion is one of the most useful methods. On what interval around the origin does your power series approximation "look good". Find a Taylor approximation for an alternating Gaussian exponential sum. The coefficients of the Taylor series are obtained by computing the residues at the poles of the Mellin transform for this sum. Do your answers agree? Does your series contain odd powers of x, even powers of x, or both? Is tan x an odd function, an even function, or neither? Do these agree? Use Desmos to plot both tan x and your power series on the same graph. This example shows how an infinite exponential sum can be approximated with a Taylor expansion in terms of the parameter in the summand. Then ask Wolfram Alpha to compute the Taylor series directly. Since no one wants to do this by hand (trust me, you don't!), use Wolfram Alpha to compute these derivatives for you (it will also evaluate them at x = 0 ). This will involve differentiating tan x 5 times and evaluating the answer at x = 0. Compute the Taylor series for f ( x ) = tan x centered at x = 0 up to the x 5 term.
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