2.3 Maclaurin Series 67 2.42 sinh x. (Never heard of it? Look it up!) Problems 2

Question

2.3 Maclaurin Series 67 2.42 sinh x. (Never heard of it? Look it up!) Problems 2.43-2.45 make a plot showing the given function and the first three partial sums of its Maclaurin series in the range -5sx 5. In calculating the partial sums you should skip terms that equal zero, so for example the second partial sum of 1-x2/2+ /24 would be 1 2/2. Make sure it's clear which curve is the function and which ones are the partial sums. Estimate visually the value of x where each partial sum stops being a good pproximation. (There is no precise correct answer for this question; just make a reasonable estimate based on looking at your plot.) 2.43 () cos x 2.44 (x) 1/(2+ cos x) 2.45 f(x) = sin (e- im In this problem you'll use the function (x) 0.5 sin()+1.5 sinx)+ 0.2 to visually nce of a Maclaurin ur

Explanation / Answer

(2.45) f(x) = sin (e-x^2)

Using the chain rule to differentiate an above equation w.r.t x, we get

d [f(x)] / dx = d [sin (e-x^2)] / dx . d [(e-x^2)] / dx

f'(x) = cos (e-x^2) (-2x e-x^2)

f'(x) = - 2x e-x^2 cos (e-x^2)

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