dis_14

Week 14 Discussion

Use the Taylor series given in Key Idea 8.8.1 to create the Taylor series of the given functions.

29. f(x) = e^x sin x (only find the first 4 terms)

Ans:

The Taylor Series of f(x), centered at c is, lim n = 0 -> ∞[(f^((n) (c))/n! (x-c)^n]

Setting c = 0 gives the Maclaurin Series of f(x): lim n = 0 -> ∞ [(f^((n) (0))/n! (x)^n]

So, f(x) = f(0) + x f’ (0) + x^2/2! f’’ (0) + x^3/3! f’’’ (0) + x^4/4! f’’’’ (0) + ⋯ + ∞

Now,

f(x) = e^x sinx 

so, f(0) = 0

f’(x) = e^x cosx + e^x sinx 

so, f’(0) = 1

f’’(x) = e^x(cosx – sinx) + (sinx + cosx) e^x

         = 2e^x cosx
so, f’’(0) = 2

f’’’(x) = -2e^x sinx + 2 e^x cosx           

so, f’’’(0) = 2

f’’’’(x) = -2e^x cosx - 2e^x sinx - 2e^x sinx + 2e^x cosx

    = -4e^x sinx                
    
so, f’’’’(0) = 0

Now, f(x)= e^x sinx = 0 + x + x^2/2 * 2+ x^3/6 * 2+ x^4/24 * 0 + ⋯ + ∞

        = 0 + x + x^2 + x^3/3 + … + ∞