Help with Math Question

Question

Write out the first four terms of the Maclaurin series of 𝑓 if

𝑓(0)=7,𝑓′(0)=7,𝑓″(0)=18,𝑓‴(0)=18

Answer 1

A Maclaurin series has the form:

f(x) = ∑₀^∞ f⁽ⁿ⁾(0)xⁿ / n! where f⁽ⁿ⁾ is the nth derivative of f. Thus:

1^st term (n = 0) is 7x⁰ / 0! = 7

2^nd term (n=1) is  7x¹ / 1! = 7x

3^rd term (n=2) is 18x² / 2! = 9x²

4^th term (n=3) is 18x³ / 3! = 3x³

The first four terms of the MacClaurin series are:

7 + 7x + 9x² + 3x³

Answer 2

So, this can simply be done by thinking in terms of antiderivatives. f will have no antiderivative. We will take the antiderivative once for f'. We will take the antiderivative two times for f'', and we will take the antiderivative three times for f'''.

So of course, f(0) = 7 will be 7. f'(0) = 7 will have the antiderivative taken one time, so it will be (7x^0+1)/(1!) = 7x. f''(0) = 18 will have the antiderivative taken two times, so it will be (18x^0+1+1)/(2!) = 18x = (18x^2)/2 = 9x^2. f'''(0) = 18 will have the antiderivative taken three times, so it will be (18x^0+1+1+1)/(3!) = 18x = (18x^2)/2 = 9x^2 = (9x^3)/3 = 3x^3.

So, the first four terms of the Maclaurin series of f are: 7 + 7x + 9x^2 + 3x^3.