Find a MacLaurin polynomial...?

Question

How would I go about writing a MacLaurin polynomial P3(x) and remainder R3(x) for f(x)=e^2x?

Answer 1

P3(x) = 1 + 2 x + 2 x^2 + (4/3) x^3

To do that, you have to find f(0) and the first three derivatives of f(x) at x=0:

f(x)=e^(2x)
f'(x)=2e^(2x)
f''(x)=4e^(2x)
f'''(x)=8e^(2x)

plugging in zero, you get:

f(0) = 1
f'(0) = 2
f''(0) = 4
f'''(0) = 8

These will be the coefficients of our polynomial. Remember, each term is in the form:

(f^(n) (0) * x^n) / n!

where f^(n) (0) is the nth derivative of f at x=0. Plugging those coefficients in and dividing by the factoral,

1/0! x^0 + 2/1! x^1 + 4/2! x^2 + 8/3! x^3
1/1 x^0 + 2/1 x^1 + 4/2 x^2 + 8/6 x^3
x^0 + 2 x^1 + 4 x^2 + (4/3) x^3
1 + 2 x + 4 x^2 + (4/3) x^3

As for your remainder, this will be the value of the next term in the polynomial at x=a where a is the value you evaluate P3(x) at. Since our derivatives were in the pattern 1, 2, 4, 8, the next one must be 16:

16/4! x^4
16/24 x^4
(1/3) x^4
(1/3) a^4

when P3(x) is evaluated at x=a.

Answer 2

P3(x) =1+2x+4x^2/2! +8x 3/3!^+16e^2ax/4! where
0