Calculus series question please help asap!!!!!!!!!!

Question

What is the maximum possible error generated in using the 5th degree Maclaurin polynomial for sin(x) for x in the interval [−0.3, 0.3]?

a). The absolute value of 0.3 raised to the 7th power over 6 factorial

b). The absolute value of 0.3 raised to the 6th power over 6 factorial

c). The value cannot be determined.

d). There is no error. The value will be exact.

Can you please take me through it? I'm having trouble plugging the values into the formula.

I know it involves Lagrange's formula.

|En(x)| ≤ M/(n+1)! * |x - a|^(n+1)

Answer 1

The sinx function has a Maclaurin series expansion

sinx = x - (x^3)/3! +(x^5)/5! - (x^7)/7!+...

This is an alternating series for all nonzero values of x, and the successive terms

decrease in size since |x|<1.

Thus we can use the Alternating Series Estimation Theorem .

The error in approximating sinx by the first three terms of its Maclaurin series is at most

(|x|^7)/(7!)= |x^7|/5040

If -.3≤ x ≤.3, then |x|≤.3 so the error is smaller than

(0.3)^7/ 5040≅ 4.3 (10)^(-8)