CAS means computer algebra system

u(x)=0.98762x-0.155271x^3+0.00564312x^5

Here is u(x) and t(x)

Again, using a CAS, find ||u(I) - sin :|| and ||t(1) - sin x ||. What gives a better approximation to sin c? u() or t()? Verify that the 5-degree polynomial Taylor approximation t(x) to sin x is given by (6.61). Solution: To define the Taylor approximation t(x) given by (6.61) we know that taylor series can be defined as f(a) + f'(a)(x a) + 1 C) (x a)2 + ... + moca (r a) +... Substituting sin(a) for f(a) yields the following 5-degree polynomial, sin a + cos a (x a) + = sin a (x a)2 + = cos a (x a)3 + sima (r a)4 + 0$ a (x a)5 To approximate we say a = 0 and receive 0+x+0 - +0+ er this simplifies to x - x + en 3! which is identical as the Taylor approximation t(x) given by (6.61) Again, using a CAS, find ||u(I) - sin :|| and ||t(1) - sin x ||. What gives a better approximation to sin c? u() or t()? Verify that the 5-degree polynomial Taylor approximation t(x) to sin x is given by (6.61). Solution: To define the Taylor approximation t(x) given by (6.61) we know that taylor series can be defined as f(a) + f'(a)(x a) + 1 C) (x a)2 + ... + moca (r a) +... Substituting sin(a) for f(a) yields the following 5-degree polynomial, sin a + cos a (x a) + = sin a (x a)2 + = cos a (x a)3 + sima (r a)4 + 0$ a (x a)5 To approximate we say a = 0 and receive 0+x+0 - +0+ er this simplifies to x - x + en 3! which is identical as the Taylor approximation t(x) given by (6.61)