Exercises IX.C: Solutions (In Part)

In exercises 1-10, find `P_n (x,f(x))` for the specified function and `n`.

1. `f (x) = x sin(x)`; `n = 6`

`P_5 (x, sin(x)) = x -{x^3}/6 + {x^5)/{5!}`, so  `P_6 (x, x sin(x))= x P_5 (x, sin(x)) = x (x -{x^3}/6 + {x^5}/{5!}) = x^2 -{x^4}/6 + {x^6}/{5!}`.
2. `f (x) = x^2 cos(x)`; `n = 6`
3. `f (x) = x sin(x) + cos(x)`; `n = 6`
`P_6 (x, cos(x)) = 1 -{x^2}/2 + {x^4}/{4!} -{x^6}/{6!}`
So `P_6 (x, x sin(x) + cos(x))= P_6 (x, x sin(x))+ P_6 (x, cos(x)) = x^2 -{x^4}/6 + {x^6}/{5!}+ 1 -{x^2}/2 + {x^4}/{4!} - {x^6}/{6!}=...`
4. `f (x) = x e^x`; `n = 6`
5. `f (x) = sin (x^3)`; `n = 9`
`P_9 (x, sin(x^3)) = P_3 (x^3, sin(x)) = (x^3 ) - {(x^3)^3}/6 = x^3 - {x^9}/6`
6. `f (x) = 1/{2 - x}`; `n = 4`
7. `f (x) = 1/{1 + x}`; `n = 4`
8. `f (x) = 1/{1 + x^2}`; `n = 8`
9. `f (x) = `arctan`(x)`; `n = 8`
Hint: Use arctan'`(x) = 1/{1 + x^2}` and the previous problem.
10. `f (x) = x/{2-x}`; `n = 4`
11. `f (x) = e^{-2x}` ; `n = 4`
12-22: For the functions `f` in problems 1-11 use the MacLaurin polynomial to estimate `int_0^1 f(x) dx`.