Question

Let f(x, y) = sin x √y.

Find the Taylor polynomial of degree two of f(x, y) at (x, y) = (0, 9).

Give an reasonable approximation of sin (0.1)√ 9.1 from the Taylor polynomial of degree one of f(x, y) at (0, 9).

Answer #1

1. This question is on the Taylor polynomial.
(a) Find the Taylor Polynomial p3(x) for f(x)= e^ x sin(x) about
the point a = 0.
(b) Bound the error |f(x) − p3(x)| using the Taylor Remainder
R3(x) on [−π/4, π/4].
(c) Let pn(x) be the Taylor Polynomial of degree n of f(x) =
cos(x) about a = 0. How large should n be so that |f(x) − pn(x)|
< 10^−5 for −π/4 ≤ x ≤ π/4 ?

Find the degree-2 Taylor polynomial for the function f(x, y) =
exy at the point (4, 0).

5. Find Taylor polynomial of degree n, at x = c, for the given
function. (a) f(x) = sin x, n = 3, c = 0 (b) f(x) = p (x), n = 2, c
= 9

Generate the Taylor Polynomial of degree 7 for:
f(x) = sin(x) + x
Where:
c = pi/4

1.
Find the Taylor polynomial, degree 4, T4, about 1/2 for f (x) = tan-inv (x) and use it to approximate tan-inv (1/16).
2.
Find the taylor polynomial, degree 4, S4, about 0 for f (x) = tan-inv (x) and use it to approximate tan-inv (1/16).
3.
who provides the best approximation, S4 or T4? Prove it.

let f(x)=cos(x). Use the Taylor polynomial of degree 4
centered at a=0 to approximate f(pi/4)

Find the quadratic approximation (Taylor Polynomial) for f(x,y)
= 2xe^(2y) near (2,0).

Find the 5th Taylor polynomial of f(x) = 1 + x + 2x^5
+sin(x^2) based at b = 0.

Find the Taylor degree 4 polynomial of ? (?) = −? ∗ ??? (?)
centered on 0 and find the interval
for which the approximation has a smaller error or than a.
???.

Determine the degree of the MacLaurin polynomial for the
function
f(x) = sin x required for the error in the approximation of
sin(0.3)
to be less than 0.001, and this approximate value for
sin(0.3).

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