Question

`Find the ?? (?) Taylor polynomial of ? (?) = (? - ?) ?? (? - ?) centered on ?. Using the Taylor inequality, find the error of the approximation if .? ≤ ? - ? ≤ ?. ?.`

Answer #1

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.
???.

1.
Use a deﬁnition of a Taylor polynomial to ﬁnd the Taylor
polynomial T2(x) for f(x) = x^3/2 centered at a = 4.
We use T1(3.98) to approximate (3.98)^3/2. Apply Taylor’s
inequality on the interval [3.98,4.02] to answer the following
question: can we guarantee that the error |(3.98)^3/2 −T1(3.98)| of
our approximation is less than 0.0001 ?

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 3 Taylor polynomial T3 (x) centered
at a = 4 of the function f(x) = (-7x+36)4/3

find the 6th order taylor polynomial for f(x) = xsin(x^2)
centered at a=0.

For the function f(x) = ln(4x), find the 3rd order Taylor
Polynomial centered at x = 2.

Let f(x) = 2/ x and a = 1. (a) Find the third order Taylor
polynomial, T3(x), that approximates f near a. (b) Estimate the
largest that |f(x)−T3(x)| can be on the interval [0.5,1.5] by using
Taylor’s inequality for the remainder.

Find the Taylor polynomial of degree 3, centered at a=4 for the
function f(x)= sqrt (x+4)

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

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).

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