Question

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.

Answer #1

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 ?

Let f(x) =(x)^3/2 (a) Find the second Taylor polynomial T2(x)
based at b = 1. x3. (b) Find an upper bound for |T2(x)−f(x)| on the
interval [1−a,1+a]. Assume 0 < a < 1. Your answer should be
in terms of a. (c) Find a value of a such that 0 < a < 1 and
|T2(x)−f(x)| ≤ 0.004 for all x in [1−a,1+a].

(1 point) Find the degree 3 Taylor polynomial T3(x) centered
at a=4 of the function f(x)=(7x−20)4/3.
T3(x)=
? True False Cannot be determined The function f(x)=(7x−20)4/3
equals its third degree Taylor polynomial T3(x) centered at a=4.
Hint: Graph both of them. If it looks like they are equal, then do
the algebra.

(1 point) Find the degree 3 Taylor polynomial T3(x) of
function
f(x)=(7x+67)^(5/4)
at a=2
T3(x)=?

Find the degree 3 Taylor polynomial T3(x) of function
f(x)=(7x−5)^3/2 at a=2. T3(x)=

Let f(x) = 1 + x − x2 +ex-1.
(a) Find the second Taylor polynomial T2(x) for f(x)
based at b = 1.
b) Find (and justify) an error bound for |f(x) − T2(x)| on the
interval
[0.9, 1.1]. The f(x) - T2(x) is absolute value.
Please answer both questions cause it will be hard to post them
separately.

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

The second-order Taylor polynomial fort he functions f(x)=√1+x
about X0= is P2=1+(x/2)-(x^2/2) using the given Taylor polynomial
approximate f(0.05) with 2 digits rounding and the find the
relative error of the obtained value (Note f(0.05=1.0247). write
down the answer and all the calculations steps in the text
filed.

Using the Taylor Remainder Theorem, what is the upper bound on
| f (x) − T3(x)|, for x ∈ [4, 10] if f (x) = 2 sin (x) and T3(x)
is the Taylor polynomial centered on 7.

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