Question

Let f(x) = 1 + x − x^{2} +e^{x-1}.

(a) Find the second Taylor polynomial T_{2}(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) - T_{2}(x) is absolute value.

Please answer both questions cause it will be hard to post them separately.

Answer #1

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. 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 ?

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 ?

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.

For this problem, consider the function f(x) = ln(1 + x).
(a) Write the Taylor series expansion for f(x) based at b = 0. Give
your
final answer in Σ notation using one sigma sign. (You may use 4
basic Taylor
series in TN4 to find the Taylor series for f(x).)
(b) Find f(2020) (0).
Please answer both questions, cause it will be hard to post them
separately.

Let f(x,y)=2ex+y. Find the second-order Taylor polynomial for
f(x,y) at the point (0,0).
Group of answer choices
2+x+y+12x2+12y2
2x+2y+x2+y2
2+2x+2y+x2+2xy+y2
2−2x−2y+x2−xy+y2
None of the above.

Find the first order Taylor polynomin of f(x,y)=x^2e^y at (0,0)
T1(x,y)=
Find the second orser Taylor polynomial of f(x,y)=x^2e^y at
(0,0)
T2(x,y)=

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).

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.

The second-order Taylor polynomial fort he functions f(x)=xlnx
about X0= 1 is P2= -1+(x-1)^2/2 using the given Taylor polynomial
approximate f(1.05) with 2 digits rounding and the find the
relative error of the obtained value (Note f(1.05=0.0512). write
down the answer and all the calculations steps in the text
filed.

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