Question

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 ?

Answer #1

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).

i) Approximate the function f(x) = cos x by a Taylor polynomial
of degree 3 at a = π/3
ii) What is the maximum error when π/6 ≤ x ≤ π/2? (this is the
continuation of part i))

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

1.)Find T5(x), the degree 5 Taylor polynomial of the function
f(x)=cos(x) at a=0.
T5(x)=
Find all values of x for which this approximation is within
0.003452 of the right answer. Assume for simplicity that we limit
ourselves to |x|≤1.
|x|≤
2.) (1 point) Use substitution to find the Taylor series of
(e^(−5x)) at the point a=0. Your answers should not include the
variable x. Finally, determine the general term an in
(e^(−5x))=∑n=0∞ (an(x^n))
e^(−5x)= + x + x^2
+ x^3 + ... = ∑∞n=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

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.

Calculus, Taylor series Consider the function f(x) = sin(x) x .
1. Compute limx→0 f(x) using l’Hˆopital’s rule. 2. Use Taylor’s
remainder theorem to get the same result: (a) Write down P1(x), the
first-order Taylor polynomial for sin(x) centered at a = 0. (b)
Write down an upper bound on the absolute value of the remainder
R1(x) = sin(x) − P1(x), using your knowledge about the derivatives
of sin(x). (c) Express f(x) as f(x) = P1(x) x + R1(x) x...

Find the MacLaurin series for f(x) = cos(5x^3 ) and its radius
of convergence.
Find the degree four Taylor polynomial, T4(x), for g(x) = sin(x)
at a = π/4.

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.

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

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