Question

Consider the function f(x)=x⋅sin(x).

a) Find the area bound by y=f(x) and the x-axis over the interval,
0≤x≤π. (Do this without a calculator for practice and give the
exact answer.)

b) Let M(x) be the Maclaurin polynomial that consists of the first 5 nonzero terms of the Maclaurin series for f(x). Find M(x) by taking advantage of the fact that you already know the Maclaurin series for sin x.

M(x)=

c) Since every Maclaurin polynomial is by definition centered at 0, the farther from zero, the less accurately it approximates its corresponding function. If you were to approximate the area you calculated in part (a) by correctly integrating M(x) found in part (b) the resulting estimate would be off by more than 0.0001 from the actual area. (That is, ∣∣∣∣∫0πf(x)dx−∫0πM(x)dx∣∣∣∣>0.0001).

Find the largest a (to three decimal places) between x=0 and x=π
so that the alternating series error bound guarantees

∣∣∣∣∫0af(x)dx−∫0aM(x)dx∣∣∣∣≤0.0001

(Give a number with exactly three digits after the decimal point.)

Answer #1

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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...

(1 point) Let F(x)=∫o,x sin(6t^2) dt F(x)=∫0xsin(6t^2) dt. The
integrals go from 0 to x
Find the MacLaurin polynomial of degree 7 for F(x)F(x).
Use this polynomial to estimate the value of ∫0, .790 sin(6x^2) dx
∫0, 0.79 sin(6x^2) dx. the integral go from 0 to .790

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.

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).

Find the error bound |?(?) − ?4(?)| for ?(?) =
(1/(?+1)) centered at ? = 0 on the intervals
[0, 0.1] and [0, 0.5].
It is using Taylor and Maclaurin polynomials, that is
the question that was given.
thats all the informa5ion i have for that problem. can
you help with this one instead.
Find the Taylor polynomial ?4 for ?(?) = ? sin ?
centered at ? = ?⁄4.

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).

1. Find the area between the curve f(x)=sin^3(x)cos^2(x) and y=0
from 0 ≤ x ≤ π
2. Find the surface area of the function f(x)=x^3/6 + 1/2x from
1≤ x ≤ 2 when rotated about the x-axis.

Consider the function f(x, y) = sin(2x − 2y) (a) Solve and find
the gradient of the function.
(b) Find the directional derivative of the function at the point
P(π/2,π/6) in the direction of the vector
v = <sqrt(3), −1>
(c) Compute the unit vector in the direction of the steepest
ascent at A (π/2,π/2)

f(x) = e x ln (1+x) Using the table of common Maclaurin
Series to find the first 4 nonzero term of the Maclaurin Series for
the function.

B.) Let R be the region between the curves y = x^3 , y = 0, x =
1, x = 2. Use the method of cylindrical shells to compute the
volume of the solid obtained by rotating R about the y-axis.
C.) The curve x(t) = sin (π t) y(t) = t^2 − t has two tangent
lines at the point (0, 0). List both of them. Give your answer in
the form y = mx + b ?...

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