Find the Taylor polynomial of degree 4 at c=1 for the equation
f(x)=ln x

and determine the accuracy of this polynomial at x=2.

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I am going to explain two methods of solving the first part. I like the second approach much better of course:

Part a) Method I-

We know that if the derivatives of all orders at x=c exist, then:
f(x)= f(c)+ f'(c)(x-c)+ f''(c)((x-c)^2/2!)+ f'''(c)((x-c)^3/3!)+...
Now let's go back to the problem. Here f(x)=ln(x) and we want the taylor series about x=1. Therefore, for example:
f(x)= ln(x) --> f(1)= ln(1) =0,
f'(x)= 1/x --> f'(1)= 1/1= 1,
....
we use the above results to write f(x)= ln(x) in the form of:
ln(x)= 0+ 1*(x-1)+...
and we stop and truncate once we find the 4th degree of x. (i.e. here it is the term including (x-1)^4 and will not go as far as (x-1)^5).

part a:) Method II-

Well, the better way in my point of view is to enjoy the taylor series of the famous functions and extend them to what we want to find. You can find the Taylor series of these functions in almost all the calculus textbooks. Trig. functions, exponentials ...

2a. Find the second order Taylor polynomial for f(x) = x^(1/3) about x = x0, x0 > 0.
2b. Show that the function G(x) is differentiable at x0 and find G'(x0).
3a. Find an expression for the lower sum L(f,D) and upper sum U(f,D).
3b. Determine the lower integral and upper integral.
Please refer to the attached images f

Use Taylorpolynomials about 0 to evaluate sin(0.3) to 4dp,showing all workings.
1)F(x)=square root 4+x and G(x)=square root 1+x
by writing square root of 4+x=2 square root 1+1/4x
and using substitution in one of the standard Taylor series, find the Taylor series about 0 for f.Given explicitly all terms up to term in x raise

1 Write the Taylorpolynomial with center zero and degree 4 for the function f(x) = e^-x
2 Determine the values of p for which the series
∞
Σ 1/(2p)ⁿ
n=1
3 Calculate the sum of the first ten terms of the series, then estimate the error

Please assist with the following problems I am having a hard time solving. Please see attached.
Determine if the following series are convergent or divergent
Find the values of x for which
and tell if the series converges or diverges when given the series
Find power series for the following function

1. For the equation ?(x)= x^(1/2)
a) Find the Taylorpolynomial of degree 4 of at c = 4
b) Determine the accuracy of the polynomial at x = 2.
2. Find the Maclaurin series in closed form of
a) ?(x)=((1) / ((x+1)^2)
b) ?(x)=ln ((x^2)+1)
3. Use the chain rule to find dw / dt, where
w = x^2 + y^2 + z^2, x=(e^t) cos t, y=(

Please see the attached file for the fully formatted problems.
1.
? Calculate the TaylorPolynomial and the Taylor residual for the function .
? Prove that as , for all .
? Find the Taylor series of f.
? What is the radius of convergence for the Taylor series? Justify your answer.
2.
? Let f:[0,1] be a bo

1) Find the equilibrium solution of the differential equation:
Dy/dt = 3y(1-(1/2)y)
Sketch the slope field and use it to determine whether each equilibrium is stable or unstable.
2) Consider the initial value problem
Y^1 = 4 - y^2, y(0) = 1
Use the Euler's method with 5 steps to estimate y(1). Sketch the field and u

Given ( INTEGRAL ln square(x)dx, as x from n to n+1 ) = ( INTEGRAL ln square (n+x)dx, as x from 0 to 1 ) = ( INTEGRAL [[ln(n+x) - ln(x) + ln(n)]square] dx, as x from 0 to 1 ),
(a) Verify that ( LIMIT (n/ln(n)) [INTEGRAL (ln square (x)dx) - (ln square (n))] as n approach to the infinity ) = 1
(b) Compute LIMIT ((n square)/