1
vote
0answers
50 views

Given a function $f$, find the largest $n$ such that $f(x)/x^n$ can be defined at $x=0$ to become differentiable there

Let $f(x) = \ln\left(\frac{x^2}{2}+1\right)+\cos x -1$. Find the largest $n\in\Bbb{N}$ such that there is $C\in\Bbb{R}$ such that: $$g(x) = \begin{cases} \frac{f(x)}{x^n} &\mbox{if } x\ne 0 \\ C ...
0
votes
1answer
25 views

Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
0
votes
2answers
42 views

How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
1
vote
1answer
32 views

Aftermath of Cauchy's mean value theorem

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
0
votes
1answer
14 views

Question about Peano form of the remainder

Let $f(x)$ be a real-valued function defined on a closed interval [a, b], differentiable on the open interval (a, b) $n-1$ times. $x_0$ belongs to [a, b]. Suppose that we ...
5
votes
5answers
76 views

Interpreting higher order differentials

I'm trying to understand Taylor's Theorem for functions of $n$ variables, but all this higher dimensionality is causing me trouble. One of my problems is understanding the higher order differentials. ...
-1
votes
1answer
38 views

Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...
0
votes
2answers
45 views

Prove an inequality (Using Taylor expansion)

Prove: $\frac{x}{1+x} < \ln(1+x) < x$. I thought a good practice would be to prove it using Taylor Expansion. Here's my try: $$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3}...$$ The n=1 ...
0
votes
0answers
20 views

solution check for approximating derivative using a Taylor expansion.

I'm wondering if there's in a mistake in either my reasoning or the given solution for the problem and was hoping to have someone double check this for me. The problem states: Let $g(2)=3$ , ...
2
votes
2answers
50 views

Trying to solve a Taylor series problem

I have a Taylor series problem, well more precisely a Maclaurin series. I am trying to find convergence of: $f(x) = e^{x^3} + e^{{2x}^3}$ Okay here goes: $$f'(x) = 3xe^{x^3} + 6x e^{{2x}^3}$$ ...
1
vote
1answer
18 views

Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
1
vote
0answers
27 views

Can one obtaining a mean value form of the Taylor series remainder using the integral remainder?

Can we show that $$(\exists \epsilon \in[0,x])\left(\int_{0}^x \frac{(x-s)^n f^{(n+1)}(s)}{n!}ds= \frac{x^{n+1}f^{(n+1)}( \epsilon)}{k!}\right)\text{ ?}$$ Thanks in advance!
6
votes
0answers
52 views

Higher Order Derivative Proof .

I would appreciate if someone could check over my proof for this question and advise me if it is correct. My attempt so far; Now as $f$ is k times differentiable , it taylor series about $x_{0}$ ...
1
vote
2answers
125 views

Find the Taylor series of $f(x) = e ^{- 1 / x^2}$

Find the Taylor series about 0, the function defined as: $f(x) = e ^{- 1 / x^2}$ if $x \ne 0$ and $f(x) = 0$ if $x=0$ and What can i conclude of the resulting? First i note that the function f is ...
2
votes
3answers
122 views

Taylor Theorem inequality

Prove that for all $f\in C^2([0,1])$ with $f(0)=f(1)=0$ and $|f''(x)| \le 1$ $$|f(x)| \le \frac{1}{2}x(1-x)$$ $\forall x \in [0,1]$.
0
votes
2answers
61 views

Solve this limit (Maclaurin or differentiate?)

I have this assignment where I should calculate the limit below: $$ \lim_{x\to0}\frac{\sin 2x}{x\cos x} $$ I can use l'Hospitals rule (because it is a "zero divided by zero"-case) and therefore ...
1
vote
1answer
67 views

differentiate arctan (maclaurin?)

I have this assignment: Differentiate this expression: $$ f(x) =\arctan \frac{x-1}{x+1} $$ There is also known that $-1 < x$ (Why is that important?). I do not know how to solve this problem... ...
2
votes
1answer
52 views

The 7-th derivative of $ x^3 \cdot\tan(2x) $ is this right

I have to find $y^{(7)}\left(0\right)$ of $y(x)=x^3\cdot\tan{(2x)}$ So my idea was to use Taylor expansion for $\tan(2x)$ to the $7$-th element and then multiply the hole thing by $x^3 $ and then ...
0
votes
1answer
43 views

Proving this Taylor-esque expansion for a $C^2$ function vanishing at 0 and 1

I am trying to prove the following (which I think is true!): if $f:[0,1]\rightarrow \mathbb{R}$ is twice continuously differentiable and $f(0)=0=f(1)$, then for every $x \in (0,1)$ there exists $\xi ...
1
vote
1answer
51 views

Taylor series expansion - application

I am working on the following: Let $f : \mathbb C \to \mathbb C$ be analytic. Suppose for all $z \in \mathbb C$ hold $f(2z) = 4f(z)$ and $f(1) = 1$. Then $f(z) = z^2$ for all $z \in \mathbb C$. I ...
2
votes
1answer
860 views

Finding the taylor series of $f(z) = 1/(1+z^2)$.

I am working on the following exercise: Find the Taylor expansion of the function $f(z) = \frac{1}{1+z^2}$ about $z = 3i$. We had the Taylor Series Theorem in the lecture: Let $D \subset ...
1
vote
1answer
110 views

How to prove that a complex power series is differentiable

I am always using the following result but I do not know why it is true. So: How to prove the following statement: Suppose the complex power series $\sum_{n = 0}^\infty a_n(z-z_0)^n$ has radius of ...
0
votes
1answer
103 views

Compute the 10th derivative

$f(x) = (\cos(5x^2) - 1 )/ x^2 $ at $x = 0$ We were given the hint to use the MacLaurin series for f(x). I get how to do it if it was just $\cos(5x^2)$ but what would I do with the other values in ...
1
vote
1answer
42 views

A basic doubt on derivatives

I have one question regarding differentiation : 1) Why in the definition of Taylor's series it requires the function to be "continuously" differentiable $m$ times in $[a,b]$? The book I am following ...
2
votes
2answers
91 views

Taylor expansion

Is there an easier way to do a Taylor expansion of $e^{u^2+u}$ than do derivatives or substitute and then use Newton's binomial? For example, expanding until the $4$th term: $$e^{u^2+u}=1+u^2+u+ ...
0
votes
2answers
84 views

derative of Taylor expansion

I'm reading this part of article about key points localization in image processing, and there is something I don't quite understand, mathematically it's this, $$D(w) = D + {\frac{\partial D}{\partial ...
0
votes
1answer
74 views

Taylor Expansion with Integral Remainder Question

I have the following question at hand and I have to admit that I am not used to integral remainder form of taylor approximation. I am still trying to work around, so a couple of hints would be useful ...
2
votes
2answers
73 views

$\frac{\sin x}{x^5} - \frac{1}{x^4} \underset{x\to 0}{\approx} \frac{-1}{6} \cdot \frac{1}{x^2}$, right?

I was reading an set of notes about Taylor series, and I came across a part I think is a typo. I want to make sure, because I want to understand this stuff correctly. Here is the relevant page of the ...
1
vote
1answer
289 views

A problem related to mean value theorem and taylor's formula

I guess I need to use Taylor's formula and the mean value theorem. I have no idea except for them. Note: honestly, this is not homework. I am studying by myself. Suppose that ...
0
votes
0answers
45 views

How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
4
votes
4answers
470 views

Find nth derivative of $\frac{x^{n}}{(1-x)^{2}}$, please?

I need to find the nth derivative of $\frac{x^{n}}{(1-x)^{2}}$ for $0<x<1$ So far, I tried the same method used for $\frac{x^{n}}{1-x}$ and here's what I got: \begin{equation} ...
6
votes
3answers
94 views

Find $f^{(1001)}(0)$

I am to find the value in 0 of 1001th derivative of the function $$f(x) = \frac{1}{2+3x^2}$$ How should I approach this kind of problem? I tried something like : $$\frac{1}{2+3x^2} = ...
4
votes
1answer
138 views

Question about a solution to a problem involving Taylor's theorem and local minimum

I've been studying "Berkeley Problems in Mathematics, Souza, Silva" and I came across this problem: Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a $C^{\infty}$ function. Assume that $f(x)$ has a ...
1
vote
0answers
146 views

Taylor series expansion example

I was reading an article and there was a snippet with a taylor series expansion as shown below: My question is, should (11) read as $F(xA+h)+(xΔA+Δh)\frac{\partial}{\partial x}F(xA+h)$ instead of ...
1
vote
2answers
52 views

Taylor series $ \sqrt{\frac{t}{t+1}}$

Could someone tell me how to calculate $ \sqrt{\frac{t}{t+1}}$ it should be $ \sqrt t - \frac{t^{\frac{3}{2}}}{2} +O(t^{\frac{5}{2}}) $
2
votes
2answers
999 views

Finding the 9th derivative of $\frac{\cos(5 x^2)-1}{x^3}$

How do you find the 9th derivative of $(\cos(5 x^2)-1)/x^3$ and evaluate at $x=0$ without differentiating it straightforwardly with the quotient rule? The teacher's hint is to use Maclaurin Series, ...
4
votes
1answer
92 views

$\frac{\mathrm d^n}{\mathrm d x^n} e^{-\frac {1}{x^2}} = 0$ at $x=0$ [duplicate]

This is an exercise from David Brannan's Mathematical Analysis. I've proved parts (a) - (c) but need help with Part (d). Any guidance appreciated. EDIT I have solved it, by induction using the ...
3
votes
1answer
298 views

Taylor expansion of an integral

I am interested in the Taylor series expansion around $t=0$ of the following expression: $$I(t)=\int_{0}^{\infty}e^{-x^2}\log\left(e^{-(x-t)^2}+e^{-(x+t)^2}\right)dx$$ Normally, I would proceed by ...
5
votes
1answer
208 views

Application for mean value theorem

$f(x)$ is three-times differentiable on $[a,b]$, how to show that there is $\varepsilon\in(a,b)$ such that $$f(b)=f(a)+\cfrac{1}{2}(b-a)[f'(a)+f'(b)]-\cfrac{1}{12}(b-a)^3f'''(\varepsilon)$$
2
votes
1answer
205 views

How to expand $x \sqrt{4 - x}$ to Maclaurin series?

Here is the task: using standard expansions, expand $f(x) = x \sqrt{4-x}$ to Maclaurin's series. I calculated derivatives up to $f^{(5)}(x)$, and got some results. Fortunately, in Maclaurin's ...
1
vote
1answer
101 views

Taylor polynomial of sin of given degree and error

Replace the following function by its taylor polynomial of the given grade, and approximate the error in the given interval: $$f(x) = \sin(x) \textrm{ by } T_3(f,x,0) \textrm{ in } |x| ...
2
votes
1answer
1k views

What is the nth derivative of $\dfrac{1}{\sqrt{1 + x^2}}$

I'm trying to find a general formula for the $n$th derivative of $$\dfrac{1}{\sqrt{1 + x^2}}$$ I got up to, \begin{eqnarray*} g^{(0)}(x) &=& g(x) \\ g^{(1)}(x) &=& \dfrac{1}{(1 + ...
1
vote
1answer
144 views

Function $k$ times differentiable $+$ root of multiplicity $k$

Problem: Consider the continuous function $f$ which is $k$ times differentiable: $f(\alpha )=f'(\alpha )=\cdots=f^{(k-1)}(\alpha )=0$ and $f^{(k)}(\alpha )\neq 0$. Assume that $\alpha$ is a root to ...
3
votes
2answers
2k views

Proof of Taylor's series expansion with two terms

I am looking for a simple direct proof of the fact that $$ \frac{\frac{f(x + \Delta x) - f(x)}{\Delta x} -f'(x)}{\Delta x} \stackrel{\Delta x \to 0}{\to} \frac{1}{2}f''(x), $$ or, ...
0
votes
2answers
97 views

Little question about finding a MacLaurin expansion for $f(x)=\frac{x^2}{1-x}$

First off all, I am sorry if my english is not perfect. I need help again for this exercise: Find Maclaurin series expansion for $f(x)=\frac{x^2}{1-x}$. That's what I did: ...