0
votes
2answers
40 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
votes
1answer
35 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
1answer
24 views

How to approximate the bounding region of a 2d differentiable mapping locally?

I have got a differentiable mapping $f:\Bbb R^2 \to \Bbb R^2$, Is the image of $f$ of a very small convex subset (e.g., a unit square) around any point, a bounded region? If it is bounded, can I ...
2
votes
1answer
24 views

Approximating error using Taylors theorem

I have used a Maclaurin series for the function $f(x) = \cos(2x)$ and have successfully produced: $\dfrac{2^n cos(\frac{n\pi}2)x^n}{n!}$ Now I want to estimate the error in approximating $\cos(2x)$ ...
0
votes
2answers
31 views

Evaluating a taylor series around a given point

So I'm having some trouble with the problem: Given that $\ln(x+1)=\sum_{n=1}^{\infty } \frac{(-1)^{n+1}}{n}x^{n}, -1<x\leq 1$, find the Taylor series of ln(x) around 3. For which x is this series ...
3
votes
1answer
65 views

Why does $\arctan(\frac{\tan \theta}{2}) \approx \frac{1}{2 - \theta} - \frac{1}{2 + \theta}$ for small $\theta$?

In answering this question, I needed to show that $\arctan \left( \frac{\tan \theta}{2} \right) \approx \frac{\theta}{2}$ when $\theta$ was small. So, naturally, I computed the first few terms of the ...
1
vote
2answers
47 views

Can a function be approximated by finite number of Taylor expansion terms outside of disk of convergence?

Suppose we have a finite number of terms for Taylor expansion of a conditionally convergent function. For example, $f=\frac1{1-x}$ with expansion $f=\sum_{n=0}^\infty x^n$. This expansion diverges ...
3
votes
1answer
76 views

Approximating $\sqrt{101}$ using Taylor series methods

I'm trying to approximate $\sqrt{101}$ using the Taylor series for the function $f(x)=\sqrt{x}$ centered at the point $x=100$. I need to obtain an approximation that is within $0.01$ of the correct ...
0
votes
3answers
40 views

Having trouble calculating approximations using Taylor polynomials

I have a problem to approximate $\sqrt{1.06}$ using a third degree Taylor polynomial. The way I learned was to pick a center that we would know the answer to that is close to the value we're trying ...
1
vote
2answers
36 views

Approximation of $x\log(x/a)$ for $x$ near a

I'm trying to see where the approximation $$(x-a) + ((x-a)^2)/2a$$ of $x\log(x/a)$ comes from (for x near a). Might be missing something very trivial but I've already tried the usual expansions ...
0
votes
1answer
50 views

Approximation of $\frac{1}{2^{N}} \binom{N}{\frac{1}{2}(N + s)} $ for big N

I have the following function that I have to approximate for big N and I really have no clue what to do - I hope that anyone can help me out: $$P_N(s) = \begin{cases} \frac{1}{2^{N}} ...
0
votes
1answer
162 views

Approximation to the square root

I was reading an article that approximated a square root operator as follows $\sqrt{1+x+y} \cong \sqrt{1+x} + \frac{1}{2}y + O(xy,y^2) $ At first glance that looks like a Taylor series expansion, ...
0
votes
1answer
36 views

A power approximation function

I am trying to construct a function that would approximate $a^b$ using Maclaurin series. Here are my reasoning: Since $$a^b=e^{b\ln a}$$ and $$e^x=\sum^{\infty}_{k=0} \frac{x^k}{k!}$$ it should ...
3
votes
0answers
297 views

Taylor Series of Integral

I'm trying to come up with the Taylor expansion of an integral expression. For simplicity, consider the toy integral $$ ...
0
votes
3answers
57 views

I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator.

Let $f(x)=\ln x$. Then the Taylor polynomial of degree 2 at $x=e$ is $P(x)=1+\frac1e(x-e)-\frac1{2e^2}(x-e)^2$ I want to show that $\left | f(3)-P(3) \right |\leq \frac1{1000}$ without a calculator. ...
3
votes
0answers
44 views

Why does the moment of approximation matter for the end result?

I am trying to wrap my head around the reason why the moment of approximation matters for the end result of my analysis. As an example, let's take an equation for which we can still find the full ...
1
vote
1answer
166 views

Thinking through a Taylor error bound for arcsine

In lecture, we went through solving a Taylor error bound for arcsine. I followed most of it except for where it talks about the odds divided by the evens divided by $2n+1$ gaining in accuracy by a ...
2
votes
1answer
142 views

Uniqueness of approximations like the Taylor polynomial

Given a function $f: \mathbb {R} ^n \to \mathbb {R} $, I am curious about the uniqueness of a $k$th-order approximation at $c \in \mathbb {R}^n $, i.e. a function $\phi(x)$ such that $$ \frac {f(c ...
4
votes
3answers
488 views

Taylor Series for $e^x$ where $x = 1$, estimating the Error

I'm trying to calculate $e$ to a certain number of digits. The Maclaurin Series expansion of $\displaystyle e^x = \sum_{n=0}^\infty \frac{x^n}{n!}$. When $x = 1$ we can approximate the value of $e$ by ...
1
vote
1answer
172 views

Find taylor polynomial that approximates e^x with accuracy at least 1.

Find Taylor polynomial at $x=0$ which approximates $e^x$ with accuracy at least $1$ for each $x \in [-2,2]$. I dont undestand these questions that involve the $n^{th}$ remainder. I know I need to ...
8
votes
3answers
300 views

difference of square roots approximation

In two of my physics courses in the past week, I've come across an approximation for the difference of two square roots for large radicands: $\sqrt{x+a}-\sqrt{x+b}\approx\frac{a-b}{2\sqrt x}$ for ...
1
vote
1answer
95 views

Approximate function from sample data

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function. I don't have function definition. It's described as a fuzzy inference system. I have the inference system and can manipulate sample data for each ...
2
votes
3answers
143 views

Precision with Taylor Expansions

when you take a 1st order taylor expansion of a function, so: $$f(a) + f'(a)(x-a)$$ does that mean that if the result is only accurate to one decimal place? so for a value a.bcd, d would be the ...
5
votes
1answer
954 views

Approximating $\arctan x$ for large $|x|$

I would like to know if there is reasonably fast converging method for computing large arguments of arctan. Until now I've came across Taylor series that converges only on interval $(-1,1)$ and for ...
1
vote
1answer
136 views

Quadratic approximation of a cost function with a Taylor expansion

See also http://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-832-underactuated-robotics-spring-2009/readings/MIT6_832s09_read_ch12.pdf, page 92. Given an instantaneous cost ...
1
vote
1answer
371 views

Using binomial theorem find general formula for the coefficients

Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion: $$ ...
3
votes
1answer
102 views

Approximating the logarithm of a Laplace transform

Suppose $X$ is a random variable on $\mathbb R_+$ with finite mean, i.e. $\mathbb E X <+\infty$. Let $F_X(t)$ be its c.d.f. and $\mathcal{L}_X(\cdot)$ its Laplace transform, i.e. ...
1
vote
1answer
101 views

Limit of a sum for which the upper limit is also in the argument of the sum - Taylor series of $e^x$

A book I'm reading claims that $\frac{1}{2}(k-1)!\sum \limits_{j=0}^{k-3} \frac{k^j}{j!} \sim (\pi / 8)^{1/2}k^{k-\frac{1}{2}}$ as $k \to \infty$. I can get most of the expression to work out nicely ...
1
vote
0answers
36 views

Approximation of a function with certain restrictions at problematic points

I can't compute a Taylor series of a function like $f(x)=\sqrt{x}$ to some order around $x_0=0$, because the derivative at that point doesn't exist. If I consider the taylor series $Tf$ at any ...
2
votes
2answers
836 views

Help finding the absolute error with $n$th degree Taylor polynomials

I am trying to estimate the absolute error in approximating $\ln 1.09$ with the $3$rd-order Taylor polynomial centered at $0$. It's been a while since I've taken the Calculus and I'm afraid I need ...
1
vote
1answer
651 views

Approximate the integral $\int_0^1 \sin(x^2) dx$

I'd like to ask if someone can please give me a little push with this assignment: Approximate the value of the integral $\int_0^1 \sin(x^2) dx$ using only $\mathbb{N}$ numbers and basic operations ...
7
votes
2answers
495 views

Approximating roots of the truncated Taylor series of $\exp$ by values of the Lambert W function

To everyone: don't bother writing up another answer, i'm giving this bounty Antonio's answer. It just doesn't let me yet (24 hours delay). If you map the nth roots of unity $z$ with the function ...
0
votes
1answer
144 views

Taylor Series. Reusing an approximation of a function

I have this function, $e^{-x}$ bounded between 0 and 1500 and I have an approximation by Taylor Series of the same function bounded between 0 and 0.5. I would like to express my function $e^{-x}$ ...
3
votes
2answers
164 views

Next term in $(1+a/n)^n \rightarrow \exp (n)$

Working on the generalized birthday problem, where you draw with replacement from $\{1,2,3, \ldots,d\}$ and look for the number of draws $n$ for which you have greater than $1/2$ chance of a match I ...