Tagged Questions
4
votes
3answers
55 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
41 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
175 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
52 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
60 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 ...
3
votes
1answer
257 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 ...
0
votes
1answer
68 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
157 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
69 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
84 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
33 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
339 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
370 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 ...
0
votes
1answer
126 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
152 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 ...
