0
votes
1answer
25 views

Approximating an interval [duplicate]

Corollary C is derived from $E_(x) = \frac{f''(x)}{2} \cdot (x-a)^2$ I'm having serious issues understanding this problem and I'm just not getting it right. I'm preparing for a test so this isn't ...
2
votes
1answer
28 views

Error formula for linearization

Can anyone shed some light on this formula? I can't find any information on it. It has three corollaries that I also need to understand:
1
vote
0answers
66 views

Sharpness of the upper bound $(1-x)^n \leq 1 + \frac{nx}{2}$

Here is a known inequality: $$(1-x)^n\leq 1+\frac{nx}{2}\qquad \text{for} \, \frac 1n\leq x\leq 1 $$ I am wondering if there is a better upper bound than this? Thank you.
1
vote
1answer
109 views

Stable algorithms from a backwards recurrence?

This is homework, so please only hints. We see that the recurrence $$I_n=\frac{1}{n}-\alpha I_{n-1}$$ can be used to compute the value of an integral, but the algorithm is unstable when $|\alpha ...
2
votes
1answer
154 views

Find derivation (dB/decade) for given amplitude characteristic of low pass filter [Hz, -]

I am trying to find derivation (differential attenuation) for frequency's 600 and 2000 Hz for given amplitude characteristic of low pass filter, which look like this: I assume, that I should ...
0
votes
1answer
30 views

Check my short proof - asymptotic approximation, which function is bigger

The goal of this exercise is to show that $\ln(n+1)-\ln(n) = O(\frac{1}{n})$ what I did is: I used the fact that if $f=O(g)$ then $\frac{f}{g}=O(1)$. $\ln(n+1)-\ln(n)=\ln(\frac{n+1}{n}) = \ln(O(1))$ ...
2
votes
0answers
54 views

Approximating a function using its integral

Question: Let $f:\Bbb R \to \Bbb R \in C^{1}, \forall \delta>0:$ $$F_\delta = \frac 1{2\delta}\int^{x+\delta}_{x-\delta} f(t) \, d(t)$$ in $[a,b]$ prove that $\forall \varepsilon>0 \exists ...
0
votes
1answer
63 views

How do I use the linear approximation of a function given a value, a, and change in x?

My book gives a few definitions/formulas for obtaining linear approximation, but I'm having trouble understanding how to use them. Heres the question: a.) Use the Linear Approximation for f(x) = ...
1
vote
2answers
128 views

Least squares problem linearization

Suppose we want to find the best coefficients $a$ and $b$ that fits the data we have according to a model of the form $$ y = a t e^{bt} \text{ or } y = a e^{bt} \text{ or } y = a \left( \frac{x}{b+x} ...
2
votes
1answer
72 views

About continued fractions as best rational approximations

I'm reading this notes about continued fractions: http://www.math.jacobs-university.de/timorin/PM/continued_fractions.pdf I had no problems understanding everything there, except one thing that has ...
0
votes
1answer
139 views

Simpson's Error Bound Estimation

The problem: I need to use Error Bound to find n (least) to the $10^{-9}$ in approximating the integral of 5e^x^2 from 0 to 1 I'm using $$Error(Sn) \le \frac{k(b-a)^5}{180N^4}$$ I found the 4th ...
29
votes
7answers
1k views

Pi Estimation using Integers

I ran across this problem in a high school math competition: "You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the ...
1
vote
1answer
24 views

The value of $w$ also has a max error of $p\%$

Suppose $\frac{1}{w}=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ where each variable $x,y,z$ can be measured with a max error of $p\%$ Prove that the calculated value of $w$ also has a max error of $p\%$ ...
0
votes
1answer
53 views

I know what I need to do but dont know how to apply: the question related to The first order approximation theorem

$\mathbf{Question:}$ Prove that $\displaystyle \lim_{(x,y)\to (0,0)} \dfrac{\sin(2x+2y)-2x-2y}{\sqrt{x^{2}+y^{2}}}=0$ $\mathbf{My\ ideas:}$ I will use the First Order Approximation Theorem. But ...
2
votes
1answer
42 views

Verifing the solution

$\mathbf{Question:}$ Let $f(x,y)=e^{\sin(x-y)}$ for $(x,y)\in \Bbb R^2$ Find the affine function that is a first order approximation to the function $f$ at the point $(0,0)$ $\mathbf{Answer:}$ ...
4
votes
1answer
264 views

Linearization of an implicitly defined function.

Problem: Given the equation: $xz^{2}+y^{2}z^{5}=19$ Also given: (3,4,1) is a solution to the equation. This point is not the only solution. 1) Find dz/dx and dz/dy (through implicit ...
0
votes
1answer
91 views

Newton-Raphson in $\mathbb{R}^n$

Let $U\in\mathbb{R}^n$ be open and $f:U\to\mathbb{R}^n$ be a $\mathcal{C}^1$ map. $\exists p\in U$ such that $\;f(p) = 0$ and $Df_{|p}$ is invertible. Define $\phi(x)= x-(Df_{|x})^{-1}f(x)$, show ...
1
vote
1answer
52 views

Properties of Lebesgue functions

If $f\in \mathcal {L}$ then there exists a sequence $\{f_k\}$ of step functions s.t. $\lim_{k\to\infty} f_k(x)=f(x)$ for almost all $x$ and $$\lim_{k\to\infty} \int|f(x)-f_k(x)|\,dx=0.$$ If I have ...
0
votes
2answers
249 views

How to show that a measurable function on $R^d$ can be approximated by step functions?

In Stein's Book: Real Analysis, Theorem 4.3 says that any measurable function $f$ on $R^d$ can be approximated by step functions. But the proof provided there only show that when $f=\chi_E$, with ...
0
votes
3answers
354 views

Approximating the integral $\int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$

How would I go about evaluating $I = \int_0^{0.1} \sqrt{1-1/2\sin^2(t)} dt$ to four decimal accuracy?
0
votes
1answer
25 views

Approximation related to resonance

Can someone help me with this problem. We have $$x(t)=N \sin (w_{0} t)+\frac{w_0}{w_1}e^{\frac{-t}{T}}\sin (w_{1}t)$$ and $w_1=(1+\frac{\delta_1}{N^2})w_0$ for some $|\delta_1|\leq 1$. I need to ...
2
votes
2answers
42 views

Formula for the pseudofrequency using approximations

We know that $w_1=\frac{\sqrt{4w_{0}^2-b^2}}{2}$ and $(1+u)^{\alpha}=1+\alpha u + \mathrm{O}\left(u^2\right)$ I need to show that ...
0
votes
0answers
137 views

Reciprocal uncertainty

In the equation $y = \dfrac{1}{L + 0.4 d}, L$ has an uncertainty of $± \,0.025$ and $d$ has an uncertainty of $±\,0.01$. I understand that the uncertainty of $L + 0.4 d$ is $0.025 + 0.4 \times 0.01 ...
1
vote
0answers
25 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
2
votes
1answer
48 views

Prove that s is finite and find an n so large that $S_n$ approximate s to three decimal places.

$$S=\sum_{k=1}^\infty\left(\frac{k}{k+1}\right)^{k^2};\hspace{10pt}S_n=\sum_{k=1}^n\left(\frac{k}{k+1}\right)^{k^2}$$ Let $S_n$ represent its partial sums and let $S$ represent its value. Prove that ...
1
vote
1answer
94 views

understanding the least squares criterion

I was given 20 data points and asked to choose the most suitable lowest degree polynomial to fit them using the least-squares criterion. I looked it up, but what i found seems far too complex or just ...
1
vote
0answers
26 views

clairification on standard deviation

I have a homework question that gives me a set of $x$ values and their respective $f(x)$ values and asks me to find the line which best fits the data. I have to do this by finding the estimated ...
2
votes
2answers
218 views

Using differentials to approximate a function

So I have a homework problem that I cannot figure out. I am supposed to approximate the value of $\sqrt{(4.98)^2-(3.03)^2}$ using differentials. What I have so far is $$f(x,y)=\sqrt{x^2-y^2}$$ ...
1
vote
1answer
65 views

How to approximate the error of a starting guess?

Given the equation $$ \frac{1}{x^4}+e^{x-100}=10^8 $$ that has one positive root > 1, formulate Newton's method for finding the root. Make one iteration with starting value = 1. Try to make another ...
0
votes
0answers
50 views

How to know number of digits that are lost from condition number?

I've got the formula for computing the condition number of a problem: Condition number = Rout / Rin where Rout is the relative error in output and Rin is the relative error in input. And if the ...
3
votes
4answers
315 views

What is the correct value of $\pi$

I have seen that: $\pi = 22/7$ $\pi = 3.14\ldots$ $\pi = 17 - \sqrt{192}$. $22/7 \gt \pi$ $22/7 \lt\pi$ My brain storming doubt was, is A = B? Is B = C? Is C= A = B? how D and E are correct or ...
1
vote
0answers
84 views

integral with bessel function represented as a series [duplicate]

Possible Duplicate: prove equality with integral and series This integral was my homework question with $p=2$ and $n=1$. I am wondering if one can get the general formula for p, or at least ...
1
vote
1answer
323 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: $$ ...
2
votes
1answer
644 views

approximation hypergeometric distribution with binomial

Let $X$ be $\rm{Hypergeometric}(2n,\ell,n)$ and $E(X)=\frac{1}{2} \ell=:\mu$. Is it possible and how to approximate the $q$-th central moment $E(X-\mu)^q$ of the hypergeometric distribution by the ...
3
votes
1answer
86 views

is the approximation of the sum true?

Someone commented under my question Calculation of the moments using Hypergeometric distribution that $$ \sum_{k=0}^l\frac{{l \choose k}{2n-l \choose n-k}(2k-l)^q}{{2n\choose n}}\sim \sum_{k=0}^l ...
0
votes
1answer
66 views

Expansions of Hermite functions

I am wondering if someone knows good references. I am looking for expansions of Hermite functions, which gives connections between rates of decay and smoothness of coefficients. Thank you for your ...
1
vote
0answers
146 views

integral with Bessel function

Let $n$ be half an odd integer, say $n=k+1/2, k \in Z$. Let $q\geq 1$. I would like to calculate (or approximate) the following integral $$ \int_0^{\infty}\left(\sqrt{\frac{\pi}{2}}\cdot 1\cdot 3\cdot ...
1
vote
2answers
330 views

Finding the second-degree polynomial that is the best approximation for cos(x)

So, I need to find the second-degree polynomial that is the best approximation for $f(x) = cos(x)$ in $L^2_w[a, b]$, where $w(x) = e^{-x}$, $a=0$, $b=\infty$. "Best approximation" for f is a function ...
1
vote
1answer
104 views

Is there a typo in Calculus:Early Transcedentals?

I just finished doing my homework on Local Linear Approximations in 3-space (Ch.13.4). In one of the problems the answer I got is different from the answer key. Problem 39. We have a function ...
2
votes
1answer
75 views

asymptotic limit at the integral

I would like to get an asymptotic limit at the following integral: for $p\ge 2, n \in N$, $t \ge 0$ $$ \int_{0}^{\frac 12 \sqrt{(n+1)!}}\left(1-\frac{t^2}{2^2(n+1)!}\right)^p \mathrm{d} t $$ I think ...
1
vote
1answer
236 views

Newton-Raphson's Method to find $\sqrt{2012}$

I am asked to find $\sqrt{2012}$ using Newton-Raphson's Method with the following recursive method $$x_{n+1} = \frac{1}{2} (x_n + \frac{a}{x_n}) $$ I notied that give same answers as using ...
2
votes
1answer
327 views

Sum of power series

Good morning, I have difficulties to find an approximation formula (or bound from the height) for the sum of the following power series $\sum \limits_{k=1}^\infty e^{-k^2}x^k$. Thanks
1
vote
2answers
603 views

Deriving the approximation formula

$f'(x) \thickapprox$ $\frac{1}{2h} [ 4f(x+h) - 3 f(x) + f(x + 2h)]$ I need to derive the approximation formula for the function above. And I need to show that it's error term is of the form ...
2
votes
0answers
139 views

approximation of the sum

I have difficulties to find an approximation formula (or bound from the below) for the following sum: $$ \sum_{k=1}^n\left( \frac{1}{35}\right)^{k-1}(n-k)!\left(k-\frac 32\right)!. $$
2
votes
2answers
153 views

approximation formula for the integral

Get an approximation formula for the following integral: $$ \sum_{k=1}^n \left( \frac{1}{35} \right)^{k-1}\int_0^{\frac{\pi}{2}}\cos^{2(n-k)+1}(y) \cdot \sin^{2(k-1)}(y) \, dy $$
1
vote
1answer
50 views

Euler's approximation of $m' = -\frac{m}{V}v$

Continuing my previous question from today - I should code a program that finds approximate value of the aforementioned differential equation. The full text of the assignment is: Water purifier with ...
1
vote
2answers
854 views

Approximating a sum of exponential distribution with a normal distribution

Here is the actual question: $A$ is random variable representing the lifespan of a component. It is an exponential law with an average of 10. Considering a system with $n$ components $A$, what is the ...
2
votes
1answer
254 views

Euler's method approximation

I'm supposed to write a program for approximating the value of function $y = y(x)$, which is given as: $$y' = \frac{1+y}{1 + x^2}$$ I also know that $y(0) = 0$. I should approximate the value for ...
1
vote
1answer
595 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 ...
1
vote
1answer
312 views

Complex-Analytic theorem similar to Runge's theorem

I'm trying to prove a result similar to Runge's theorem and Mergelyan's theorem (link at the bottom of the previous link), but without the condition of analyticity. The problem is as follows: Let γ : ...