1
vote
1answer
23 views

Approximating zeros on an interval

I'm writing a program for my AP Calculus class, and I'm trying to write an equation solver that approximates the zeros of functions. Right now it can take symbolic derivatives and evaluate functions. ...
1
vote
2answers
65 views

Approximate a definite integral to three decimal places: $\int_0^2 \frac{dx}{\sqrt[3]{64+x^3}}$.

I try to expand function $$\frac1{\sqrt[3]{64+x^3}}$$ using Maclaurin series. So, $f(x) = 64{(1+ \frac{x^3}{64})}^{-1/3}$. I expand it and I get ...
4
votes
1answer
70 views

Applications of the Exponential Integral?

this is my first time asking a question on here so please forgive me if I have made any formatting mistakes. I have the integral $f(x) = \int_0^\infty \frac{e^{-t}}{x + t} \; dt$ and I have shown the ...
0
votes
1answer
40 views

Solving an ODE in Maple via Picard approximations

I am trying to use Maple to apply Picard approximations to the following ODE: $$ 2xyy' + x^2 - y^2 = 0,\ y(0) = 1 $$ I transformed it into the form $y' = f(x, y)$: $$ y' = \frac {y^2 - x^2} {2xy} ...
2
votes
0answers
42 views

Finding the cubic near minimax approximation for sin(x) on (0,pi/2)

I am really stuck here. Here is the question that I have. Find the cubic near minimax approximation for $f(x)=\sin(x)$ on $(0,\pi/2)$. So I defined $h(x)=ax^3 + bx^2 + cx + d - sin(x)$ The max ...
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
32 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
80 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
121 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
300 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
35 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
56 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
76 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
146 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
84 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
165 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 ...
30
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
54 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
323 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
93 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
53 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
288 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
445 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
43 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
172 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
96 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
246 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
51 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
322 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
85 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
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: $$ ...
2
votes
1answer
677 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
88 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
68 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
153 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
355 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
106 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
249 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
345 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
655 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
155 views

approximation formula for the integral

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