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21 views

solve equation involving digamma function

I have the following equations that I need to solve. $$ \psi(\alpha)-\psi(\alpha+\beta)=X_0 \\ \psi(\beta)-\psi(\alpha+\beta)=Y_0 $$ $X_0$ and $Y_0$ are known constants. Is there a way to atleast ...
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1answer
38 views

Arc Length in two dimensions by integration

I'm really at the end of my wits on this problem. Basically I'm trying to find arc length. The vector-valued function is: $R=\langle t,\sqrt{t}\rangle$ and $t\ge0$. We're looking for the length of ...
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0answers
40 views

Computer Algebra Systems for Experimental Mathematics (especially Integer Relations with PSLQ)

I would like to use a computer algebra system to do some experimental mathematics, particularly Integer Relation problems using the PSLQ algorithm. I know that Maple has a PSLQ implementation, but ...
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1answer
23 views

Most accurate linear approximation for two lines

Consider two lines defined by: $$\begin{aligned}y_1 &= m_1 x + b_1\\y_2 &= m_2 x + b_2\end{aligned}$$ where for the sake of argument, the domain of both lines is the same and everything is ...
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0answers
30 views

Euler's method and its error

A student asked me how to approximate the error of Euler's method, and the book he is using said "If a value of a function is approximated with $n$ steps, the error is proportional to ...
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3answers
53 views

Where did the linear approximation/linearization formula come from?

Where did the linear approximation/linearization formula come from? I understand that it takes root in the point-slope form and slope intercept form of a linear equation, but I don't understand where ...
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1answer
26 views

Determine the values of x for which the linear approximation is accurate to within 0.1.

So I've got a function: $$\frac{1}{(1+2x)^4}$$ with its linear approximation: $$1-8x$$ For all values of $x$ where the linear approximation is accurate within $0.1$, then surely we subtract the ...
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1answer
26 views

Use economisation to find linear approximation to x^2-x-1?

I've been given the solution to this question... It uses chebychev, and you get: $1/2(2x^2-1)-2x-1/2$ So the Chebyshev economisation polynomial is $-2T1 -1/2 T0$ I can see the logic in how this ...
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45 views

Questions About The Linear Algebra Behind Least Square Approximation

I am working on a few linear algebra problems, and I am stuck. I was hoping to get some directions on this site. Before I state my questions, here's the necessary context: Given a matrix $X$ ...
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1answer
25 views

How do I do Linearization at a point that lies on a curve?

I keep applying the formula to the info given but I keep getting lost/weird answers. Can someone please help me? I know $L(x)=f(a)+f'(a)(x-a)$ question Y(x) satisfies $x^2y^2 + xy = 6$. Point (x,y) ...
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1answer
363 views

Linear approximation of cos(28 degrees)

Evaluate cos(28 degrees) using linear approximation. I have done this so far, but the answer seems to be incorrect and I can't figure out why. ...
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1answer
25 views

Write an equation that approximates this relationship. Y = Seconds of daylight in day, X = a range of days

Disclaimer: This is a project for a math class. (who does math for fun anyways? Jk I actually enjoy math when I understand it, and not so much when I feel lost, but I digress ) The problem roughly ...
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1answer
38 views

Is best approximation from a linear subspace a linear map?

Let $X$ be a strictly convex Banach space, and $Y \subset X$ a closed subspace. Then for any $x \in X$ there exists a unique $y \in Y$ that minimizes the distance to $x$, i.e. a best approximation of ...
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0answers
20 views

Iterative approximation of non-constant values in linear equation

The issue regards an algorithm for iterative approximation of unknown transaction values. For each iteration (each day), we are give the total revenue of all transactions for that day, and we have the ...
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2answers
69 views

Linear approximation formulas for $x$ not near $0$

I am following MIT's calculus videos and I have noticed that when dealing with linear approximations, the professor calculates a set of approximation "formulas" for $x$ near $0$ like $1+x$ for $e^x$ , ...
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0answers
27 views

Correct terminology for polylines, their segments, knots, etc.

Background: piecewise-linear continuous functions $f(x_k)=y_k$ with fixed set of knots $x_k$ with restrictions on the angles between adjacent segments. The translator who dealt with my paper, ...
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2answers
129 views

Regression analysis on temperature/sensor data

Looking for a solution to what I thought should be an easy problem, but has me running in circles somehow... I'm working with two sets of data: he first set is raw values from a sensor ...
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0answers
69 views

Reason for the name “Ring of dual numbers”

The ring of dual numbers over a field $k$ is defined as the quotient $$k[\varepsilon]/\varepsilon^2.$$ I was reading this question with an interesting answer about some of their basic properties ...
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2answers
953 views

Linear approximation to ln(x) at x = 1, then estimate ln(1.08)

I know that the derivative of $\ln(x)$, or log of whatever base (x) = $(1/x)$ *the original function. If x is a more complicated expression, then the derivative would be $(x'/x)*f(x)$. If I knew the ...
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3answers
1k views

Linear approximation to $y = \sqrt{1-x}$ at $x=0$, then approximate $\sqrt{0.9}$ and $\sqrt{0.99}$

How do I find this? I know that the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$. Here, I would plug in $(1-x)$ instead of $x$. When $x = 0$, the slope would evaluate to $\dfrac{1}{2}$. I got ...
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1answer
73 views

Which number is larger if f'(x) is a differentiable increasing function for all x?

Suppose is a differentiable increasing function for all x. Which number is the larger and why? or ? I believe f(x) must be concave up everywhere since the derivative is increasing, but I am not ...
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1answer
104 views

Use least squares to estimate coefficients in a linear system containing noise

Problem I have a linear system of the form, $y=Ax+v$, where $v$ is noise. I need to use least squares to estimate the coefficients of the matrix $A$. Atempt I made the assumption that the error, $v$, ...
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0answers
79 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
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1answer
334 views

Relative error; absolute error divided by real value or approx value?

I have the following function; $$f(x) = -\frac{x}{2x + 4}\cdot v_r \Rightarrow \hat{f}(x) = -\frac{x}{4}\cdot v_r$$ Because $x$ is very small we can approximate $f(x)$ to $\hat{f}(x)$. Now the ...