# Tagged Questions

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### Can someone explain in general what a central difference formula is and what it is used for?

Topic- Numerical Approximations
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### Relationship between Lagrange interpolation and Taylor expansion

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h_{-1}$ and $x_{1} = x_0 + h_1$ with $h_1, h_{-1}$ > 0. Given a smooth function f, and an approximation to $f'(x_0)$ given by the ...
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### How do I construct such a numerical method for solving ODE?

I am asked to expand $x(t+h)$ and $x(t+2h)$ around $t$ up to the rest term of the third order, find $A, B, C \in \mathbb R$ such that $$x'(t)=\frac{Ax(t)+Bx(t+h)+Cx(t+2h)}{h} + O(h^2)$$ and based on ...
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### Numerical Differentiation Given Set Of Values

Given the values $f(0),f(h),f(2h)$ and $f'(h)$ , I need to find a numerical differentiation of highest approximation order to approximate $f''(0)$. Usually I'd use Taylor expansion , but I need to ...
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### Finite differences coefficients

I'm interested in deriving a forward finite difference approximation for the gradient of a function, $f(x)$, at the point $x = x_i$ using $k+1$ points. If the spatial domain is uniformly discretized, ...
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### 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 ...
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### Taylor polynom, residual for symmetric values

When creating the taylor polynom for a $C^3$-function around a certain point i get the formula $f(z+h)=f(h)+hf'(z)+\frac{h^2}{2}f''(z) + \frac{h^3}{6}f'''(z) + R$ Now lets say I create the polynom ...
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### Use Taylor polynomials with remainder term to evaluate the following limits $\large\frac{e^x-x-1}{x^2}$

My work: Since $\large e^x=\sum\limits_{j=0}^\infty \frac{x^j}{j!}$, then $\large\frac{e^x-x-1}{x^2}=\sum\limits_{j=2}^\infty \frac{x^{j-2}}{j!}=\sum\limits_{d=0}^\infty \frac{x^{d}}{(d+2)!}$. (Let ...
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### Taylor expansion: first derivative approximation with third order

About first derivative approximation with third order. Let $$f'(t)=\frac{(2t+h)\cdot{f(h)}-4t\cdot{f(0)}+(2t-h)\cdot{f(-h)}}{2h^2}+R.$$ Show that $$R=\frac{f'''(\xi)\cdot{(3t^2-h^2)}}{6}$$ and ...
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### Numerical Analysis best estimate on polynomial order

I need to determine the best integer value of $k$ for the equation: $$\arctan(x) = x + O(x^k) \text{ as x\to 0}$$ Taylor's Theorem with Lagrange Remainder would ...
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### Numerical Analysis and Big O

How can I show that $e^x -1$ is not $O(x^2)$ as $x\to0$ I'm not sure where to start. We can use Taylor's Theorem with remainder: e^x = \sum\limits_{k=0}^n\dfrac{x^n}{n!} ...
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### How to choose a numeric approach for derivates

I would like to find the derivate from some combined logistic and exponential functions that all describe the same data numerically. $$f'(t)=\frac{f(t+h)-f(t)}{h}$$ seems not the best choise for ...
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### estimating the error of $\sin(x) = x$ with Taylor's Theorem

I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places? Here is ...
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I'm working on the last problem in an assignment, and need some guidance on what to actually start by doing. The question is asking me to use taylor expansion to determine the leading error term ...
How can I proof that backward Euler method has consistency order 1? Implicit function theorem states that for a sufficiently small $h$, $$\vec{y}_1 = \vec{y}_0 + h f(t_1,\vec{y}_1)$$ has a unique ...