# Tagged Questions

58 views

### Second difference $\to 0$ everywhere $\implies f$ linear

Exercise 20-27 in Spivak's Calculus, 4th ed., asks us to show that if $f$ is a continuous function on $[a,b]$ that has $$\lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^2}=0\,\,\,\text{for all }x,$$ then ...
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### How good an approximation to the derivative is an arc-length based approximation?

Note - my original definition below was wrong. I hope this replacement is better. The usual approximation to $f'(x)$ with step size $h$ is $D_h(f, x) = \frac{f(x+h)-f(x)}{h}$. This has so many nice ...
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### Show that the approximation to $f$'($x_0$) has discretization error $O$($h^2$)

We Define 3 grid points $x_{-1}$, $x_0$, $x_1$ with $x_{-1}=x_0-h$ and $x_{1} = x_0 + h$ with $h$ > 0. Given a smooth function f, show that the approximation to $f'(x_0)$ given by the centered ...
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### How do you solve part (b) to this polynomial interpolation question?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the polynomial interpolation method. (b) Assuming $f(x)\in C^3$, evaluate ...
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### How do you answer these questions regarding the Taylor series method?

(a) Approximate $f'(x_0)$ and $f''(x_0)$ using the values $x_0-h$, $x_0$ and $x_0 + \alpha h$ $(0 < \alpha)$ by applying the Taylor series method. (b) Assuming $f(x)\in C^3$, evaluate the ...
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### Derivatives as Linear Approximations

I have always thought of the fact that a derivative is a linear approximation as being nothing more than that- an approximation. But is there an epsilon-delta meaning behind that? Is there a stronger ...
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### 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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### Approximate $|x|$ with a smooth function

I am trying to get the derivative of $|x|$, and I want that derivative function, say $g(x)$, to be a function of x. So it really needs the |x| to be smooth (ex. $x^2$); I am wondering what is the ...
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### How to evaluate the derivate of a hypergeometric function w.r.t. one of its parameters?

I have to numerically evaluate the derivative of the hypergeometric function w.r.t. its first and second parameters $\large\frac{\partial}{\partial a}{_2F_1}\left(a , b ,c;z\right)$ and ...
Consider a random variable $z$ with a Gaussian distribution : $$\mathbf{z} \sim \mathcal{N} ( \mathbf{m}, \mathbf{V} )$$ Where $\mathbf{m}$ and $\mathbf{V}$ are mean and variance parameters. ...