# Tagged Questions

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### Prove there's $M>0$ such that: $f(x)\le Mx^2$

Let $f:[-1,1]\to\mathbb{R}$, three-times differentiable function and $f(0)=0$, $f(x)\ge 0$ for all $x\in[-1,1]$. Prove there's $M>0$ such that $f(x)\le Mx^2$. Hint: use Taylor formula. So ...
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### Series expansion of square root function

Could I please know how to expand: $$\sqrt{4-3x^2}$$ I simplified it to $\sqrt{1-\frac34x^2}$ but the question is if I let $x = x^2$, what will the coefficient of, say $x^5$ or $x^{10}$ be in the ...
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### Expand a sin(x^3) in Maclaurin's series and find a 30th derivative at (0)

I have a big task and problems with it. I have to expand this function in Maclaurin's series. $$cos(x^{3})$$ I tried expand it as $$\sum_{n=0}^\infty (x^n)/n!$$ but it's for $cos(x)$. So i don't know ...
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### Taylor expansion with non integer exponents in the rest

Consider the function: $$f(x)=\sqrt[3]{8x^2+4x+1}$$ 1) Find $a,b,\alpha,\beta$ such that: $$f(x)=ax^\alpha+bx^\beta+o(x^{-1/3})$$ 2) Find $A=f([0,+∞[)$ and prove that $f:[0,+∞[\rightarrow A$ is ...
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### Functions and their Taylor polynomials

Given a function $f$ from $\Bbb{R}$ to $\Bbb{R}$, we define $P_{f,n,a}$ to be the Taylor polynomial of $f$ of degree $n$ at $a$ (if the function itself is clear from the context, we simply write the ...
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### Taylor expansion - what order would be preferred?

Let say you want to calculate the following limit: $$\mathop {\lim }\limits_{x \to 0} \left( {\frac{1}{{1 - \cos x}}\ln \left( {\frac{{\sin x}}{x}} \right)} \right)$$ Obviously, Taylor Expansion ...
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### Finding Taylor's expansion for $f(x) = \sqrt{1 + x} -\sqrt{ 1 - x}$

I know I have to find the derivatives of $f(x)$ (i.e. $f'(x)$ ..) but I'm confused about what to do afterwards .
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### Gamma Type Integral

I was hoping someone could help me with a question I came across recently: essentially it's a gamma type integral that your asked to evaluate/reduce: ...
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### How does a taylor series of a binomial function equals a trigonometric function? [closed]

Any proof or derivation for the sinx and cosx function would be help. Image taken from http://en.wikipedia.org/wiki/Taylor_series
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### Coefficients of series

Suppose that i have a function $f(x)=\sum_{i=0}^{\infty}a_ix^i$ with radius of convergence $r_f>0$ and that i want to write $f$ in a form $f(x)={e^{g(x)}}$ where $e$ is natural logarithm base and ...
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### Derivative of a little-o remainder

If we have a $\phi: \mathbb{R} \times \mathbb{R}^n \times \mathbb{R} \to \mathbb{R}$, $\phi = \phi(t, \mathbf{q},\alpha)$ one-parameter group of infinitesimal transformation which is $\mathcal{C}^2$ ...
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### Approximating arcsin from above

I am very new to function approximations, and I am interested in approximating arcsin with a function $f$, s.t. $f(x) \geq \arcsin(x)$ for all $x$. Taylor series would give me a function which is ...
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### Difference in limits because of greatest-integer function

A Problem : $$\lim_{x\to 0} \frac{\sin x}{x}$$ results in the solution : 1 But the same function enclosed in a greatest integer function results in a 0 ...
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Assume a vector-valued function, for example ${\bf f}=(f_1, f_2)$, where $$f_1(x,y)= x^2+3xy$$ $$f_2(x,y)= 2xy+y^2$$ (here f is column vector, x, y are variables) Assume that each $f_i$ is a ...
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### Laurent series for an infinite sum of functions

I have an infinite sum of analytic functions that is guaranteed to converge for every $x$, except for $x=0$: $$g(x) = \sum_{n=1}^\infty f_n (x)$$ I want to expand the ...
I have this function, $e^{-x}$ bounded between 0 and 1500 and I have an approximation by Taylor Series of the same function bounded between 0 and 0.5. I would like to express my function $e^{-x}$ ...