AsafHaas
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Given the approximation $\cos(x)\approx 1$, how small must $x$ be to have $\frac{1}{2}\cdot 10^{-8}$ accuracy?
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5 votes

By taylor's theorem, it holds that $\cos(x) = 1 + R_1(x)$. In the Lagrange's form of the remainder, there exists $c$ such that $R_1(x) = \frac {(-\cos(c))} {2!} \cdot x^2$. $|R_1(x)| = |\frac {(-\cos(...

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Log with $\sqrt x$ base
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4 votes

$$\log_a(x) = \frac {\log_b(x)} {\log_b(a)}$$ for any $b$. therefore: $\frac 1 2 \log_{\sqrt 2}(x-2) = \frac {log_2(x - 2)} {2 \log_2(\sqrt 2)} = \frac {log_2(x - 2)} {2 \times \frac 1 2} = \log_2(x-...

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Concluding divergence based on first and second derivatives
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3 votes

You can prove that $f$ is not bounded, since then it will follow that $f$ is increasing and unbounded and therefore diverge to $\infty$ as $x \to \infty$. There is a sneek peek at the proof's outline ...

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How to solve this trigonometric equation $\sin (x)-\cos (x)=0$?
3 votes

Recall that $\sin(x) = \cos(\frac \pi 2 - x)$. Therefore your equation is: $$\cos(x) = \cos(\frac \pi 2 - x)$$ And from here we get: $x = \frac \pi 2 - x + 2\pi k \Rightarrow 2x = \frac \pi 2 + 2 \pi ...

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Finding the value of $\lim_{x\to 0} \frac{1}{x^3} \int_0^x\frac{t\ln(1+t)}{t^4+4}dt$
3 votes

Notice that both the numerator and the denominator tends to 0 as $x \to 0$. You can use L'Hôpital's rule, and then the numerator's derivative is... Good luck! :)

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Determine values of $ \ p , \ q \ $ if we have$ \ (3 \mathbb{Z}+1) \cap (4 \mathbb{Z}+2) =(p \mathbb{Z}+q) \ $
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2 votes

You can use the Chinese Remainder Theorem. Since $3$ and $4$ are coprime, the set of equations: $$x = 1 \mod 3 \\ x = 2 \mod 4$$ has a solution which is unique modulo $3\cdot4 = 12$ By observing we ...

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Cardinality of the set of all polynomials with coefficients in $\mathbb R$
2 votes

Notice that the cardinality of polynomials of degree $0$ (Only free coefficients) is $|\mathbb{R}| = \mathfrak{c}$ (We just map such polynomials to their free coefficients). The cardinality of ...

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Riemann sum for integral using subintervals
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2 votes

If I understood your question correctly, you want the Riemann sum of $f(t)$ for the partition $P = \{0, 1/100, 2/100, ..., 99/100, 1\}$ This is, by definition: $$\sum_{i=1}^{100} f(c_i) \frac 1 {100}$...

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Dimension of Image/Kernel of Linear Transformation
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2 votes

The transformation is a linear transformation. In order for a transformation $T$ to be a linear transformation, it has to implement 2 conditions: $$(1) \hspace{0.2cm} T(v + w) = T(v) + T(w) \\ (2) \...

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Taylor series for $f(x) = cos (x)$
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2 votes

The Lauren remainder in Lagrange form is $$R_n(x) = \frac {f^{(n+1)}(c)} {(n+1)!}x^{n+1}$$ for some point $c$. So seeking for an $n$ for which $|R_n(x)| \lt 0.1$ means: $|\frac {{cos}^{(n+1)}(c)} {(...

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Does $\sum_{n=1}^\infty \frac{\cos^2(n\pi)}{n\pi}$ converge or diverge?
1 votes

Notice that $\cos(n\pi) = (-1)^n$ and therefore $\cos^2(n\pi) = 1$, and try using the Limit Comparison Test.

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an $n$-order cyclic group generated by $a$ has $a^{kn}=e$
1 votes

An easy conclusion form Lagrange's theorem is that if $|G| = n$ and $a \in G$ then $a^n = e$. Proof: Since $<a> \le G$, By Lagrange's theorem $|<a>| = o(a) \mid |G| = n$. Therefore $n = \...

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Construct a matrix whose column space contains vectors (1, 1, 0) & (0, 0, 1) and row space contains vectors (2, 5) & (1, 2).
1 votes

Well as you said you can take $$A = \begin{bmatrix} 1 & 0 \\ 1 & 0 \\ 0 & 1 \end{bmatrix}$$ and then $Row(a) = span\{(1,0), (0, 1)\}$ is the entire space, ...

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Linear Algebra: $\mathcal{B}=\lbrace (e,1),(1,e) \rbrace$ is a basis of $U$ under new rules of addition and scalar multiplication.
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1 votes

No, it does not suffice to say that $e^x$ yields only positive numbers, or that $\mathcal{B}$ yields only elements in $U$. You need to show that every element in $U$ is a linear combination of the ...

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Translation for $if \exists i \in [m]$ s.t. $x_i = x$
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1 votes

This is saying in English: "if there exist an $i$ in the set $[m]$ such that $x_i = x$, then $hS(x) = y_i$. Otherwise, $hS(x) = 0$."

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boolean algebra, why is this try to simplify wrong?
1 votes

For the derivation, notice that there are 4 possible states of $a$ and $b$: $$a = 0, b = 0 \\ a=0, b=1 \\ a=1,b=0 \\ a=1, b=1$$ Now, the expression $(a*b)$ is true for the fourth state, the ...

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Give the equivalence classes of the relation aRb if and only if $a^4 ≡ b^4 \pmod {30}$ on the set $\{1,2,3,...,15 \}$
1 votes

The equivalence class of a $x \in A$ ($A$ is a set) is defined as: $$[x] = \{a \in A | aRx \}$$ Meaning it is the set of all items in $A$ related to $x$. Your question is to find all of those ...

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Differentiating constant piecewise function with interval dependent on x
1 votes

No. Take $$f(x) = \begin{cases} 1 & x \in \mathbb{Q} \\ 0 & \text{otherwise} \end{cases}$$ And $C =1$. Now your $g(x)$ is exactly $f(x)$ which is the Dirichlet function, and is not ...

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Find the characteristic polynomial $P_A(\lambda)$ of this matrix
1 votes

So the characteristic polynomial is: $$ p(\lambda) = \begin{vmatrix} -1-\lambda & 1 & 1 \\ 1 & -1-\lambda & 1 \\ 1 & 1 & -1-\lambda \\ \end{...

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Proving linear independence of a basis from coordinate vectors
1 votes

First, recall this facts about coordinate vectors: $$[v_1 + v_2]_B = [v_1]_B + [v_2]_B$$ $$[\alpha v]_B = \alpha[v]_B$$ $$v = 0 \iff [v]_B = 0$$ Now we can prove your claim. Let $\alpha_1, \alpha_2, ....

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$\lim\limits_{n\to\infty}a^{n}\left(\frac{n+1}{n}\right)^{n^{2}}$ converges for $a$ in what range?
1 votes

You can use the root test for sequences: Take the sequence in absolute value: The limit of the sequence with an $n$'th root is $$L = \lim_{n \to \infty}\sqrt[n] {|a_n|} =\lim_{n \to \infty} |a| (1+\...

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Find the Volume using Polar Coordinates Multivariable Calculus
1 votes

First I must say that $z = x^2 + y^2$ is a paraboloid and not a cone, just for being exact. Now for the problem: You need to calculate the volume between the paraboloid and the sphere in the domain ...

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Proof for hermitian matrices
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1 votes

As hinted in the comments, consider the matrix $A^*A$. Let us calculate its trace: $tr(A^*A) = \sum_{i=1}^n \sum_{j=1}^n ((A)_{i,j}(A^*)_{j,i}) = \sum_{i=1}^n \sum_{j=1}^n a_{i,j} * \overline {a_{i,j}}...

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Find the integral $\int_{0}^{\infty} \frac{\log(x)\operatorname{arccot}(x)}{\sqrt{x}}\,dx$
0 votes

It seems that the integral does not converge on $[1, \infty]$ for example, therefore not on $[0, \infty]$: $\lim_{x \to \infty} \frac {\frac {\log(x)\arctan(x)} {\sqrt x}} {\frac 1 {\sqrt x}} = \lim_{...

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When does a limit of a sequence equal both its lim sup and lim inf?
0 votes

It is equivalent. If $\limsup_{n \to \infty} a_n = \liminf_{n \to \infty} a_n = L \in \mathbb{R}$, then the set of partial limits will only contain $L$ (If it contains anything else: say bigger than $...

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Which statement is false about the ring $(\mathbb{Z}_{10}, +, \cdot)$?
0 votes

Consider the isomorphism $\varphi: \{0, 2, 4, 6, 8\} \to \mathbb{Z}_5$ given by $0 \to 0, 2 \to 2, 4\to4,6\to1,8\to3$.

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If $h|mn$ and $\gcd(m, n)=1$ then $h=dd^\prime$ where $d|m$ and $d^\prime|n$
0 votes

Eventually I figured out a proof without using the Fundamental Theorem of Arithmetic: Denote $\gcd(m, \alpha ) = u$. $u|m \Rightarrow \gcd(u, n)=1$. $u|\alpha \Rightarrow \alpha = \beta u$. Now ...

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Finding Inflection points
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Inflection points are defined as passing from convexity to concavity, which is equivalent as passing from an increasing $f^\prime$ to a decreasing $f^\prime$ for differentiable functions, its extremas....

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Proving a property of the closure of a set
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0 votes

You can show again the contrapositive: assume $\exists \epsilon_0 > 0: B(x, \epsilon_0) \cap M = \emptyset$. can you proceed showing that $x \notin \overline{M}$? (Hint: you need to show that $\...

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Want to clarify and check DFA and NFA attempt
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0 votes

I afraid that your answer for (a) is incorrect... Check for example that the word $w = 111$ is accepted by your automata, but $n_1(w) = 0 \pmod{3}$. Label the states in your answer to (a) as: $$q_0: ...

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