# All Questions

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### Inequality in recurrence relation

I'm having a mental block understanding what is probably a simple inequality in a guess and check example for a recurrence relation. Would someone please explain to me how they obtain the inequalities ...
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### Proving theorems using the Compactness theorem

We say an infinite set $S$ is closed under $\wedge$ if for all $a,b$ $\in S$ so $a\wedge b \in S$. I need to prove that if S is closed under $\wedge$ and for all $a \in S$ we know is that $a$ is ...
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### If $f$ is differentiable in $(a,b)$ then $\frac{1}{f}$ is differentiable at $(a,b)$, provided $f(a,b)\neq0$

"Suppose that $f$ is a differentiable function at $(a,b)$. Prove that $\frac{1}{f}$ is differentiable in $(a,b)$, provided $f(a,b)\neq0$" We were given the following definition of differentiability: ...
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Suppose that the dollar cost of producing q appliances is c(q)=1000-0.05q+0.3q^2 And the demand function is given by p=20-0.025q Q1: compute the marginal cost when the quantity is equal to 1 Q2: ...
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### Subtracting scaled projection matrix from identity matrix

I am trying to understand what the following operation signifies. $$\rm W_n=I-2u_n u_n^H/u_n^Hu_n$$ where I and $u_n$ is described in section 6.3.4.2.3 of this document. My question is, what does ...
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### Image: proof re fractional exponents

Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?
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### Any power of $x$ from $1$ to $n$, inclusive, starts with a $9$
For any positive integer $n$, prove that we may choose a sufficiently long string of $9$s for a positive integer $x$ so that any power of $x$ from $1$ to $n$, inclusive, starts with a $9$. I was ...