# Tagged Questions

This tag is in the process of being removed; please do not add this tag to new questions. Rather, use tags that describe the subject of your question, like [calculus], [algebra-precalculus], etc. For more information, see meta: http://meta.math.stackexchange.com/questions/16425.

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### Word Problem, Calculus estimation homework

Roger runs a marathon. His friend Jeff rides behind him on a bicycle and clocks his speed every 15 minutes. Roger starts out strong, but after an hour and a half he is so exhausted that he has to ...
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### How do I find the sum of the infinite geometric series?

$$2/3-2/9+2/27-2/81+\cdots$$ The formula is $$\mathrm{sum}= \frac{A_g}{1-r}\,.$$ To find the ratio, I did the following: $$r=\frac29\Big/\frac23$$ Then got: $$\frac29 \cdot \frac32= \frac13=r$$ and ...
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### Equation of a line with a positive gradient [closed]

Two straight lines passing through the point (0,2) are tangent to the graph of the function y=1-x^2. Find the equation of the line with a positive gradient.
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### Ambiguous Limits in Area Determination

I am to find the centroid of the area bounded by the curve $y=8x^3-24x+11$, the $x$-axis and the line $x=-1$. Now I know that the centroid requires me to find the area under the curve first. I have ...
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### Find all values of $x$ at which the tangent line to the given curve has intercept $y= 2$

Find all values of $x$ at which the tangent line to the given curve has intercept $y = 2$ I am confused about the $y$-intercept $2$ the function $$f(x) = \frac{(2x + 5)}{(x + 2)}$$ The derivative ...
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### Diagonalization with the given eigenvalue and its vector

Let $-3$ be an eigenvalue of a $3\times3$ singular matrix $P$ and $$P\begin{bmatrix} 5\\ 3\\ -2 \end{bmatrix}=\begin{bmatrix} -20\\ -12\\ 8 \end{bmatrix}.$$ Then find whether $P$ is ...
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### Invertible Linear Maps Proof [closed]

1) Suppose $V$ is finite dimensional and $S$, $T$, $U \in L(V)$ and $STU = I$. Show $T$ is invertible and $T^{-1} = US$. 2) Suppose $V$ is finite dimensional and $R$, $S$, $T \in L(V)$ are such that ...
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### Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
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### How to find the width of a path around a rectangle, given the area of the path?

A man built a walk of uniform width around a rectangular pool. If the area of the walk is 165 square feet and the dimensions of the pool are 17 feet by 11 feet, how wide is the walk? How should I ...
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### In which direction to round the answer, if it represents maximal population that could be infected?

I'm calculating the maximum number of a population that could be infected. I have an answer of $240.0729395$. Should I round this to $241$ because that is the max? Everyone else says $240$ is ...
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### What is the greatest speed he can reach with an acceleration of 5.00 g before blacking out?

A jet fighter pilot wishes to accelerate from rest at $5.00$ G to reach Mach-3 (three times the speed of sound) as quickly as possible. Experimental tests reveal that he will black out if this ...
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### Norm of difference of two squares of matrices

Let $x,y$ be square matrices and $c$ be any scalar. Is it true that $\Vert x^2 \Vert - c^2 \Vert y^2 \Vert = \Vert x - cy \Vert ^2$? If this is true then I'm done with the proof of a theorem on ...
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### Describing an open interval I centered at c, $I \subseteq (a, b)$

Entire question: Let (a,b) be an open interval of Real numbers and let $c \in (a,b).$ Describe an open interval I centered at c such that $I \subseteq (a,b)$ I didn't quite get where I should've ...
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### Please check my proof on: $\sim$ is an equivalence relation $\Leftrightarrow S<G$

Problem: Let $\emptyset\ne S\subset G$, where $G$ is a group, and define a relation on $G$ by $a\sim b\Leftrightarrow ab^{-1}\in S$. Show that $\sim$ is an equivalence relation if and only if $S$ is a ...
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### Area of a Curved Surface

Find the area of the part o the surface $z=xy$ that lies within the cylinder $x^2+y^2=1$. I'm not sure how to set up the surface integral to compute this.
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### How to introduce flat cost of flow over a node using mixed integer programming.

In the set up for the program we have a graph where we are trying to minimize the cost of sending flow over the arcs. I have formulated the following linear program. \begin{array}{ll} \text{minimize} ...
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### Number of open sets in a metric space

I have got the following question which I could not solve: can a metric space have exactly 36 open sets? I believe if the metric space is finie, then it has to be discrete and so the number of open ...
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### Decompose a real symmetric matrix

Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ ...
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### Transform gradient to reference element

Minimal example of the problem My attempt I think this is not a linear solution like $$\nabla u = \nabla A_K x + \nabla b_K$$ which must be wrong because $A_K$ is a ...