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.

learn more… | top users | synonyms

1
vote
1answer
39 views

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 ...
2
votes
6answers
248 views

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 ...
3
votes
1answer
43 views

Area enclosed between half lines in polar space

I don't know if the anwser to my question is obvious because I cannot find any explanation anywhere on google. Question The blue region $R$ is bounded by the curve C with equation $r^{2} = ...
1
vote
1answer
23 views

Simple question (hopefully) on unitary method

In India we have an exam called NEST. I gave it today, and this was a question I encountered: Lactobacillus sp. and Streptococcus sp. are two bacterial species responsible for curdling milk. One ...
0
votes
1answer
27 views

Trig question, inequality

How can I find the following product using elementary trigonometry? Suppose $0 \lt x \lt \frac{\pi}{2}$ is an angle measured in radians. Use the trigonometric circle and show that $\cos(x) \le ...
-1
votes
1answer
12 views

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.
1
vote
1answer
24 views

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 ...
1
vote
3answers
208 views

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 ...
1
vote
3answers
47 views

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 ...
-1
votes
2answers
29 views

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 ...
2
votes
4answers
40 views

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 ...
0
votes
1answer
14 views

Linear Operators Injectivity and Surjectivity

Suppose T $\in L(P(R))$ is such that T is injective and deg Tp $\leq$ deg p for every nonzero polynomial p $\in P(R)$. Prove that T is surjective and that deg Tp = deg p for every nonzero p $\in ...
7
votes
1answer
265 views

Galois over Galois

I am working on this exercise: If $E$ is an intermediate field of an extension $F/K$ of fields. Suppose $F/E$ and $E/K$ are Galois extensions, and every $\sigma\in Gal(E/K)$ is extendible to an ...
0
votes
1answer
23 views

How to know when a line is parallel to the xz-plane

What are some features of the equations of a line that is parallel to the xz plane, but does not lie on the plane, and is not parallel to any of the axes? So far all I got: -dot product of plane's ...
1
vote
2answers
107 views

Determine the number of solutions of nonlinear system without solving.

$x^2-y^2+2y=0$, $2x+y^2-6=0$ I need to determine the number of solutions without solving it. There is a hint that a graph can help but I am still not sure how to go about this. Thanks
1
vote
1answer
1k views

Vector equation and parametric equation for a line segment

Suppose that $P = (1,1,7)$ and $Q = (8,6,1)$. Inside parenthesis are x-coordinate value, y-coordinate and z-coordinate. The question is to find vector and parametric equation for a line segment. Now I ...
4
votes
2answers
539 views

To show that Fermat number $F_{5}$ is divisible by $641$.

How can I show that Fermat number $F_{5}=2^{2^5}+1$ is divisible by $641$.
1
vote
1answer
76 views

How to find the second derivative of an implicit function?

We know from multivariable calculus that if $y(x)$ is a function given implicitly by the equation $F(x,y) = 0$, then $$ \frac{dy}{dx} = -\frac{F_x}{F_y} \tag{1} $$ This is quickly proved by applying ...
2
votes
2answers
141 views

Prove that every odd natural number divides some number of the form $2^n - 1$ [duplicate]

Suppose that $m$ is an odd natural number. Prove that there is a natural number $n$ such that $m$ divides $2^n -1$. I have absolutely no idea how to tackle this; any assistance would be welcome.
1
vote
1answer
29 views

How to find an equation of the plane, given its normal vector and a point on the plane? [duplicate]

I have a question regarding vectors: Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
3
votes
1answer
696 views

Moment generating function of two independent variables

The moment generating functions of two independent variables $X$ and $Y$ are $M_X(t)=\exp(2e^t-2)$ and $M_Y(t)=\left(\frac34e^t+\frac14\right)^{10}$. What are (a) $P(X+Y=2)$; (b) $P(XY=0)$; ...
1
vote
0answers
29 views

Not lebesgue integrable function?

I want to consider the function $f:[-1,1]\times [-1,1]\rightarrow \mathbb R:f(x,y)= \begin{cases} \frac{xy}{(x^2+y^2)^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases} $ And I have ...
8
votes
1answer
210 views

Maximal Ideals and Maximal Subspaces in normed algebras

This is a kind of "prove or give a counter-example" question, and I'm having some difficults with it: By a maximal ideal $I$ of an algebra $A$, we mean an ideal $I\neq A$ which is not properly ...
1
vote
4answers
1k views

What's wrong with my aproach to solving this equation with multiple logarithms?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
2
votes
3answers
3k views

how to prove vector norm equivalence in finite dimensional space($\mathbb{R}^{n}$)?

In most of the vector norm material, it was mentioned that the following inequalities can be proved, but no one provided the proof: $$\lVert x\rVert_2\le\lVert x\rVert_1\le\sqrt{n}\lVert x\rVert_2;$$ ...
7
votes
3answers
437 views

Homotopy equivalence of two different gluings of $B^n$ and an arbitrary space $X$

Let $f, g: S^{n-1} \to X$ be a pair of homotopic continuous maps. Let $X \cup_f B^n$ and $X \cup_g B^n$ be the respective adjunction spaces (pushouts of $B^n \hookleftarrow S^{n-1} \rightarrow X$). I ...
0
votes
1answer
35 views

Relations that are: reflexive but not transitive; transitive but not symmetric; symmetric but not reflexive

I have an incomplete answer to my question. Can anyone help me answer the last two parts. My question is: Find example of a set $S$ and three relations $R_1$, $R_2$, $R_3$ on it such that ...
0
votes
3answers
24 views

Transforming a linear congruence equation into an equivalent one

Is the equation $x \equiv -6 \bmod 5$ identical to $x \equiv 4 \bmod 5$ or to $\equiv 1 \bmod 5$? Generally what is the best way to convert negative constant into positive? Do we have a formula for ...
0
votes
2answers
32 views

How to quickly solve simple linear congruence equation

How to solve $97x \equiv 1 \mod 61$ in a quick way? I tried to solve it in the Diophantine equation form $97x + 61y = 1$ by using Euclidean algorithm and got the sample result of $x = 22$ and $y = ...
1
vote
1answer
1k views

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 ...
0
votes
1answer
35 views

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 ...
0
votes
1answer
915 views

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 ...
0
votes
1answer
30 views

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 ...
0
votes
0answers
21 views

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 ...
0
votes
1answer
33 views

Find the number of positive integer $a \leq n$ such that $(a,n) = (a+1,n) = 1)

For every positive integer $n$, let $$A_n = \{a \in \mathbb{N} \mid 1 \leq a \leq n \mid gcd(a,n) = gcd(a+1, n) = 1\}$$ Evaluate $\mid A_n\mid$ Assume that $n$ has the factorization ...
4
votes
1answer
19 views

Solving Differential equations with Laplace transform

$\displaystyle y''+4y'+3y=e^{-t}$, given $\displaystyle y(0)=y'(0)=1$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ $\displaystyle [s^2\bar y-sy(0)-y'(0)]+4[s\bar ...
2
votes
1answer
37 views

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 ...
3
votes
1answer
47 views

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.
1
vote
2answers
27 views

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} ...
0
votes
2answers
35 views

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 ...
2
votes
2answers
53 views

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 $A$ ...
0
votes
0answers
6 views

Transform gradient to reference element

Minimal example of the problem My attempt I think this is not a linear solution like \begin{equation} \nabla u = \nabla A_K x + \nabla b_K \end{equation} which must be wrong because $A_K$ is a ...
1
vote
0answers
12 views

Convergence of this priori error in FEM?

Problem My attempt I think h is the size of the mesh. C is a constant which probably depends on the size of the mesh, I think. I think the error converges linearly and dependent on the size of ...
1
vote
1answer
26 views

rearrange $t - (m-q)^2 = v - (m-p)^2$ for quadratic formula form $ax^2 + bx +c = 0$ solving for $q$

I have the equation $t - (m-q)^2 = v - (m-p)^2$ which I would like to rearrange to be able to apply the quadratic formula, and solve in terms of $q$. Accordingly, it needs to be in the form: $ax^2 ...
0
votes
1answer
26 views

shortest point on a line segment from point out side the line

from the above pic I found the value x from line (p1,p2) and point a using y=mx+b and imaginary red line which is perpendicular to black line having slope -1/m and the intersecting point x. the ...
2
votes
1answer
87 views

Uniform convex space

Please I want to know if this space $$H^1_{0,p}([0,+\infty))=\lbrace u, u\in AC([0,+\infty)), u(0)=u(+\infty)=0,\sqrt{p}u'\in L^2\rbrace$$ where $p>0$, $p\in L^1((0,+\infty))$ ...
0
votes
2answers
271 views

what is the maximum number of non loop edges that can exist in an undirected graph

please tell me a equation to find maximum number of non loop edges that can exist in an undirected graph. for example if vertices are 10 then how many non loop edges can exist?
1
vote
3answers
38 views

solve $-(x_m - x_q)^2 = -(x_m - x_p)^2$ in terms of $x_q$

I have an equation, $-(x_m - x_q)^2 = -(x_m - x_p)^2$ which I want to solve in terms of $x_q$. I can see (by using a number line) that $q$ can have two solutions: $x_q = x_p$ or: $x_q = 2x_m-x_p$ ...
-1
votes
0answers
255 views

What is a bi-rhombus? [closed]

Can anyone tell me what a bi-rhombus is? I need it for my school project.
0
votes
0answers
34 views

Linear maps and linearly independent sets

Suppose $v_1,...,v_m$ is a linearly dependent list of vectors in $V$. Suppose also $W \neq \{0\}$. Prove there exist $w_1,...,w_m$ in $W$ such that no $T\in L(V, W)$ satisfies $Tv_j = w_j$ for $j = ...