For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.

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3answers
30 views

Integration by substitution exam question help

$$\int_1^2 x(2x-3)^4 \, dx\\ U = 2x - 3$$ I have rearranged to get $dx = dU/2$. So I am now at $\int xU^4 \, dU$ I am not quite sure what to do with the $x$ as it is not cancelled out as I thought ...
-1
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1answer
19 views

Solving a logarithmic equation with variables on each side

Okay, so while doing a problem for my calculus class I was required to graph two functions in order to see where they intersect, as according to my teacher there is no way to solve it analytically. ...
-1
votes
2answers
24 views

Evaluate $\lim _{n\to \infty }\left(n+1\right)\cdot I_n$

We have $I_n=\int _0^1\:\frac{x^n}{x+1}dx$ , and we need to find $\lim _{n\to \infty }\left(n+1\right)\cdot I_n$
1
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1answer
21 views

solve the integral bu substitution

I tried substituting $x=sin\ u$ but I didn't get nowhere, can someone just give me a hint how to solve this integral? $$\int\frac{dx}{(x^2-1)^2}$$
0
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1answer
24 views

Minimize the norm of w.

Why is the same minimize the norm of w and to minimize ${1 \over 2} \cdot |w|^2$? I have tried to derive the norm but the result is the following $${1 \over {2 \cdot |w|}}$$
7
votes
3answers
145 views

For what functions is $y'' = y$?

What functions $y = f(x)$ have the property that $f(x) = f''(x)$, i.e. what functions have the same integral and derivitive? I could think of $ce^x$ and $ce^{-x}$ (where $c$ is a constant), but are ...
0
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0answers
9 views

How do I find the rate of change in area of rotating loop?

Say we have a rectangle of side lengths $a$ and and $b$ that is rotating with one of its sides $a$ about a fixed axis with angular speed $v$. Consider a line $l$ a distance $x$ from the axis of the ...
1
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0answers
16 views

Calculate the right Riemann sum to approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5]$.

I'm attempting to calculate the right Riemann sum and approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5] = [a, b]$. $$\sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta ...
2
votes
1answer
23 views

Inequality which is true for almost every n

In my assignment I have to prove the following statement: Let $l$ be a natrual number. Prove that for almost every $n$ the following inequality is true: $n\lt\sqrt{n ^ 2 + l}\lt n+1$ I chose to ...
4
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3answers
41 views

Evaluating $\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$ by substitution

$$\int_{\sqrt{2}}^{\sqrt{5}} \frac{x^3}{\sqrt{x^2-1}} dx$$ $u^2 = x^2 - 1$ I have worked out that $dx = du$ and that $u = x - 1$ so, $\int\frac{x^3}{u} du$ - but I'm stuck at this stage. Any ...
4
votes
2answers
70 views

What's so special about radians? (Differentiation) [duplicate]

It seems to me that radians have lots of very special properties that allow us to do maths with trigonometric functions. When I first came across radians, I was led to believe that they were designed ...
0
votes
2answers
23 views

limit of sequnce in real numbers

For every tow real numbers $a$ and $b$,with this condition that $0<a<b$ ,define sequence ${x_{n}}$ to the following: $x_{1}=a$ $x_{2}=b$ ${x_{n}}=\frac{ x_{n-1}+ x_{n-2} }{2}$ (for ...
-5
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0answers
27 views

a problem on functions [on hold]

Given $$f(x)= \sin \Big(\frac{\pi}{3[x-x^2]}\Big) $$ Where $2<x<3$, and $[ \; \;]$ is GIF (the geatest integer function). I am looking for the derivative of $f(x)$. Please, is there a ...
1
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3answers
83 views

Proving wrong the fundamental theorem of calculus

So today I have a car zig-zaggging, and I want to compute its average orientation (as compared to the horizontal) at each time (blue arrows). The first idea that comes in mind is : just take the start ...
0
votes
2answers
18 views

Finding the roots and the rescaling of an equation

This question is taken from Hinch's book on perturbation. I need to find the rescalings $x=\delta X$ and the roots of the equation $\epsilon^2x^3+x^2+2x+\epsilon=0$ I have found to possible ...
2
votes
2answers
76 views

Prove that $a^2+b^2+c^2\geq [2(a-b)^2(b-c)^2(a-c)^2]^{1/3}$

Mathematica seems to know that this statement is true, yet I am struggling to prove it. Possible useful inequalities are Minkowski and the geometric mean. Using the geometric mean inequality I can ...
3
votes
2answers
46 views

How do I rigorously show that $f(x, y) = \frac{x}{2|x|\sqrt{|x|+|y|}}$ is continuous when $x, y \neq 0$?

For the function $f : \mathbb{R}^2 \to \mathbb{R}$ to be continuous, I need to show that for some given $\epsilon > 0$, there exists a $\delta > 0$ so that if $||z - z'|| < \delta$, then ...
0
votes
0answers
14 views

the examples of subspace embedding which are not Oblivious

For the definitions of Oblivious Subspace Embedding, please refer to the 1st page of paper http://arxiv.org/pdf/1308.3280v1.pdf. Then, can any one show the examples of subspace embedding which are ...
0
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0answers
7 views

the difference and similarity between 'subspace embedding' and ' dimension reduction'

Can someone show me the difference and similarity between 'subspace embedding' and 'dimension reduction' using the mathematical definition?
1
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1answer
21 views

First order non-linear ordinary differential equation

The following ODE is given: $a\pm\sqrt{b+c*(x(t)+d))}=e*x'(t)+f*x(t) $ from Matlab I'm able to get a solution for the differential equation (actually two solutions, one for the + and one for the - ...
-2
votes
1answer
26 views

Integration using Substitution [on hold]

Firstly, I know that the graph of function, $f$ must cut the x-axis at least once such that the definite integral will equal to zero so I can apply Roelle's theorem somewhere. For b (i), letting $u ...
1
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3answers
32 views

Finding the area of a triangle with integration

I want to compute the area of the triangle with vertices $(0,0), (1,0), (1,1)$ by parameterizing the line segments parallel to the hypotenuse of the triangle. For example, the length of the segment ...
0
votes
2answers
60 views

Is $\sqrt{x}$ uniformly continous in $\mathbb{R}^+$?

We are given this function: $f:R^+\rightarrow R,x\rightarrow \sqrt{x}$. We need to prove that this function is uniformly continuous. My proof is this one but i'm not sure is it complete and right. ...
1
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2answers
30 views

Can be $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$ uniform convergence?

We have $f_n:[0,1]\rightarrow\mathbb{R},\:f_{_n}(x)=x^n \cdot\ e^x$. I don't know how we can find the pointwise convergence...This sequence can be a uniform convergence? and explain your argument.
-3
votes
2answers
62 views

How many rabbits left [on hold]

The number of rabbits in a farm increases at a rate proportional to the number of rabbits at a certain time. The number of rabbits doubled to 10000 from the beginning of the year 1985 until the ...
1
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1answer
30 views

Integration Properties

I have always had a mental block towards this property and would be truly grateful if someone would please help me. $$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$$ Consider $$f(x) = x, for ...
1
vote
1answer
59 views

Proving that $\int \delta \dot{x} dt = \delta x$

Everytime I've seen this I've assumed it was true. It seems plausible. But I would like to rigorously prove it. I think this is correct, but I would like another opinion because my mathematics is very ...
1
vote
1answer
61 views

Integration with Limits

Find $$\displaystyle \lim_{n \to \infty} \int^{1}_{0}(x^{n}+(1-x)^{n})^{\frac{1}{n}}dx$$ Now, the answer is $$\dfrac{3}{4}$$ Now, the solution was hinted like this: using the property ...
0
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0answers
12 views

Limits of Taylor POlynomials over $k$-tuples?

Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix $\textbf{a}$ $\in E$, and suppose $\textbf{x}$ $\in R^{n}$ is so close to $\textbf{0}$ that the points \begin{equation*} ...
0
votes
2answers
25 views

Finding the average value of a function over an interval.

I'm having trouble finding the areas between the intervals, mainly because I know that in order to use the mean value theorem for integrals, you (probably) need an equation for f(x). But since they ...
4
votes
3answers
157 views

Show that the series is absolutely convergent

The series is $$\sum^\infty_{n=2} \frac{(-1)^n}{n(\ln(n))^3}$$ I tried the ratio test which did not do anything. I also tried the root test which gave me $$\frac{-1}{\sqrt[n]{n}\cdot (\ln(n)^3-n)}$$ ...
0
votes
2answers
29 views

Using implicit differentiation with a fraction

How do I solve this? What steps? I have been beating my head into the wall all evening. $$ x^2 + y^2 = \frac{x}{y} + 4 $$
2
votes
1answer
20 views

Finding the marked values of x on a graph

I would assume that since $x_3$ is the local maximum(or absolute maximum) on the graph of $f$ prime, that it would be the greatest on the graph of $f.$ However, this problem is online, and in ...
2
votes
2answers
22 views

Taylor polynomial manipulation

Find $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{k}$ This is in a section in my book on Taylor polynomials/Taylor series so I assume we have to find some way to manipulate Taylor polynomials to get this. ...
0
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2answers
42 views

Find $f'(2)$, where $f(x) =\frac{h(x)}{x}$.

Consider the function $h(x)$, for which $h(2) = 4$ and $h'(2) =-3$. Find $f'(2)$ for the function $f(x) = \frac{h(x)}{x}$. Progress: I know that $h(x)/x$ is equivalent to $h(x) x^{-1}$; should I ...
1
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1answer
31 views

Continuity in a function defined only at one point [duplicate]

This might be a silly question, but if I have a function defined at only one point. Is the function continuous at that point?
0
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0answers
23 views

How fast is this dot moving when the angle $θ$ between the beam and the line through the searchlight perpendicular to the wall is $π/6$?

A searchlight rotates at a rate of $4$ revolutions per minute. The beam hits a wall located $11$ miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per ...
-1
votes
1answer
38 views

$f(x+1)=xf(x)$ and $g(x)=\log f(x)$, finding $g''(N+1/2)-g''(N)$

My try: $$f(x+1)=xf(x)\implies f(x+N)=x^Nf(x),N\in\mathbb N$$ Because: $$f(x+N)=xf(x+N-1)=x^2f(x+N-2)=...x^Nf(x+N-N)=x^Nf(x)$$ Now: $$\log f(x+N)=N\log x+\log f(x)\\ g(x+N)=N\log x+g(x)\\ ...
0
votes
0answers
20 views

can any one tighten $|e_i^TXe_j|$?

Suppose we have the symmetric matrix $X\in R^{m\times m}$ with its 2-norm $\|X\|_2\leq m$. Then I can get that, for each entry of $X$, $|X_{ij}|=|e_i^TXe_j|\leq\|e_i\|_2\|X\|_2\|\|e_j\|_2=m$, where ...
0
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0answers
27 views

Area of a surface of revolution about the y-axis-

I'm trying to find the area of a surface of revolution generated by the curves $$y=x^3,\quad x=1,\quad x=2, \quad\rm{around} \quad y=-1 $$ \begin{array}{lcl} A &=& 2\pi \int_1^2 {(y + 1)\sqrt ...
0
votes
1answer
27 views

How can I find the radius and interval of convergence of $\sum_{n=0}^\infty {(-1)^n(x^{n+1}) \over \sqrt{n}+3} $

$$\sum_{n=0}^\infty {(-1)^n(x^{n+1}) \over \sqrt{n}+3} $$ I think I'm supposed to take the absolute value of the sum and apply the ratio test, but I'm not sure.
0
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0answers
16 views

For which $(x_1,x_2)$ is this a solution to the minimal surface equation?

Let $u(x_1,x_2):=arcosh(\sqrt{x_1^2+x^2})$ then I want to find out for which $(x_1,x_2)$ this is a solution to the minimal surface equation in two dimensions that you can find for example here. ...
0
votes
4answers
29 views

How should I interpret a plus superscript in limit notation?

I am doing some calc refresher problems and I found this notation... How should I interpret that? As x approaches positive two? What does the + mean?
0
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0answers
28 views

Formulae for area between shrinking circles? [on hold]

What formulae arise to describe the changing distance and area between two non-moving but shrinking circles? What occurs as they approach an infinite distance?
2
votes
1answer
28 views

equality between variable and integral

I received the following question as part of my homework: Let $f(x)$ be a continuous function onto $[0,1]$. $f(x)\le\frac{1} {2\sqrt{x}}$ for every $0<x\le1$. Prove that x=0 is the only solution ...
3
votes
2answers
49 views

Find the following indefinite integral: $\int (x^2+6x+5)^{12} (x+3) \ dx$

The solution I got was $(1/13)(x^2+6x+5)^{13} + C$ I am not sure if I am correct though and would like help. Thanks!
-2
votes
0answers
21 views

Gauss Chebyshev formula [on hold]

Use Gauss Chebyshev formula with $n=3$ to approximate the value of the integral. $$\int \frac{x^4}{\sqrt{1-x^2}}dx$$ from -1 to 1. Also compare the result with true value, where the zeros and the ...
3
votes
2answers
70 views

Equality of a quadratic function

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ an arbitrary function and $g: \mathbb{R}\rightarrow \mathbb{R} $ a quadratic function with the following property: For any $m$ and $n$ the equation ...
1
vote
0answers
47 views

The derivative of $D_x$, the differential operator? [on hold]

I was thinking about how the derivative could also be an operator and I came upon the question: What is the derivative of the differential operator? I'm very much interested in what the answer to the ...
2
votes
3answers
42 views

Sum of power series using derivation or integration

could anyone help with this question? $$\sum_{n=1}^{\infty}\frac{(x-\frac{1}{2})^{n+1}}{n(n+1)}$$ I have to find sum of this power series using differentiation or integration. Thanks a lot!