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

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1answer
16 views

Integration using Substitution

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 ...
0
votes
2answers
19 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
51 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
27 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.
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1answer
21 views

How many rabbits left

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
23 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
42 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
43 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 ...
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0answers
11 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
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2answers
24 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
118 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
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1answer
23 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
18 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
21 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. ...
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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
29 views

Continuity in a function defined only at one point

This might be a silly question, but if I have a function defined at only one point. Is the function continuous at that point?
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0answers
20 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
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1answer
37 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)\\ ...
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0answers
14 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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1answer
21 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 ...
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1answer
24 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.
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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. ...
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4answers
27 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
27 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
26 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
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2answers
46 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!
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votes
0answers
18 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
66 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 ...
0
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0answers
43 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
39 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!
2
votes
1answer
49 views

How to prove that $|a_{1}+2a_{2}+…+na_{n}| \leq 1.$

Let $$f(x)=\sum_{k=1} ^{n}a_{k}\sin(kx)$$ where $n \in \mathbb{Z^+}$ and $a_{k} \in \mathbb{R}$ for each $k=1,…,n.$ Suppose that $ \vert f(x)\vert \leq \vert \sin(x) \vert $ for every $x.$ Prove ...
1
vote
2answers
41 views

Evaluate $\lim_{x \to \infty} xe^{-x}+be^{-x}$

could someone help me to evaluate:$\lim_{x \to \infty} xe^{-x}+be^{-x} $ I know that the second half of the equation goes to zero because it is basically $b/(e^{\infty})$. but the first half of the ...
0
votes
1answer
21 views

Integration of step functions

I've managed parts (a) and (b) fairly easily, but c is causing me a real headache. I've seen the Cauchy-Schwartz inequality before, but I've hit a roadblock because I've no idea whether or not I can ...
0
votes
1answer
28 views

What is $\sum_{i,k=1}^n x_k^2x_i^2$?

I stumbled over the double sum $$\sum_{i,k=1}^n x_k^2x_i^2$$ and was wondering whether this is anything that can be expressed in terms of the euclidean norm? Maybe it is even $||x||^4$ but I am not ...
0
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0answers
52 views

Random walk, expectation & variance, joint probability, approximation question [on hold]

Consider the following random walk on a plane: The walk commences at the origin and at each timestep, a step of unit length is taken in a random direction $\theta$ (measured relative to the positive x ...
-1
votes
1answer
35 views

Partial Derivative Question involving arcsin

Suppose $u(x,y) = \sin (( x \sin^{-1} (y))$ Find ${\partial u \over \partial x}, {\partial u \over \partial y}, {\partial^2 u \over \partial x^2}, {\partial^2 u \over \partial y^2}$ and ${\partial^2 ...
0
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1answer
29 views

Uncontinuous, riemann-integrable function

What is an elementary example of a uncontinuous but riemann-integrable function that possesses an explicit antiderivative?
0
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1answer
35 views

Why is the rotation not zero and the divergence zero in the figures below?

Why is the rotation not zero and the divergence zero in figure 1 and figure 2 below? Figure 1 Figure 2
2
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1answer
40 views

Calculate area of the region formed by $f(x)= x^3-x^2$ and x-axis

What is the area of the region formed by the graph of $f(x)=x^3-x^2$ and the $x$-axis in the interval $[0,3]$? Did I do this right? I get $$\int_0^3x^3-x^2\,dx$$ giving me the answer of $45/4 = ...
0
votes
1answer
34 views

calculus, slope of a tangent line with given point

What is the slope of the tangent line $f(x) = x^3 -(1/3)x^2$ at $(1, 2/3$) I already did the first derivative, which gave me, $3x^2-2/3x$ so am i right for just doing 3(1)^2-(2/3)(1) = 7/3 = 2 ...
2
votes
0answers
34 views

General solution of $ty'+2y=4t^2$

Should we left the general solution of the differential equation $t\frac{dy}{dx}+2y=4t^2$ as $t^2y=t^4+c$ instead of $y=t^2+c/(t^2)$ ($c$ is an arbitrary constant)? Does the solution $y=t^2+c/(t^2)$ ...
2
votes
0answers
21 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
3
votes
2answers
56 views

Finding $G'(2)$ when $G(x) = \int_0^x h(t) \ dt$

Let $$G(x) = \int_0^x h(t) \ dt$$ This is the graph of $h$. I need to find the value of $G'(2)$. I've looked around and read about the Fundamental Theorem of Calculus, but I haven't found ...
2
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0answers
79 views

Finding $\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$

Finding $$\int \frac{\sin\sqrt{\frac{x}{2}}}{\sqrt{x\cos\sqrt{x}}}dx$$ This is a homework. I tried to solve it by assuming $x=u^2$ but after that the integrals become not simple, so I don't know how ...
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0answers
18 views

change in gradient$\frac nx$

Ok this is my first question so sorry if I've formatted it incorrectly. I understand the shape of a $\frac1x$ graph https://www.wolframalpha.com/input/?i=1%2Fx What I am wondering about now is if I ...
2
votes
1answer
66 views

How is the concept of the limit the foundation of calculus?

My casual study of mathematics and calculus introduced me to the notion that calculus didn't find a firm foundation until Cauchy, Weierstrauss (et al) developed set theory some ~100 years after Newton ...
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vote
1answer
26 views

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. $$\iiint_E ...
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0answers
25 views

Increasing/ decreasing functions

We are given a random variable x with a pdf f(x) and F(x) is its distribution function. Let $$r(x) = \frac {xf(x)} {1-F(x)} $$ Then for $x< e^{\mu} $ and $$f(x) = \frac {e^ {1/2(\log x - \mu)^2}} ...
4
votes
3answers
75 views

How do I compute this integral?

I'm wondering how to compute the integral $$ \int_2^3\int_0^\sqrt{3x-x^2}\frac{1}{(x^2+y^2)^{1/2}}\,\mathrm{d}y\mathrm{d}x. $$ Clearly it is too complicated to do it directly, so I'm guessing you have ...
0
votes
1answer
30 views

What is the difference between uniform convergence and dominate convergence theorem?

I saw that both have aim to change limit with integral... that's the part that interests me most. I saw in some cases where we couldn't use uniform convergence, we use dominate convergence theorem to ...