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

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0
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0answers
14 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 ...
0
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0answers
31 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 ...
1
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1answer
24 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 ...
4
votes
2answers
57 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
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1answer
24 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 ...
5
votes
5answers
66 views

Why is $f_n(x) = x^n$ not uniformly convergent on $(0, 1)$?

Definition of uniform convergence: For all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that $d(f_n(x), f(x)) < \epsilon$ for all $n > N \in \mathbb{N}$ and all $x \in (0,1)$. ...
1
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3answers
27 views

Divisibility of integers by integers

We are given a number $$K(n) = (n+3) (n^2 + 6n + 8)$$ defined for integers n. The options suggest that the number K(n) should either always be divisible by 4, 5 or 6. Factorizing the second bracket ...
4
votes
2answers
70 views

Prove the Dirac Delta Function satisfies $ x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = -\delta(x) $

$ x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = -\delta(x)$ I've been told that this answer involves integration by parts. I began like this: $\int x\frac{\mathrm{d} \delta(x)}{\mathrm{d} x} = ...
2
votes
2answers
30 views

Limit involving inverse tan function

I solved the following limit using L'Hospital's rule, but can't seem to solve it without using L'Hospital's. $$\lim_{x\to\infty} \frac{e^{-1/x^2}-1}{2\arctan x-\pi}$$ I would like a hint as to how to ...
0
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2answers
35 views

Prove these two functions have the same coefficients

We have $\displaystyle p(x) = \frac{1}{1-x}\cdot \frac{1}{1-x^3} \cdot \frac{1}{1-x^5} \cdot \ ...$ and $\displaystyle q(x) = (1+x)\cdot(1+x^2) \cdot (1+x^3)\cdot \ ...$. Let's say that these two ...
1
vote
4answers
28 views

Find $\sup_{x\in[0,1]} \frac{x}{x^2+n^2+1}$

We have $f_n:[0,1]\to \mathbb{R},\:f_n(x)=\frac{x}{x^2+n^2+1}$ and we need to prove that is uniform convergence using formula: $\lim _{n\to \infty } \sup_{x\in[0,1]} |f_n(x)-f(x)| =0$ First ...
0
votes
2answers
53 views

Convergence of $\sum_{n=1}^\infty (2n^{10}+4n^5+1)/(4n^{15}+4n^{12}+5)$

Test whether the following series converges: $$\sum_{n=1}^{\infty}\frac{2\cdot n^{10}+4\cdot n^5+1}{4\cdot n^{15}+4\cdot n^{12}+5}$$ $1.$ I know that it does not make sense to use the root ...
-1
votes
2answers
39 views

Prove: As $n$ grows if $a_{n+1}-a_n$ converges so does $\frac{a_n}n$

Let there be a sequence $a_n$ such that $a_{n+1}-a_n\rightarrow c\in \mathbb{R}$. Prove that $\frac{a_n}{n}\rightarrow c$. By definition there exists $n_0$ : $n\ge n_0$ such that ...
3
votes
2answers
32 views

If two real polynomials $f(x)$ and $g(x)$ of degrees $m \geq 2$ and $n \geq 1$ satisfy $f(x^2+1)=f(x) g(x) ~~\forall~~x \in \mathbb R, $ then :

If two real polynomials $f(x)$ and $g(x)$ of degrees $m \geq 2$ and $n \geq 1$ respectively satisfy $$f(x^2+1)=f(x) g(x) ~~\forall~~x \in \mathbb R, $$ then : $(A)~ f$ has exactly one real root $x_0$ ...
1
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1answer
16 views

Let $f(x) = |x − 1| + |x − 2|^3$ for $x\in \mathbb{R}$ . Then examine the differentiability of $f$ at x = 1 and x = 2 and rest of $x\in \mathbb{R}$ .

Let $f(x) = |x − 1| + |x − 2|^3$ for $x\in \mathbb{R}$ . Then examine the differentiability of $f$ at x = 1 and x = 2 and rest of $x\in \mathbb{R}$ . derivative of the function f at a: ...
3
votes
1answer
33 views

At which points the tangent lines of the function $y=\cos x$ are parallel to $-\frac{1}{2}x+1$?

I did the following: $$\cos' x=-1/2 \\ -\sin x=-1/2 \\ \sin x = 1/2$$ So, I guess that in this case I have to find values such that $\sin x = 1/2$, these values are $\pi/5$ and $5\pi /6$ (I know ...
1
vote
2answers
27 views

Limit problems in two variable function

How would one find the $\alpha$ and $\beta$ for which $$\frac{x^{\alpha}y^{\beta}} {\sqrt{x^2 + y^2}} \to 0$$ as $(x,y) \to (0,0)$ ? I understand the $\epsilon$-$\delta$ definition of a limit but ...
3
votes
2answers
51 views

Proving that the delta function is the derivative of the step function.

I want to prove $\frac{\mathrm{d} }{\mathrm{d} x}\Theta =\delta (x)$ using this representation of the delta function: $\delta(x)= \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{ikx}dk $ This should be ...
0
votes
1answer
22 views

directional derivative problem

for a point M(4,1) and a function $z = x y^2 - (x^2/y^3)$ I was tasked with finding a directional derivative in the direction which creates a 30 degree angle with the $x$ axis....I find it a little ...
0
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1answer
28 views

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions :

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions : $ f(x+t,y) = f(x,y) + ty~~;~~f(x,t+y) = f(x,y) + tx~~;~~f(0,0) = K$. Then $\forall ~~x,y \in \mathbb R, ...
2
votes
2answers
39 views

Evaluate if $f_{_n}$ converge uniformly or not

We have $f_n:[1,2]\to \mathbb{R},\:f_n(x)=\frac{x^n}{x^n+1}$ and we have to see if the convergence is uniform or not. From what I understand we need to prove that $\lim _{n\to \infty } ...
0
votes
0answers
13 views

Fining the angular bounds of a triple integral function

This problem requires the taking of a triple integral over a region. I believe it's most useful to convert to cylindrical coordinates, which I did. However, I could not find the theta bounds due to ...
1
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3answers
45 views

Evaluate the definite integral $ \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$

Evaluate the integral: $\displaystyle \int_{\pi/6}^{\pi/2} \frac{\cos(x)}{\sin^{5/7}(x)}\, dx$ (using substitution) Here's my attempt at solution: u = $\sin^5(x)$ $du = 5\sin^4(x) \cdot \cos(x) ...
-3
votes
1answer
30 views

Questions on Indefinite Integration 2 [on hold]

Sorry to bother you guys, I have difficulty starting these questions. I have the answers, but I just can't seem to start the questions off. Really appreciate your efforts in helping me out. Thank you ...
-1
votes
1answer
24 views

Computing definite integral with u-substitution [on hold]

How to compute $$\int_{0}^{\sqrt{3}} \frac{dx}{\sqrt{4-x^2}}$$ and $$\int_{1}^{2} \frac{dx}{3+x^2}$$ using only $u$-substitution?
0
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4answers
32 views

Questions on Indefinite Integration [on hold]

Hi would like to know who yall will go about integrating this function. I have tried by substituting with $u=2(x-1)^{0.5}$ for question 1
0
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0answers
9 views

How can I do this line integral using stoke's theorem??

$$ \int_{C}{(y^2+z^2)dx}+(x^2+z^2)dy+(x^2+y^2)dz $$ where C is the intersection of hemisphere $x^2 + y^2 + z^2 = 2ax, z \geq 0$ and $x^2 + y^2=2bx $ where 0 < b < a. Compute line integral ...
0
votes
2answers
30 views

Evaluate the indefinite integral $\int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{1}{x^2} \sin\left(\frac{6}{x}\right) \cos\left(\frac{6}{x}\right) \, dx $ (using substitution) The answer is: $\frac {1}{24} ...
1
vote
1answer
14 views

Green's theorem in divergence form and its line integral?

$$ \int_C F \times da $$ $$ k\iint_R \operatorname{div} F \ dx \, dy $$ Hi Let $F$ be two-dimensional vector field. State a definition for the vector-valued line integral so that your definition ...
3
votes
0answers
42 views

Trigonometric integral of $f(x)=(x^2)(\sin(x^2))$. [duplicate]

I've tried with the chain rule and $u$-subtitution ($u=\sqrt{x}$) but I get nothing. Can you help me please? $$\int (x^2)(\sin(x^2)) \ dx$$
2
votes
3answers
38 views

Evaluate $\int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $

Evaluate the indefinite integral: $\displaystyle \int \frac{\sec(11 x) \tan(11 x)}{\sqrt{\sec(11 x)}} \, dx $ (using substitution) The answer is: $\frac{2}{11} \sqrt{sec(11 x)} + C$ I don't get ...
1
vote
1answer
17 views

If I take partial derivatives of function using chain rule, how do I remove variable from the partials in the equation?

I have the function $$f(tx,ty)$$ and I want to take the partial derivative of this with respect to $t$. So set $x'=xt$ and $y'=yt$. I applied chain rule and got $$\frac{\partial f}{\partial t} = ...
1
vote
3answers
26 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
1
vote
0answers
20 views

3D Fourier Transform - Angle between $\mathbf{k}$ and $\mathbf{r}$

The definition of the Fourier transform for three dimensions is $$\mathcal{F}[f(\mathbf{r})](\mathbf{k})=\int e^{-i\mathbf{k}\cdot \mathbf{r}}f(\mathbf{r})\,d^3 r$$ If the function $f(\mathbf{r})$ ...
1
vote
1answer
28 views

$K$ is a region in $\mathbb{R}^2$ where the area is $5$

Say that $K$ is a region in $\mathbb{R}^2$ where the area is $5$. Let B = \begin{pmatrix} 3 & 8 \\ 4 & 6 \end{pmatrix} Find the area of the region B$K$. Any starting hints? Is it possible ...
3
votes
2answers
58 views

Can someone give me a counterexample to understand why this definition of limit is wrong?

Could someone give me a counterexample to understand why this definition of $\lim_ {x\to a} f(x) = L$, does not work? $\forall \delta>0 \exists \varepsilon>0$ such that, if ...
-3
votes
0answers
20 views

Basics of determining if limit exists [on hold]

Find the limit or determine that it does not exist $\lim_{x\rightarrow c}\sqrt{x}$, for $c\geq0$
0
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1answer
40 views

Does the following converges?

Does the following converges? $$\int_{-\infty}^{+\infty}\frac{\sin x}{1+\cos^2 x}dx$$ Thanks ahead.
1
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2answers
18 views

Set up the integral for the volume found by rotating the region bounded by the curves $x=y+1$ and $x=-y^2+2y+3$ about the line $x=-1$

Set up the integral for the volume found by rotating the region bounded by the curves $x=y+1$ and $x=-y^2+2y+3$ about the line $x=-1$. Thanks! I don't know how to solve it about the line x=-1. I could ...
-2
votes
0answers
36 views

$\lim_{x \to \infty} x^nf(x)$ [on hold]

Given $\int_0^\infty x^n f(x) < \infty , \, n = 0, 1, \dots$ and $f(x)$ is decreasing, positive for all $x$. Show that $$\lim_{x \to \infty} x^n f(x) = 0.$$
0
votes
1answer
23 views

Quotient rule for derivatives..am I making this to complicated

This is a straight forward question.. When I have something like 10/x (i.e basically whenever the numerator is just a number with no variables) and I need to take the derivative I go through the ...
0
votes
1answer
27 views

area of a bounded region

Find the area of the region bounded by $$f(x)=x^3+x^3+1$$ and $$g(x)=x^2 + x-1$$ I do know how to get the area of a bounded region, my problem now is that when I tried getting the graph of this region ...
0
votes
1answer
16 views

When is a series of sums the sum of the series?

In general, if $\Sigma_n (a_n+b_n)$ converges, then it may not be that $\Sigma_n a_n$ and $\Sigma_n b_n$ converge; for example, consider $\Sigma_n (1/n-1/n)$. If instead we know $\Sigma_n a_n$ and ...
0
votes
1answer
14 views

Integral Car Traffic Problem for $f(t) = 50*t*sin(\sqrt{t})$

So I have a question for this traffic function $f(t) =50t*sin(\sqrt{t})$ where f(t) is the rate at which cars pass through an intersection from noon(t=0) until 5pm (t=5). The question is asking to ...
0
votes
2answers
53 views

How do i evaluate the following integral?

Hi I was wondering if someone can help me evaluate the following integral. Show that if $-1 < x < 1$, then $$\int_{0}^{\pi} \frac{\log{(1+x\cos{y})}}{\cos{y}}dy= \pi \arcsin{x} $$ thank you ...
2
votes
2answers
47 views

Differential Equation: $\text dy/\text dx = x/y$

Consider the differential equation $\text dy/\text dx = x/y$ a) Write an equation for the line tangent to the solution curve that passes through the point $(1,2)$ Would it be correct to just use ...
2
votes
0answers
41 views

On a problem about Rolle's theorem

Let $f:[1,3]\to\mathbb R$ be a continuous function such that $\int_1^2 f(x)dx=2$, and $\int_1^3 f(x)dx=3$, then there exists a real number $c\in(2,3)$ such that $$ \int_1^c f(x)dx=cf(c) $$ Note. I ...
0
votes
0answers
16 views

Calculus scenario involving instantaneous and speed (sequences) [on hold]

The scenario is nearly always the same as Wilie is standing at the end of a road that is 1 kilometer long, and there at the other end is that Roadrunner, he’s just standing there, sticking his tongue ...
1
vote
3answers
43 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
0
votes
1answer
41 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$?

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...