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

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3
votes
2answers
140 views

Proving that $\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos (\tan (x))-\cot (x))\cot (x) \, dx=\frac{\pi(e-2)}{2e}$

I think one of the ways of doing it is by the use of the differentiation with parameter. Do you see an easy way of calculating it by real methods? $$\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos ...
41
votes
0answers
952 views

Generalization of Liouville's theorem

As proposed in this answer, I wonder if the answer to following question is known. Let $E = E_0$ be the set of elementary functions. For each $i > 0$, inductively define $E_i$ to be the closure ...
-1
votes
0answers
24 views

How do we know which quantity to be integrated with respect to which? [on hold]

Suppose I write z = x1y1 + x2y2 + ... + xnyn So, if x and y are not discrete values, but continuous functions, then would z be written as the integral of x.dy or would it be y.dx or something else ...
8
votes
3answers
208 views

Evaluate $\int_0^1\frac{x^a-x^{-a}}{x-1}dx$

I have heard that: $$\int_0^1\frac{x^a-x^{-a}}{x-1}dx=\frac1 a-\pi\cot(\pi a)$$ when $-1<a<1$. How would I prove this? That doesn't have an elementary indefinite integral, but the definite ...
2
votes
1answer
19 views

Transition matrix for coordinate

Suppose that $ (U,x^1,x^2,...,x^n) $ and $ (V,y^1,y^2,...,y^n) $ are two coordinate charts on a manifold.Then $$ {\partial \over \partial x^j}=\sum_i {\partial y^i \over \partial x^j } {\partial \over ...
144
votes
11answers
58k views

What is the result of infinity minus infinity?

What is $\infty - \infty$? Is it $\infty$ or $0$ or what?
3
votes
0answers
106 views

Find the derivative of the following integral

Find the derivative of $f(x)= \int_x^0 \frac{\cos(xt)}{t} dt$. My first reaction was to apply the FTOC, but I don't believe I can do this because $\frac{\cos(xt)}{t}$ is not defined at $t=0$ and thus ...
0
votes
1answer
27 views

Solving intersection points

How do you solve these functions for intersection points? $2^x = 3-x$ Do you use natural log first to get the x or do you need to use other approaches?
2
votes
1answer
44 views

Derivative exists by first principles but undefined when using chain rule

Consider the function $h$ defined by \begin{align} h(z,y)=(z^3+y^3)^{\frac{1}{3}} \end{align} Then \begin{align*} h_z(0,0)&=\lim_{t\rightarrow 0}\frac{(t^3)^{\frac{1}{3}}}{t}\\ &=1 ...
0
votes
1answer
56 views

How to compute an integral?

I am reading the lecture notes. I am trying to understand the prove of Lemma 0.0.1.1 on page 4. From line 3 to line 4 in the proof of Lemma 0.0.1.1., how to prove that $$ \int_{F^{n-1}} ...
3
votes
4answers
56 views

Integration using trig substitution or substitution

I was trying to review calculus integration techniques before my differential equations class. I came across $\int \frac{1}{\sqrt{1-2x^2}}\,\mathrm{d}x$. I can't exactly figure out a good way to solve ...
1
vote
1answer
406 views

Question on using chain rule or product rule to find Jacobian of function with matrices times a vector…

Suppose we have a function consisting of a series of matrices multiplied by a vector: $$f(x) = ABb,$$ where $x$ is a vector containing elements that are contained within $A, b$, and/or $b$, $A$ is a ...
0
votes
1answer
22 views

Intervals of the monotony and the extreme values [on hold]

What are the intervals of the monotony and the extreme values of the this function: $$f(x)=-2x^2+3x-1?$$ I have tried to find $\alpha$ and $\beta$ but I couldn't. Could somebody help me.
1
vote
0answers
37 views

Questioning the differentiability of $f(x,y)$

$$f(x,y)=\begin{cases} y- \frac{e^{x^2+y^2}-x^2-y^2}{x^2+y^2},& x^2+y^2 \neq 0. \\ -1, & x=y=0 \end{cases}$$ I keep runnung into trouble with these types of questions. The way I do them is ...
-1
votes
1answer
41 views

Convergence of $E_n=(-n,n]$ [on hold]

If $E_n=(-n,n]$ is a sequence of sets, does it converge to $(- \infty , \infty)$ or $(- \infty , \infty ]$ or $[- \infty , \infty ]$? What is the proof?
8
votes
2answers
277 views

A double series $\frac13 \sum_{j=1}^{\infty}\sum_{i=1}^{\infty}\frac{(i-1)! (j-1)!}{(i+j)!}H_{i+j}$ giving $\zeta(3)$

Here is a symmetric rational double series giving Apery's constant: $$ \frac13 \sum_{j=1}^{\infty}\sum_{i=1}^{\infty} \displaystyle \frac{(i-1)! (j-1)!}{(i+j)!} H_{i+j} = \zeta(3) $$ where ...
0
votes
1answer
31 views

Derivative of Logistic Function

Given a softmax function: $y_i = \frac{e^{z_i}}{\sum\limits_j e^{z_j}}$ With partial derivative: $\frac{\partial y_i}{\partial z_i} = y_i (1 - y_i)$ And a cross entropy function: $C = -\sum\limits_j ...
6
votes
0answers
105 views

prove that $\displaystyle \sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right]$

Using the relation $2(1-\cos x)<x^2,x\neq 0$ or otherwise, prove that $$ \sin (\tan x)\geq x\;\forall x\in \left[0,\frac{\pi}{4}\right] $$ My Attempt: Let $f(x) = \sin (\tan x)-x$. Then ...
0
votes
3answers
29 views

Prove that the following functions is differentiable on $(-1,1) \times \mathbb R$

$$f(x,y)=\begin{cases} \frac{\tan x}{x}+y, & 0<|x|<1 \\ 1+y,& x=0 \\ \end{cases}$$ Prove that it is differentiable on $(-1,1) \times \mathbb R$. I use the Frechet definition of ...
-1
votes
1answer
39 views

Probability Cauchy density function distribution Continuous random variable

Suppose a gun sprays paint in a uniform back-and-forth movement, covering an angular sweep between forty five degrees to either side, onto a screen 100cm away. Assume that the distribution of paint at ...
1
vote
1answer
80 views

Rolle's Theorem

The question reads: "Find a continuous function $f$ and an interval $(a,b)$ such that $f(a)=f(b)$, but there is no number $c$ in $(a,b)$ such that $f'(c)=0$. Explain why your function doesn't violate ...
0
votes
1answer
38 views

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$

If $u=e^x \cos y \text{ and } v=e^x \sin y$ transform the following: $w_{xx}+w_{yy}=0.$ I was hoping that someone would maybe be familiar to this $w$ function that is stated, because this is the only ...
0
votes
1answer
38 views

Factorial Series Compute

Given $\sum_{n=2}^{\infty}\sum_{j=2}^{\infty}\frac{1}{(j^n)(j!)}=a+be$ where $a$ and $b$ are integers, find $a$ and $b$.
1
vote
1answer
25 views

Is it correct to use the sum of squared differences (SSD) to determine if two lines are similar?

I have two lines ($L_{1}$ and $L_{2}$): $A_{1}x+B_{1}y+C_{1} = 0$ and $A_{2}x+B_{2}y+C_{2}=0$. I want to know if they're similar to the point that they could be the same line ...
4
votes
3answers
502 views

Why doesn't small perturbations of a matrix decrease its rank?

I want to show that if $F:\mathbb{R^n}\to\mathbb{R^m}$ has a derivative of rank = r at a point $p$, then there is a neighbourhood of that point where the rank of the function doesn't decrease, i.e. ...
5
votes
3answers
151 views

Integrating and indefinite integral any possible way

How do I integrate the following: $$\large \displaystyle\int_0^{\infty}\frac{x^4e^x}{(e^x-1)^2} \, dx$$ I have tried everything from integrating by parts to simply expanding the denominator, but it ...
0
votes
2answers
35 views

Differentiation using Chain Rule

Find $\frac{dy}{dx}$ if $y=7+5^{x^2+2x-1}$. So far I have done $\frac{dy}{dx}=(5^{x^2+2x-1})'$. Now, the RHS can be found by $(e^{\ln 5\cdot (x^2+2x-1)})'=e^{\ln 5\cdot (x^2+2x-1)}(x^2+2x-1)'\ln ...
1
vote
1answer
36 views

Differentiable not $C^1$ and Darboux property

Are there any differentiable not $C^1$ function $f: [0,1] \to \mathbb{R}$, $f'(0)<f'(1)$ such that there exists $c \in (f'(0),f'(1))$ that $f'$ doesn't reach value $c$? Classical example of ...
11
votes
6answers
8k views

Why does the harmonic series diverge but the p-harmonic series converge

I am struggling understanding intuitively why the harmonic series diverges but the p-harmonic series converges. I know there are methods and applications to prove convergence, but I am only having ...
2
votes
4answers
59 views

How to find inverse Laplace transform?

Please I am trying to solve inverse Laplace transform of $ \frac 6 {(s^2+4)^2}$ If anyone has some idea to help me please share it with me. I have been trying to do like this but it's not working, ...
3
votes
1answer
27 views

How to solve (0.1 - 10.3 + 5.132)/12.8 and round off correctly?

I've recently learnt the rules about rounding off when adding/subtracting and when multiplying/dividing. I know that when you add/subtract, the number of decimal places in the result should equal the ...
0
votes
1answer
38 views

$t\in (0,1)$ and $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$. Show that if strict inequality holds for even one $t$, then it holds for all $t$.

This is a part of a solution to a problem in showing that if $f$ is continuous and satisfies the condition $f([x+y]/2)\lt [f(x)+f(y)]/2$, then $f$ is convex. Let $t\in (0,1)$. We have the weak ...
0
votes
1answer
62 views

How was this differentiated?

How red-circled function with 1/D is equal to green-circled? Note: D is equal to dy/dx. Update: Complete pic
8
votes
3answers
155 views

Is this epsilon-delta proof that $\sin(x)$ is continuous circular?

Prove that $\lim_{x\to a}\sin x=\sin a$, where $a$ is any real number. Solution 13 here: https://www.math.ucdavis.edu/~kouba/CalcOneDIRECTORY/preclimsoldirectory/PrecLimSol.html#SOLUTION13 ...
-1
votes
0answers
17 views

Proof of Boundedness theorem [on hold]

Boundedness of a continuous function $f$ on a closed interval $[a,b]$ is proved using Borel's theorem. For that, for a given $\epsilon >0$ ,it shows $f(a)-\epsilon < f(x) < ...
0
votes
0answers
20 views

Computing Partial Derivatives (basic)

When I am asked to compute a partial derivative of $f_x$ for $f(x, y)=x \ln(xy)$, I treat this the same as $\frac{d}{dx} (x \ln(xy))$ which I then just simply apply the chain rule and get $\ln (xy) ...
8
votes
1answer
276 views

Computing $\int\frac{7x^{13}+5x^{15}}{(x^7+x^2+1)^3}\,dx$

Compute the indefinite integral $$ \int\frac{7x^{13}+5x^{15}}{(x^7+x^2+1)^3}\,dx $$ My Attempt: $$ \int\frac{7x^{13}+5x^{15}}{x^{21}(x^{-7}+x^{-5}+1)^3}\,dx = ...
6
votes
1answer
198 views

Integration of $\displaystyle \int\frac{1}{1+x^8}\,dx$

Compute the indefinite integral $$ \int\frac{1}{1+x^8}\,dx $$ My Attempt: First we will factor $1+x^8$ $$ \begin{align} 1+x^8 &= 1^2+(x^4)^2+2x^4-2x^4\\ &= (1+x^4)^2-(\sqrt{2}x^2)^2\\ ...
7
votes
5answers
154 views

Evaluation of $ \lim_{x\rightarrow \infty}\left\{2x-\left(\sqrt[3]{x^3+x^2+1}+\sqrt[3]{x^3-x^2+1}\right)\right\}$

Evaluate the limit $$ \lim_{x\rightarrow \infty}\left(2x-\left(\sqrt[3]{x^3+x^2+1}+\sqrt[3]{x^3-x^2+1}\right)\right) $$ My Attempt: To simplify notation, let $A = ...
0
votes
0answers
79 views

If $f^{-1}(x)$ is continuous, is $f(x)$ also continuous?

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a one-to-one function with $f(\mathbb{R})=\mathbb{R}$. If $f^{-1}(x)$ is continuous $\forall x\in\mathbb{R}$, prove or disprove that $f(x)$ is continuous ...
7
votes
1answer
87 views

bound for $a,b,c$ in $\mid ax^2+bx+c \mid \leq 1\;\forall x\in \left[0,1\right]$

Consider $a,b,c\in\mathbb{R}$ such that $\mid ax^2+bx+c \mid \leq 1\;\forall x\in \left[0,1\right]$. Prove that $|a|\leq 8\;\;,\mid b \mid \leq 8$ and $\mid c \mid \leq 1$. My Attempt: Set $x = ...
8
votes
2answers
312 views

evaluation of $\int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)dx$

Compute the indefinite integral $$ \int \cos (2x)\cdot \ln \left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)\,dx $$ My Attempt: First, convert $$ \frac{\cos x+\sin x}{\cos x-\sin x} = ...
-3
votes
1answer
55 views

why the integral of $\frac{dy}{y} =\ln(y)$?

I mean if I differentiate $\ln(y)$ the result will be $\frac{dy}{y}$ ? . What I know the diffential of $\ln(x)$ = $\frac{1}{x}$ right?. And following this idea what is going to happen if we ...
1
vote
2answers
855 views

Ratios of similar definite integrals

I have ended up with the following ratio of two definite integrals \begin{equation} \frac{\int_{x_1}^{x_2}\alpha T I_0 e^{-\alpha l} d \lambda}{\int_{x_1}^{x_2} T I_0 e^{-\alpha l} d \lambda} ...
-5
votes
1answer
30 views

Antiderivation, application problem. [on hold]

If the driver of a car want to increase the speed of 40 km / h to 100 km / h to travel a distance of 200 meters, what is the constant acceleration due stay? I only know that I need to use indefinite ...
6
votes
3answers
546 views

How to recognize an improper integral?

I think $\displaystyle \int ^{4} _0 \frac{\sin x}{x} dx$ is not an improper integral. Is this true? If not true, then how can one recongnise improper integrals in general?
3
votes
2answers
78 views

Evaluation of $\int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx =-\frac{\pi^4}{15}$ and $\int_{-\pi}^{\pi} \log(2\cos{\frac{x}{2}}) dx =0$

In the following encyclopedia, http://m.encyclopedia-of-equation.webnode.jp/including-integral/ I found the relations below \begin{eqnarray} \int_{0}^1 \frac{1}{x} \log^3{(1-x)}dx ...
1
vote
5answers
119 views

Help with why $\displaystyle\lim_{n\to\infty}\frac{13n^3+2n^2+6n\log(n)}{n^3}=13$

Reposting this since I apparently posted in the wrong website. Anyway, I just want to know, step by step, how did the guy reach 13 here: $$\begin{align*} \lim_{n\to\infty}\frac{T(n)}{f(n)} &= ...
3
votes
3answers
67 views

If a statement is true for every element in a sequence, would it be true at limit when $n \to \infty$?

The answer is obviously No, with an example: $\frac{1}{n} > 0$ for all n, and $lim \frac{1}{n} = 0$ (I can prove this using delta-epsilon method or just draw a picture). But I can't wrap my head ...
7
votes
4answers
405 views

Can you show me a good approach for taking the limit of this function?

I tried to use binomial expansion, but I didn't get the same result. I would like to know how to approach this please. I know the answer is $\sqrt{e}$. My problem is : $$\lim\limits_{x\to 0} ...