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

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7
votes
2answers
495 views

Calculus Paradox. I mean, what's wrong with what I think? [closed]

Is not calculus based on the paradox that the closest point to a point A is a distinct point B which is the point A itself? For example, if we consider the limit, $$ \lim_{x\to2} \frac{x^2-4}{x-2} $$...
2
votes
1answer
30 views

How to detect inflection point in a non-differentiable part of a curve

In a $W$ shaped curve, is the middle point a maximum or an inflection point? In that regard how to differentiate mathematically between a maximum (or minimum) and an inflection point at a non-...
0
votes
0answers
22 views

The existence of such function satisfies the infinitum condition

Let $t\in [0,+\infty)$ be a real number, I am looking for two functions satisfies: $f$: $[0,1]\to[0,1]$ such that $f(0)=0$, $f(1)=1$, and $f(t)>0$ if $t>0$. $f$ is lower semicontinuous ...
2
votes
3answers
139 views

Proving $\lim\limits_{x \to \infty} xf(x)=0$ if $\int_{0}^{\infty}f(x) dx$ converges.

Let $f(x)$ be a monotone non-increasing function such that $\int_{0}^{\infty}f(x) dx$ converges. Prove: $\lim\limits_{x \to \infty} xf(x)=0$. My question is, why can't I simply contradict any other ...
0
votes
3answers
92 views

How to integrate : $\int \sqrt{\tan^2x +2}dx$ [closed]

How to integrate : $\int \sqrt{\tan^2x +2}dx$ Please guide what to substitute or any approach as I am not getting any clue on this , thanks .
1
vote
1answer
137 views

Solve this question involving temperatures?

So I am given 2 formulas: $$ \frac{dT}{dt}=-k(T_t-T_s)$$ Where $\frac{dT}{dt}$ rate at which the object's temperature is changing $T(t)$ is the temperature of the object at time $t$ $T(s)$ is the ...
3
votes
5answers
156 views

How to find the MacLaurin series of $\frac{1}{1+e^x}$

Mathcad software gives me the answer as: $$ \frac{1}{1+e^x} = \frac{1}{2} -\frac{x}{4} +\frac{x^3}{48} -\frac{x^5}{480} +\cdots$$ I have no idea how it found that and i don't understand. What i did is ...
1
vote
1answer
58 views

The behavior of $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ near $0$, where $\alpha \ge 1$.

Consider $\alpha \ge 1$. Let $f(x)=\alpha x+x^2\operatorname{sin}1/x$ for $x\neq 0$ and let $f(0)=0.$ In order to find the sign of $f'(x)$ when $\alpha \ge 1$ it is necessary to decide if $2x\...
4
votes
1answer
116 views

Is there an English version of Johann Bernoulli's integral calculus lectures?

The name of lectures of integral calculus written by Johann or Jeans Bernoulli (he is called by both names as far as I know) might be " lecciones mathematicæ de calculo integral"; I must mention that, ...
1
vote
2answers
118 views

Integrating Square Roots Containing Multiple Trigonometric Functions and/or Numbers

When trying to calculate arc length of a curve I frequently come across problems that I do not know how to integrate, such as: $$ \int{\sqrt{16\cos^2{4\theta} + \sin^2{4\theta}} d\theta} $$ Which in ...
1
vote
1answer
222 views

Proof of Product Rule for Sequences using definition of infinitesimal and properties of infinitesimal sequences.

I have been trying to understand this proof for the product rule of sequences, where the author makes use of some properties for infinitesimals, to prove this theorem. This is quite a long question, ...
2
votes
3answers
69 views

L'Hopitals Rule and division by zero?

My question concerns division by zero. Let's say there are two functions, $g(x)$ and $f(x)$ that approach $0$, as $x\to t$, also assume their derivative w.r.t. $x$ is finite as $x\to t$. Using L'...
1
vote
1answer
40 views

Solving a pair of ODEs

I'm trying to solve a pair of ODEs for which I've obtained a solution. However, my problem is that my answer is slightly different from mathematica's answer. $$ \frac{dA}{dt} = \theta - (\mu + \...
0
votes
0answers
67 views

Application of Calculus: Manufacturing Costs

I'm having some trouble with some calculus. Here's the problem: A manufacturer has agreed to produce x thousand 10-packs of high quality recordable compact discs and have them available for ...
3
votes
0answers
84 views

Question on complete metric spaces and whether the following is a complete metric space:

Let $ S \subset C^2([0,1])$(set of all two-times differentiable functions on $[0,1]$), which satisfy $$f(0)+f(\frac{1}{2})+f(1)=0.$$ Question :Is $ (S,d)$ is a complete metric space, where $d$ is ...
2
votes
1answer
59 views

Computing the integral of $-1/f''$

I think this is a very silly question but I have some problems nonetheless. If I know that $g'=-\frac{1}{f''}$, is then $$ g=(f')^{-1}? $$
8
votes
1answer
770 views

Polynomials which satisfy $p^{2}(x)-1 = p(x^{2}+1)$

Can we find a polynomial $p(x) \in \mathbb{R}$ such that $\text{deg}\ p(x)>1$ and which satisfies $$p^{2}(x)-1=p(x^{2}+1)$$ for all $x \in \mathbb{R}$. This question can be very well identified ...
4
votes
3answers
89 views

Evaluation of $\int\frac{1}{x^2.(x^4+1)^{\frac{3}{4}}}dx$

Evaluate the integral $$\int\frac{1}{x^2\left(x^4+1\right)^{3/4}}\,dx$$ My Attempt: Let $x = \frac{1}{t}$. Then $dx = -\frac{1}{t^2}\,dt$. Then the integral converts to $$ -\int \frac{t^3}{(1+t^...
2
votes
3answers
122 views

Did Wolfram|Alpha mess up $\int 1 - \frac{1}{1-e^{-x}} \mathrm{d}x$ or did I?

I want to calculate $$\int 1 - \frac{1}{1-e^{-x}} \mathrm{d}x$$ Provided that $x>0$. Substituting $u=1-e^{-x}$, we get: $$\int \frac{1 - \frac{1}{u}}{1-u} \mathrm{d}u = -\int \frac{1 - \frac{1}{...
3
votes
1answer
103 views

Solving linear differential equations

Find the general solution for the following equation: $$\frac{dy}{dt}+2ty=\sin(t)e^{-t^2}$$ Find a solution for which $y(0)=0$ First I found the integrating factor which is $e^{t^2}$ Multiplying ...
0
votes
1answer
63 views

When does function $(\log_b(x))^p$ change its curvature?

Consider $(\log_b(x))^p$ where $b$ is a constant $>1$; $x, p \in \mathbb R_+$. As we increase the value of $p$ (starting from 1), at specific value of $p$, the curve changes its shape from ...
-5
votes
3answers
83 views

In the real domain, are there any theorems or definitions that state all functions are differentiable? [closed]

I want to ask about basic theory of calculus, say differentiation. We know that not every function can be integrable, but as far as I know all functions are differentiable in the real domain. My ...
2
votes
2answers
68 views

One integral involving integrals exponential and logarithmic function

Is there a closed-form solution for the integral $$ \int_{0}^{\infty}\log_{2}(1+ax)\cdot e^{-bx} \; \mathrm dx $$ with $a, b \geq 0$? If there is no closed-form solution, whether there is an ...
1
vote
2answers
84 views

How to see that $e^{x(1+x/3)} \le (1+x)^{(1+x)}$ for very small x>0?

I am reading a research paper and one of the calculation states that $e^{x(1+x/3)} \le (1+x)^{(1+x)}$ for very small x>0. Is this true? How to prove it? Thank you very much.
4
votes
5answers
151 views

$1+2+3=\int_{0}^{\infty}t^3e^{-t} dt$?

I'm reading Ivanov's: Easy as Pi. In the cover of the book, there is a formula: $$1+2+3=\int_{0}^{\infty}t^3e^{-t} dt$$ It's not clear to me if the formula has any relevance or if it is a joke. I ...
1
vote
1answer
51 views

Show that the expression holds for any $x\gt 1$

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x \gt 1$, $$\int \limits_1 ^x [u]([u]+1)f(u)du=2\...
-1
votes
2answers
53 views

Vector parametrization of a surface intersection

How does one parametrize the following curve in 3-space to $\vec{g}(t): [a, b] \to \mathbb{R}^3$: the intersection of $x^2+y^2+z^2=a^2$ and $x+y+z=0$ ? What I could come up with is as follows: $\...
0
votes
3answers
123 views

Derivative of a logarithm from first principles

The usual example where learning about the derivative is obtaining it for $f(x)=x^2$ from first principles (see this for example). I am stumped on how use first principles to obtain the derivative of ...
3
votes
0answers
70 views

A integral inequality

Let $g\in C_0^\infty((-1,1))$.Prove $\forall t\in (-1,1)$,$${g^4}\left( t \right) \le 16\int_{ - 1}^1 {\left( {{{\left| {g'\left( s \right)} \right|}^2} - \frac{{{g^2}\left( s \right)}}{{4{{\left( {1 -...
1
vote
4answers
138 views

Find $\int\frac{x-1}{x^2-5x+6}dx$. Why my solution is different from book?

I'm learning single variable calculus right now. Right now trying to understand integration with partial fraction. I'm confused in a problem from sometime. I think I'm doing right but answer in my ...
1
vote
7answers
88 views

Find min & max of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$

Problem: Find the maximum and minimal value of $f(x,y) = x + y + x^2 + y^2$ when $x^2 + y^2 = 1$. Since $x^2 > x$ (edit $x^2 \geq x$) for all $x \in \mathbb{R}$, $f$ is bowl-ish with a minimal ...
4
votes
2answers
110 views

Apparent discrepancy between change of variables in one versus multiple dimensions.

My freshman calculus book gives the change of variables formula in one dimension and then eight chapters later gives it in $n$ dimensions. But when it generalizes to $n$ dimensions it requires the ...
10
votes
4answers
383 views

An arctan integral $\int_0^{\infty } \frac{\arctan(x)}{x \left(x^2+1\right)^5} \, dx$

According to Mathematica, we have that $$\int_0^{\infty } \frac{\arctan(x)}{x \left(x^2+1\right)^5} \, dx=\pi \left(\frac{\log (2)}{2}-\frac{1321}{6144}\right)$$ that frankly speaking looks pretty ...
4
votes
3answers
126 views

Spivak tough limit proof-verification

Suppose there is a $\delta > 0$ such that $f(x) = g(x)$ when $0 < |x - a| < \delta$. Prove that $\displaystyle \lim_{x\to a} f(x) = \lim_{x \to a} g(x)$. $|f(x) - L| < \epsilon$ for $|x - ...
1
vote
1answer
63 views

Even and odd integrals

Find the definite integral $$\int_{-2}^{2} \Big(2f(x) + 3g(x)\Big)dx$$ where $f(x)$ is an even function such that $$\int_{0}^{2} f(x)dx = 3$$ and $g(x)$ is such that $$\int_{-2}^{4} g(x)dx = -3 ...
0
votes
2answers
102 views

solving a system of equations (3 equations, 3 variables)

I have 3 equations and 3 unknown variables as follows $$\frac{\beta}{1-\alpha}x=y^{\alpha-1}-z$$ $$\left(1+\frac{\beta}{1-\alpha}\right)x=\frac{1}{\sigma}\left(\alpha y-\rho\right)$$ $$x\left(\...
3
votes
3answers
286 views

One Step Forward from Gaussian Integral

Now to solve the integral $ \int_0^\infty e^{-x^2} \, dx $ has become a simple task for us. But how can we solve this integral: $$\int_0^\infty e^{-x^3} \, dx $$
0
votes
1answer
86 views

Doubt related to $\int_0^\infty \frac{\sin ax}{x}dx =\frac{\pi}{2}$

It is well known that $\int_0^\infty \frac{\sin ax}{x}dx =\frac{\pi}{2}$. Is this true if $a=0$? I don't think so. Can someone confirm this. Thanks.
0
votes
1answer
36 views

Gaussian optics

We are given $\frac{n_1}{l_0}+\frac{n_2}{l_i}=\frac{1}{R}(\frac{n_2s_i}{l_i}-\frac{n_1s_0}{l_0})$ $l_0=\sqrt{R^2+(s_0+R)^2-2R(s_0+R)cos(\phi)}$ $l_i=\sqrt{R^2+(s_i-R)^2+2R(s_i-R)cos(\phi)}$ $h= R\;...
0
votes
2answers
156 views

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$

If $f:X \to Y $ is continuous then $f^{-1}(\emptyset)= \emptyset?$ (I know the inverse of a closed set of a continuous function is closed, but is this a must?) And does the following apply to all ...
0
votes
2answers
23 views

Proof for special concave form of functions

If $f(x)$ is a non decreasing concave function between $0$ and $1$ for $x\ge 0$, then for $a>1$ I am confident that $af(x)>f(ax)$, but I am not quite sure how to prove it. Any help would be ...
4
votes
1answer
110 views

Find a formula for $f''$ in terms of $f$, where $f\gt 0$ and $(f')^2=f-\frac{1}{f^2}.$

Problem: Suppose that a function $f \gt 0$ has the property $$ (f')^2=f-\frac{1}{f^2} $$ Find a formula for $f''$ in terms of $f$. Hint: Use Theorem 7. Theorem 7: Suppose that $f$ is ...
3
votes
1answer
249 views

Find the rate of change in height of water level.

A cone shaped container has a diameter of $0.6m$ and height of $0.5m$. Water is poured into the container with a constant rate of $0.2m^{3}s^{-1}$. Calculate the rate of change in height of the water ...
2
votes
3answers
378 views

Double integral problem: $\int_0^\pi\int_x^\pi \frac{\sin y}{y} dy\, dx$

Calculate: $$\int_0^\pi \int_x^\pi \frac{\sin y}{y} dydx$$ How to calculate that? This x is terribly confusing for me. I do not know how to deal with it properly.
7
votes
5answers
176 views

$\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}$ equals $\frac{\pi}{5050}$

For $a\geq2$,if the value of the definite integral $\int_{0}^{\infty}\frac{dx}{a^2+\left(x-\frac{1}{x}\right)^2}$ equals $\frac{\pi}{5050}$.Find the value of $a$. Substituting $x-\frac{1}{x}=t$ does ...
1
vote
4answers
70 views

Integrating for a solution in terms of an natural logarithm

Evaluating the following integral: $$\int_1^2 \frac2{1-3x}\ dx$$ why do you have to take the factor of $-2/3$ out when evaluating the integral?
3
votes
1answer
57 views

Why does Abel's identity imply either $W = 0$ or $W \neq 0$ everywhere?

Let $y_1$ and $y_2$ be solutions to the linear differential equation $A(x)y'' + B(x)y' + C(x)y = 0 $ and let $W = W(y_1, y_2)$ be the Wronskian of the solutions. Why does Abel's identity $\...
1
vote
0answers
20 views

variance of an integral of the sum of lognormal random variable

(My first post here. If anything is wrong or unclear, please let me know.) This arises when I attempted to price an option with a jump diffusion. Here is the specific variance I am trying to solve: ...
2
votes
2answers
108 views

If $f(x,y)= x^{x^{x^{x^y}}} + (\log x)(\arctan(\arctan(\arctan(\sin (\cos xy-\log (x+y)))))$ Find $D_2f(1,y)$

Here we $D_2 f(1,y)$ means we have to calculate the partial derivative w.r.t $y$, so I have applied one short tricks that I have put $x=1$ in the equation then $f(1,y)= 1+0=1$ so the $D_2(f(1,y)=0$. ...
-1
votes
2answers
54 views

Evaluation on a basis of gaussian integral

Knowing that $$\int_{-\infty}^\infty e^{-x^2} dx= \pi^{\frac{1}{2}}$$ Find: $$\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{\frac{-x^2}{2}} dx$$ And my question is how does this help if have the ...