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

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2
votes
1answer
45 views

Displacing a cosinus curve and keeping zero slope

I am doing parametrization and in that regard I need to offset/displace one end of a cosinus curve up along the y-axis. The function I'm looking at is f(s)=cos(2*pi*s) with s ∈ [0..1] such that it ...
0
votes
2answers
55 views

Show that the function is increasing

Let $q>1$ and $f(h)=\dfrac{(1+h)^{q}-1}{h}$ for $h>0$. How can I show that the function $f$ is increasing on $(0,\infty)$? The problem I come across is showing that the derivative ...
0
votes
1answer
48 views

Cubic MacLaurin $e^{x^2}$

Find the Cubic MacLurin expansion of e^{x^2}. First, I tried the sub $t=x^2$ and used the regular expansion for $e^t$. But that was wrong. Can I not do non-linear substituions? My calculations: ...
1
vote
1answer
189 views

Computing “radius” of the Intersection of a Circle and an Ellipse

I've been stuck on the following problem for awhile now. Does anyone have any ideas as to how to get a solution? Suppose $r > 0$ is a real number. The circle $x^2 + (y + 4)^2 = r^2$ has radius ...
18
votes
2answers
361 views

$\int_0^1f(x)dx=1, \int_0^1xf(x)dx=\frac16$ minimum value of $\int_0^1f^2(x) dx$?

Let $f(x)\geq0$ be a Riemann integrable function, and $$\int_0^1f(x)\,\mathrm dx=1, \int_0^1xf(x)\,\mathrm dx=\frac16.$$ Find the minimum value of $\int_0^1f^2(x)\,\mathrm dx$ Cauchy-Schwarz ...
8
votes
2answers
259 views

Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
0
votes
2answers
141 views

Fundamental group of torus by van Kampens theorem

So I am currently going through some lecture notes where the fundamental group of a torus is calculated by van Kampen's theorem: The torus is decomposed into its characteristic fundamental polygon ...
1
vote
1answer
74 views

Implicit partial derivative of a spherical cap

Consider a spherical cap, for which the base radius is $a$ and the height is $h$. Then, the surface area and volume is (these equations can be found on Wolfram Mathworld) $A(a,h) = \pi(a^2 +h^2)$, ...
0
votes
3answers
102 views

What is $f(x)$? [duplicate]

I have the following question. When $x > 0$, $f(x)$ is differentiable and satisfies that $$f(x)=1+\frac{1}{x}{\int_{1}^{x}}f(t)dt.$$ Then What is $f(x)$? Thanks for your help.
3
votes
1answer
59 views

Is there a reason that sine substitution is preferred to cosine substitution?

When integrating by trigonometric substitution that has $\sqrt{a^2-x^2}$ in the integrand, the general recommended approach is to make the substitution $x = a \sin\theta$, and this substitution is the ...
7
votes
4answers
373 views

The closed form of $\int_0^{\infty} \frac{\log(\cosh(x))}{x} e^{-x} \ dx$

An integral I discussed last days in a chat, and it looks like a hard nut since after some manipulations of the initial form we reach an integral where the integrand is expressed in terms of ...
3
votes
1answer
90 views

Area of projected surface

Suppose I have a surface given by $z=f(x,t)$, $a\le x \le b$ and $c\le t \le d$. How can I find the area of the surface projected onto the $(x,z)$ plane? We may assume appropriate monotonicity ...
4
votes
2answers
270 views

What is the sum of this? $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$

I'm in trouble with this homework. Find the sum of the series $ 1 + \frac12 + \frac13 + \frac14 + \frac16 + \frac18 + \frac19 + \frac1{12} +\cdots$, where the terms are the inverse of the positive ...
0
votes
1answer
63 views

Bounds and uniqueness of a transcendental equation

Let $p\in[0,1]$ and $\rho(x): [0,1] \rightarrow [0,\infty)$ such that $$\int_0^1 dx \rho(x) = 1.$$ I'd like to investigate the following transcendental equation: $$\frac{1}{2p} = \int_0^{1} dx ...
3
votes
3answers
922 views

Show bounded and convex function on $\mathbb R$ is constant

How can we show that a bounded and convex function on $\mathbb R$ is constant? Derivatives are of no use since the function does not have to differentiable. I saw an answer here I think a while ago ...
0
votes
2answers
81 views

Find the area of the region of the given curves

Find the area of the region between the curves $y=x^{2013}$ and $y=x^{2014}$. How do you define the upper and lower bound for this case? I found the point of intersection where $x=1$. So is it from ...
0
votes
1answer
31 views

Calculus of vector functions: arc length and speed

Hi! I am trying to study or a test in my calc3 class by doing some online problems, but I am not quite sure how to solve this one. I thought the correct answer was ...
4
votes
0answers
91 views

Convection-diffusion-reaction problem

I seek to solve to the system $$ \frac{\partial \phi_{a}}{\partial t} = D_{a} \frac{\partial^{2} \phi_{a}}{\partial x^{2}} - v_{a} \frac{\partial \phi_{a}}{\partial x} + \mathfrak{K}_{b}\phi_{b} ...
1
vote
1answer
88 views

Flux of a vector field.

In $\mathbb{R}^3 \setminus \{0\}$, it's given the vector field $$\vec{E}(\vec{r}) = g(r) \vec{r}$$ where $g$ is some function of class $\mathcal{C}^\infty$ defined in $[0, + \infty[$, $\vec{r} = ...
6
votes
1answer
109 views

Help needed in solving a system of DE

The system of DE is: $$\frac{dI}{db}=-\frac{b}{c}\frac{dJ}{db}-\frac{2ab+1}{2c}J$$ $$\frac{dJ}{db}=\frac{b}{c}\frac{dI}{db}-\frac{2ab-1}{2c}I$$ Assume that $a$ and $c$ are constants and both $I$ and ...
4
votes
6answers
410 views

Continuous function?

Consider the function $f(x)=[x]$ on the interval $[0,2]$ where $[x]$ denotes the largest integer less than or equal to x. Is this function continuous? I cant find a reason for it not to be, although ...
0
votes
1answer
34 views

Cosine proof of vectors

Hi this is the excerpt from the book I'm reading Proof: We will prove the theorem for vectors in $\Bbb R^3$ (the proof for $\Bbb R^2$ is similar). Let ${\bf v}=(v_1,v_2,v_3)$ and ${\bf ...
3
votes
1answer
188 views

Motivation and Derivation of the Riccati Equation Transformation

Given a Riccati Equation which is differential equation of the form: $$ \frac{dy}{dx} = a_0 (x) + a_1 (x)y + a_2 (x)y^2 $$ It is well known that the transformation: $$ y = -\frac{1}{a_2(x)} ...
1
vote
3answers
105 views

How can I determine the value of this limit? $\lim\limits_{t \to -\sqrt{8}} \frac{t + 2\sqrt{2}}{3t^2 - 24}.$

Evaluate the following limit: $$\lim_{t \to -\sqrt{8}} \frac{t + 2\sqrt{2}}{3t^2 - 24}.$$
1
vote
1answer
101 views

How to find the following integral? $\int\tfrac{x}{\sqrt{1+3x^2}}\mathrm dx$

Find: $$\int\dfrac{x}{\sqrt{1+3x^2}}\,\mathrm dx$$ I can't fully integrate this, I get $1/x+\sqrt3 x$ and then I don't know what to do, not sure if I even started it correctly, thanks in advance. ...
1
vote
4answers
85 views

calculation the Integral $ \int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx$

How can i calculate the Integral $\displaystyle \int_{0}^{1}\frac{\tan^{-1}(x)}{1+x}dx$ $\bf{My\; Trial::}$ Let $\displaystyle I(\alpha) = \int_{0}^{\alpha}\frac{\tan^{-1}(\alpha x)}{1+x}dx$ Now ...
2
votes
1answer
124 views

minimum and maximum of $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$

we are asked to find the minimum and maximum of the function$f:A \to A$ $f(x,y)=\sin(x)+\sin(y)-\sin(x+y)$ Where $A$ is the triangle bound by $x=0$,$y=0$ and $y=-x+2\pi$ I'd like someone to review ...
2
votes
2answers
244 views

What is the “proof” for this vector calculus theorem besides intuition?

I'm reading Div, Grad, Curl, and All That, and one of the exercises reads as follows: Instead of using arrows to represent vector functions, we sometimes use families of curves called field lines. ...
0
votes
1answer
61 views

Finding the integral of a trig function using a matrix

It can be shown that Ɓ = {1, $\cos(t)$,…$\cos(6t)$ and Ƈ = (1,$\cos(t)$,…$\cos^6$(t)} span the same subspace of Ƈ(ℝ) a. Use an appropriate change of coordinate matrix to find $cos^6$(t) in terms of ...
1
vote
1answer
33 views

Rate of surface increase of a sphere related to its volume increase

I'm trying to solve problems in the Thompson's classic "Calculus made easy" (not so easy by the way). One of this problems is to find the rate increase of the surface of a sphere which volume is ...
2
votes
3answers
72 views

How do I find $\lim\limits_{x \to 0^+}{\sin^2x \over e^{-1/x}}$?

Find the limit $$ \lim_{x \to 0^+}{\sin^2x \over e^{-1/x}} $$ Have tried to apply the L'Hospital's rule, but the denominator stays always the same $\left( {d \over dx}e^x=e^x \right)$ and brings ...
1
vote
1answer
45 views

Integrate with given substitution

I've been given the following definite integral: $$\int_0^{1/2}\dfrac{1}{(1-x^2)^{3/2}}dx$$ I've been asked to solve it using the substitution $x = \sin u$. My attempts are not yielding the right ...
1
vote
0answers
60 views

Can this integral be expressed in terms of elementary functions? $\int\frac{\ln(2+x^2)}{1+x^2}dx.$

If the following primitive can be computed, can someone show me the way ? $$\int\frac{\ln(2+x^2)}{1+x^2}dx.$$ I tried substituting $x$ with $\tan(t)$ but the integral didn't get much simpler.
0
votes
0answers
33 views

is this a valid counter-example - function is not locally invertible

Let $S_n$ be the set of all symmetric matrices with real entries of size $n$x$n$. We are asked if the function $f:S_n \to S_n$, $f(A)=A^2$ is locally invertible for every $A$ (Using the Inverse ...
1
vote
1answer
99 views

Does $\int_0^{\infty } \cos \left(e^x\right) \, dx$ converge?

So, there he is $$\int_0^{\infty } \cos \left(e^x\right) \, dx$$ Mathematica says it is a convergent integral, but I need some sort of a proof. How do we know that it is actually convergent? I've ...
1
vote
3answers
28 views

Distance of points in $\mathbb R^3$ using vectors

Why is this $\mathbf{w}-\mathbf{v}$ and not $\mathbf{v}+\mathbf{w}$? I understand $\mathbf{v}+\mathbf{w}$ means $\mathbf{w}$ vector is placed at end of $\mathbf{v}$. However book didn't explain well ...
3
votes
0answers
18 views

$\sqrt{X}$ where $X$ is a positive definite matrix is smooth $C^{\infty}$ [duplicate]

I'm trying to prove the following statement. Let $P_n \subset Mat_{nxn}(\mathbb R)$ be the set of all symmetric positive definite matrices with real entries of size $n$x$n$. Let $\sqrt{}:P_n \to ...
1
vote
1answer
29 views

An example of a function which is (locally) above its tangents but is not convex

I am looking for an example of a function $f:[a,b]\rightarrow\mathbb{R}$, continuous in $[a,b]$, differentaible in $(a,b)$, such that for any $a<t<b$ there is some open interval $t\in I\subseteq ...
1
vote
1answer
39 views

Solve definite Integral with given substitution

I've been given the following integral $$\int_a^b f(x) \,dx,$$ where $f(x) = \sqrt{16 - x^2}$ and $[a, b] = [-4, 4]$ I've been given the instruction to solve the definite integral with the ...
8
votes
5answers
349 views

Calculate $\lim\limits_{y\to{b}}\frac{y-b}{\ln{y}-\ln{b}}$

How can we find $\displaystyle \lim_{y\to{b}}\frac{y-b}{\ln{y}-\ln{b}}$ without using: (a) L'Hôpital's rule, (b) the limit $\displaystyle \lim_{h \to 0}\frac{e^h-1}{h} = 1$, and (c) the fact that ...
2
votes
1answer
173 views

Definite integral problem $\int_0^{\infty} \frac{\sin(nx)e^{-anx}}{x^2-\pi^2}\,dx$

$$\int_0^{\infty} \frac{\sin(nx)e^{-anx}}{x^2-\pi^2}\,dx$$ $n$ is an integer and $a>0$. I came across this integral while solving an another problem but I have no idea about evaluating it. I ...
0
votes
0answers
54 views

Find the value of the integral $\int^\infty_0 \frac{\cos(ax)}{1+x^2} dx$ [duplicate]

I have to compute the value of the integral $$ \int^\infty_0 \frac{\cos{(ax)}}{1+x^2} dx $$ It may help that $$\int_0^\infty e^{-tx}\cdot \sin(x)dx = \dfrac{1}{1+t^2}$$ but i can't find the link ...
7
votes
1answer
114 views

Prove that $f(x)=O(x)$ as $x\to 0$

Let $f$ be a function defined on $R$. for any absoulutely convergent series $\sum_{n=1}^{\infty}a_{n}$,the series $\sum_{n=1}^{\infty}f(a_{n})$ converges,Prove that $$ f(x)=O(x)\qquad (x\to 0) $$. I ...
0
votes
4answers
60 views

How can I find this limit?

The question is: $$\lim_{x\rightarrow0} \frac {(x+1)^{1/3}-1} x$$ Obviously I have to rationalize it such that the x in the denominator is removed, but I'm not sure how to do so. Thanks.
0
votes
0answers
47 views

Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
1
vote
2answers
127 views

Differential and Infinitesimals

In a calculus textbook I have (Calculus, Stewart), it states that for a differentiable function $y=f(x)$, the differential of the function is defined as $$dy=f'(x) dx.$$ It states that $\Delta ...
2
votes
4answers
197 views

Calculation of a limit+

For $\alpha>1$ how can I show that $\lim\limits_{n\to \infty}\dfrac{\pi}{n}\left( \ln \left(\dfrac{\left(α-1\right)^{2}\left(α^{2n}-1\right)}{(α+1)(\alpha-1)}\right)\right)=2\pi\ln(\alpha)$ ...
0
votes
0answers
54 views

Solve double integral in polar coordinates.

Hello I have to solve this double integral using polar coordinates: $$\iint_D x^2+ 3y^2 {\,\rm d}x {\,\rm d}y$$ where the domain D is { -1≤x≤1, $-√1-x^2$≤ y ≤$√1-x^2$}. I did solved using the ...
5
votes
5answers
231 views

Find the following integral (most likely substitution)

$$\int_0^1 \frac{\ln(1+x^2)}{1+x^2} \ dx$$ I tried letting $x^2=\tan \theta$ but it didn't work. What should I do? Please don't give full solution, just a hint and I will continue.
1
vote
2answers
65 views

function - area under the curve

Trying to find area under the curve for the function $f(x)= 4x^3 + 3x^2 - x + 1$ on the interval $[1,2]$. My answer is: $$\int_{1}^2 4x^3 + 3x^2 - x + 1 dx = ...