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

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2
votes
1answer
31 views

$f(t) = \cos t^{-1} + \int_t^\infty \frac{1}{\tau^2 + f(\tau)^2} d\tau$ implies the integral is $O(\frac{1}{t})$

The following is a quote from "asymptotic methods in analysis" by de Bruijn (p. 136). If we know that the real function $f(t)$ satisfies the relation $$f(t) = \cos t^{-1} + \int_t^\infty ...
1
vote
2answers
35 views

Extract a variable from a formula

my maths a a bit rusty and I need to extract a variable from a formula. It's needed for a project about air quality in order to convert data from sensors to an index. The formula is : $$\left ...
1
vote
2answers
64 views

How do I solve ? $\mathop {\lim }\limits_{x \to 0} \frac{{2\cos x}}{{x + 3}}$

When I see $2\,cosx$ do I assume they want me to change it to a value? If that is the case, what value would that be?
3
votes
3answers
163 views

Limit evaluation problem

I've got this limit and I can't figure out the right steps to find the result. (I know the result). $\lim_{x\to-\infty}e^{x}(x+1)^{n}, n\in \Bbb N$ Any hints?
5
votes
1answer
323 views

How do you determine the points of inflection for $f(x) = \frac{e^x}{1+e^x}$?

$$f(x) =\dfrac{e^x}{1+e^x}$$ I know we can find points of inflection using the second derivative test. The second derivative for the function above is $$f''(x) = \dfrac{e^x(1-e^x)}{(e^x+1)^3}$$ I have ...
1
vote
3answers
286 views

Derivative of $x^2\sqrt{1+x}$

Given that $f(x)=x^2\sqrt{1+x}$, show that $f'(x)=\dfrac{x(ax+b)}{2\sqrt{1+x}}$ where $a$ and $b$ are constants to be found. I first tried using the product rule: ...
3
votes
3answers
68 views

calculus question attempt

find the maximum value of the function $$y = 15 \sin x -8 \cos x $$ attempt at a solution: deriving: $y' = 15\cos x +8\sin x $ equating to zero and doubling by $ 1/\cos x$ (Im not sure this is ...
0
votes
1answer
25 views

Is the following graph having two local minima

https://www.desmos.com/calculator/abuvb1zdkb I think yes, the main question i think is of the definition of neighbourhood For a function with domain $(-\infty, -3)\cup (3, \infty)$ $ $ Is -3 in ...
2
votes
4answers
6k views

Find tangent line of curve that intersects point.

How do I find the tangent line of the curve $y=x^2$ that intersects the point $(8,2)$?
1
vote
1answer
54 views

If $f(x) = x^{\alpha}\cdot \ln(x)$ and $f(0)=0$, for which $\alpha$ Rolle's Theorem is applicable?

If $f(x) = x^{\alpha}\cdot \ln(x)$ and $f(0)=0$. Then the value of $\alpha$ for which Rolles Theorem is applicable, is $\bf{My\; Try::}$ Rolle,s Theorem is Applicable in $x\in \left[\; ...
1
vote
1answer
121 views

Evaluation of $\int\frac{1}{2014^x+2015^x}dx$

Evaluation of $\displaystyle \int\frac{1}{2014^x+2015^x}dx$ $\bf{My\; Try::}$ Let $\displaystyle I = \int\frac{1}{2014^x+2015^x}dx = \int\frac{2014^{-x}}{1+\left(\frac{2015}{2014}\right)^x}dx$ Now ...
1
vote
2answers
97 views

Choosing the right $\delta$ for uniformly continuous function

I'm reading a proof for the claim $g(x)$ is uniformly continuous. It comes down to: $\forall x,y>B:\left|g(x)-g(y)\right|\le \left|x-y\right| + \frac{\varepsilon}{2}$ The auther claims $\delta = ...
0
votes
4answers
105 views

Definite integral $\int_{-64}^{1}\frac{dx}{x^{1/3}}$

I am having some trouble with a problem very similar to this in my study guide, how can I start, the $-64$ is really intimidating to me.
6
votes
5answers
226 views

Alternate Proof for $e^x \ge x+1$

This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it: Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) ...
1
vote
1answer
82 views

Evaluation of $\int_0^\infty \frac{(x^2+y^2)^{-s/2}}{e^{2\pi y}-1}\cos(s \arctan(y/x))dy$

$$\mbox{Does the integral}\quad \int_{0}^{\infty}{\left(x^{2} + y^{2}\right)^{-s/2} \over {\rm e}^{2\pi y} - 1}\, \cos\left(s\arctan\left(y \over x\right)\right)\,{\rm d}y\quad \mbox{converge or ...
11
votes
2answers
453 views

Integral $\int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx$

Calculate the following integral: \begin{equation} \int_1^{\sqrt{2}}\frac{1}{x}\ln\left(\frac{2-2x^2+x^4}{2x-2x^2+x^3}\right)dx \end{equation} I am having trouble to calculate the integral. I ...
32
votes
8answers
6k views

Why is $\int_{0}^{\infty} \frac {\ln x}{1+x^2} \mathrm{d}x =0$?

We had our final exam yesterday and one of the questions was to find out the value of: $$\int_{0}^{\infty} \frac {\ln x}{1+x^2} \mathrm{d}x $$ Interestingly enough, using the substitution ...
3
votes
3answers
167 views

Explanation of line element formula $dl^2 = dx^2 + dy^2$

I found this in a physics textbook without justification: $$dl^2 = dx^2 +dy^2,$$ where I presume that $l = \sqrt{x^2+y^2}$. Why is this so? By my calculations I obtain $$ dl = \dfrac{\partial ...
1
vote
1answer
67 views

Understanding the notation of a book when derivating

I'm trying to understand the notation that the book uses. The book says $(1)$ $y=a\cdot \sin x$ and then the derivate of $(1)$ is $(2)$ $\frac{d^2y}{dx^2}=-a \cdot \sin x$ I don't get what to do ...
15
votes
3answers
381 views

Need help with $\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx$

I need you help with this integral: $$\int_0^\infty\frac{\log(1+x)}{\left(1+x^2\right)\,\left(1+x^3\right)}dx.$$ Mathematica says it does not converge, which is apparently false.
3
votes
0answers
61 views

Maximize or find an upper bound of the function $kx^{k-1}\exp(-\mu(x^k-x))$

I was programming some random variable simulation using the acceptance-rejection method and I encounter with the Weibull$(k,\lambda)$ distribution. This random variable is posible to simulate with ...
11
votes
2answers
160 views

Trigamma identity $4\,\psi_1\!\left(\frac15\right)+\psi_1\!\left(\frac25\right)-\psi_1\!\left(\frac1{10}\right)=\frac{4\pi^2}{\phi\,\sqrt5}.$

I heuristically discovered the following identity for the trigamma function, that I could not find in any tables or papers or infer from existing formulae (e.g. [1], [2], [3], [4], [5], [6]): ...
2
votes
0answers
115 views

Definite Integral involving matrices

We have a definite integral of the form given below $ f(t) = \int_0^1 e^{\alpha X(t)} \frac{dX(t)}{dt} e^{(1-\alpha) X(t)}\,d\alpha \tag 1$ Given Data in the question $X(t)$ is a ...
1
vote
1answer
94 views

Matrix - Commutative property

I have a rotation matrix represented as $R(t)=e^{B(t)},\tag 1$ where $B(t)$ is a skew symmetric matrix (since any rotation matrix can be expressed as a matrix exponent of a skew symmetric matrix), ...
2
votes
1answer
41 views

Trouble isolating a variable in a simple equation.

I am trying in vain to isolate $t$ in the equation $(t + \tau)^\alpha - t^\alpha = \beta$ with $t > 0$ and $\alpha$, $\beta$ and $\tau$ all real. Since I didn't get any useful result, I solve the ...
3
votes
5answers
2k views

Best practice book for calculus

I tried with every inch in me to not ask a question such as this but I just couldn't resist asking this. What is the best Calculus practice book? I tried looking around but couldn't find a ...
1
vote
1answer
44 views

Complement of a set

Suppose I have a set defined by: \begin{align} A=\{(x,y): \min_z f(x,y,z) >d \} \end{align} what is the compliment of this set: Is it \begin{align} A^c=\{(x,y): \min_z f(x,y,z) \le d \} ...
5
votes
0answers
68 views

How to compute product integrals?

From the wikipedia article about product integrals I can see that if our function is scalar, then to compute type I product integral we can just take exponential of a usual integral: $$\prod_a^b ...
0
votes
1answer
108 views

about calculus books.

I need a calculus book that has all the details or is closest to it. If one book for calculus would not be enough to understand concepts, kindly recommend books that don't have overlapping concepts. ...
2
votes
0answers
227 views

Calculus book advice [duplicate]

I'm reading Thomas Calculus now but I don't think it includes Mellin transform or Riemann-Stieltjes integration... Can you recommend an advanced calculus book which includes all of this stuff?
3
votes
5answers
81 views

Integrate $\int \left(A x^2+B x+c\right) \, dx$

I am asked to find the solution to the initial value problem: $$y'=\text{Ax}^2+\text{Bx}+c,$$ where $y(1)=1$, I get: $$\frac{A x^3}{3}+\frac{B x^2}{2}+c x+d$$ But the answer to this is: ...
1
vote
4answers
1k views

calculus, self-study..recommendations?

I've been trying to educate myself in various areas of mathematics. I have been out of any formal math education for quite some time and so I brushed up on some basic (really, really basic) stuff ...
0
votes
3answers
72 views

Area of the region: $\;x ≥ 0; \;−x\sqrt3 ≤ y ≤ x\sqrt3;\,\;(x−1)^2 + y^2 ≤ 1$.

Can anyone please explain how to set up the needed integral? I need to calculate the area of the region given by: $x ≥ 0,$ $-x\sqrt3 ≤ y ≤ x\sqrt3,$ $(x−1)^2 + y^2 ≤ 1$.
19
votes
6answers
2k views

Proof of $\int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}$

Numerically it seems to be true that $$ \int_0^\infty \frac{\sin x}{\sqrt{x}}dx=\sqrt{\frac{\pi}{2}}. $$ Any ideas how to prove this?
7
votes
6answers
899 views

How to calculate limit of a function having factorial in denominator

For $n$ tending to infinity find the following limit $$2^n/n!$$ i have a feeling that it is multiplication of many numbers with the last one turning to 0 but the 1st one is finite so limit should be 0 ...
3
votes
2answers
100 views

Definite integral into indefinitie series

Convert $\displaystyle \int_0^1 e^{x^2}\, dx$ to an infinite series.
0
votes
1answer
79 views

Show That $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ And $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ Are Orthogonal Trajectories

Show that the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ are orthogonal trajectories if $A^2< a^2$ and $a^2-b^2 = A^2+B^2$. What I've ...
0
votes
0answers
33 views

Square a linear ODE

Assuming that I have a linear ODE without any singularities over the complex numbers $$\sum_{k=0}^{n} g_i(x) y^{(k)}(x)=0.$$ Now I substitute $\sqrt{f}:=y$ into this differential equation and square ...
1
vote
1answer
50 views

Intervals on which function is increasing and decreasing

Let $p(x)=x^5-q^2x-q$ , where $q$ is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below. We compute ...
0
votes
0answers
44 views

Derivation using Ito calculus?

I am reading the paper "Coupling Wiener processes by using copulas" by P. Jaworski and I've come across a statement I cannot reproduce. Let $L^{-}$ and $L^{+}$ be differential operators acting on ...
1
vote
3answers
291 views

Differential equation $\sin \theta \frac{dr}{d \theta}+r\cos \theta =\tan \theta,0<\theta<\pi/2$ [closed]

This problem has been stumping me for over an hour how can I set it up, I think I have done it wrong over and over. Solving for $r$.
1
vote
1answer
49 views

Prove that the length of segment on tangent is constant for $y=\frac a2\ln{\frac{a+\sqrt{a^2-x^2}}{a-\sqrt{a^2-x^2}}}-\sqrt{a^2-x^2}$

Prove that the length of segment of tangent from point of tangency to the point where it cuts the y-axis is constant. $$y=\frac a2\ln{\frac{a+\sqrt{a^2-x^2}}{a-\sqrt{a^2-x^2}}}-\sqrt{a^2-x^2}$$ ...
9
votes
2answers
238 views

$f'$ exists, but $\lim \frac{f(x)-f(y)}{x-y}$ does not exist

Suppose $f$ is differentiable at $a$, i.e. $\lim_{x\to a}\frac{f(x)-f(a)} {x-a}$ exists. I wondered whether it was necessarily true that $$\lim_{\substack{x,y\to a\\x\neq y}}\frac{f(x)-f(y)}{x-y} ...
0
votes
2answers
48 views

Calculus long division $\int\frac{y^4+3y^2-1}{y^3+3y}\ dy$

I have a problem like this in my homework and want to see how to go by doing this problem. I understand the long division, but cannot get the partial fraction part. $$\int\frac{y^4+3y^2-1}{y^3+3y}\ ...
4
votes
1answer
215 views

Can a function have a strict local extremum at each point?

A problem given in Spivak's Calculus text is to show that a function $f:[a,b]\to \mathbb{R}$ cannot have a strict local maximum at each point. I will sketch the proof below the fold. My question is: ...
1
vote
1answer
22 views

When is $f=\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^5-3x+\ln5$ decreasing $\forall\; x$?

When is $$f=\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^5-3x+\ln5$$ decreasing $\forall\; x$? Diffrentiating: $$f'=5\left(\frac{\sqrt{p+4}}{1-p}-1\right)x^4-3$$ If $f$ is decreasing, $f'<0$: ...
0
votes
2answers
84 views

Using integral test for $\sum 1/i^2$

It's been a while since I've done an Integral, but am required to relearn them for a class. Could anyone help me with the integral of $\dfrac{1}{i^2}$? Wouldn't the answer be $-1/i$ ? Context I'm ...
0
votes
2answers
707 views

How to evaluate $\int 1/(1+x^{2n})\,dx$ for an arbitrary positive integer $n$?

How to find $$\int\dfrac{dx}{1+x^{2n}}$$ where $n \in \mathbb N$? Remark When $n=1$, the antiderivative is $\tan^{-1}x+C$. But already with $n=2$ this is something much more complicated. Is there a ...
-1
votes
2answers
170 views

If a continuous real function is additive, then it is linear

I have to prove the following problem Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x + y) = f(x) + f(y),\ \forall x,y \in \mathbb{Z}$. Then $f$ is a linear ...
0
votes
2answers
131 views

If a sequence of integers converges, is its limit an integer?

Let ${a_{n}}$ be a convergent sequence in $\mathbb Z$. Is it true that $\lim_{n\rightarrow \infty}{a_{n}}\in \mathbb Z$? Remark This is false if $\mathbb Z$ is replaced by $\mathbb Q$, because then ...