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

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1
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1answer
62 views

Integrating $(2x+1)^{-2}$

How do I integrate the following expression: $$(2x+1)^{-2}$$ I should end up with something like this: $$\frac{-1}{2(2x+1)}$$
1
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2answers
42 views

a proof of constants are null from a given inequality

Problem: given constants $a,b\text{ and }c$, and a variable $x$, assume that for all $x\in\mathbb{R}$ holds that $|ax^2+bx+c|\le|x|^3$, then proof that $a=b=c=0$ My try: substitute $x=0$ into the ...
3
votes
2answers
73 views

Derivatives of trig polynomials do not increase degree?

Let $c = \cos x$ and $s = \sin x$, and consider a trigonometric polynomial $p(x)$ in $c$ and $s$. The degree of $p(x)$ is the maximum of $n+m$ in terms $c^n s^m$. Is it the case that repeated ...
3
votes
4answers
868 views

Limit of sum is sum of limits

$$ \lim_{x\to a } [ f(x) + g(x) ] = \lim_{x\to a } f(x) + \lim_{x\to a } g(x) $$ $$ \lim_{x\to a } f(x) \ \text{ and } \ \lim_{x\to a } g(x) \ \text{ exist}. $$ I wanted to know that if $f(x)$ ...
0
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2answers
74 views

Taylor series $(x+1)^{\frac{1}{3}}$

Complete the Maclaurin polynomial of degree three for $(x+1)^{1/3}$. I have completed the first two derivatives of this function and thus have coefficients but am not certain how to put them into ...
1
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5answers
67 views

Rearrange the equation and solve for $y(x)$ [closed]

How do I solve for $y$? $$ \ln\left(\frac{y-1}{y+1}\right)= x^2 $$
3
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2answers
142 views

Integral Involving Dilogarithms

I came across the identity $$\int^x_0\frac{\ln(p+qt)}{r+st}{\rm ...
0
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1answer
81 views

Calculus of variations: big-O notation?

I have a formula in my text-book $$y(x+C) = y(x) + \frac{dy}{dx}C + O(C^2)$$ Can someone explain this formula?
0
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1answer
23 views

Calculus Maximal Domain

Hi just a quick question regarding maximal domain. Find the maximal domain of the following function $f\left(x\right)=\frac{64-x^6}{\left(64-x^2\right)\left(64+x^2\right)\left(64+x^3\right)}$. What ...
3
votes
0answers
95 views

The difference between an affine k-simplex and a rectilinear k-simplex

The notion of rectilinear k-simplex appears in Theorem 10.27 of Rudin's book "Principles of Mathematical analysis", then what is the definition of a rectilinear k-simplex? I read the proof of Theorem ...
0
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2answers
43 views

Two-dimensional Taylor linearisation

I have to perform a first order taylor expansion of a function $f(\vec{x}) = f(x+u,y+1)$ at the point $\vec{a} =(x,y)$. My solution reads $$ f(\vec{x}) \approx f(x,y) + \left( \begin{matrix} ...
1
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1answer
3k views

Find delta with a given epsilon for $\lim_{x\to-2}x^3 = - 8$

Here is the problem. If $$\lim_{x\to-2}x^3 = - 8$$ then find $\delta$ to go with $\varepsilon = 1/5 = 0.2$. Is $\delta = -2$?
6
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3answers
4k views

Finding $\delta$ with given $\varepsilon$ for $f(x)=4x^2 +x +3$ and $x\to3$

$f(x)=4x^2 +x +3$ and the limit as x approaches $-3$ of $f(-3)= 36$, Find $\delta$ such that $0<|x+3|<\delta \longrightarrow |f(x)-36|<.003$ I have tried: $|(x+3)(4x-11)|<0.003$ ...
5
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4answers
642 views

Spivak's Calculus (Chapter 5, Problem 41): Proof that $\lim_{x \to a} x^2 = a^2$

In Chapter 5, Problem 41, Spivak provides an alternative way to prove that $$\lim_{x \rightarrow a} x^2 = a^2\,\,,\,\,a > 0$$ Given $\,\epsilon > 0\,$ let $$\delta = \min\left\{\sqrt{a^2 + ...
4
votes
2answers
197 views

Integral of the Von karman equation

What is the result of this integral, and how can I proceed: $$ \int_{-\infty}^{\infty}{c_{1} \over\left(1 + c_{2}\,x^{2}\right)^{5/6}}\, \cos\left(x\tau\right)\,{\rm d}x\,,\qquad c_{1}, ...
3
votes
2answers
60 views

For what values of $x$ does the geometric series $(2+x)+(2+x)^2+(2+x)^3+\cdots$ converge?

I am stuck on this geometric series question: For what values of $x$ would the infinite geometric series $(2+x)+(2+x)^2+(2+x)^3+\cdots$ converge? Formula for series: $$S_n=\frac{a(1-r^n)}{1-r}$$ ...
0
votes
2answers
92 views

Find all values $c$ such that $(x+1)/(x^2+2cx+4)$ has domain R

Word for word: Find all values of $c$ such that $f(x)=\frac{x+1}{x^2+2cx+4}$ has a domain R I really don't know where exactly to start. I'm not sure what it means by "a domain R"
1
vote
2answers
39 views

Overall difference in percent

I want to calculate the total difference in % between two investments {A,B} in the following scenario: In year t=0 revenue A is 70 % smaller than revenue B. Every year the revenue from A further ...
2
votes
5answers
444 views

How to solve a limit using L'Hôpital's rule

This is my limit: $$\lim_{x \to 1} \left[(1-x)\tan\left(\frac{\pi x}{2}\right)\right] $$ My mind is probably playing games on me right now but can you help me?
0
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2answers
21 views

Polynomial division to function multiplication on ODE with Separable Method

I want to solve the ODE below with the Separable Method. I know I need to see the product of $f(y)$ and $f(x)$, but I don't remember the algebra needed to see it on the polynomial division: ...
0
votes
1answer
84 views

Finding distance using rates of change — best approach?

The question: A man drives from state $A$ to state $B$ going $60 \frac{miles}{hour}$. Then he returns from state $B$ to state $A$, driving $45 \frac{miles}{hour}$. His total driving time is $2.5 ...
2
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3answers
64 views

Infinite Limit Question

I am just starting limits, really stumped on this one. How do I approach this? $$\lim_{x\to -\infty} (x-2)(x-3)$$
0
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4answers
56 views

Integration, ball throw simple example

I thought that I could do it like this. Given that $$g=9.8m/s^2$$ $$\int -9.8 \, dt=v_0-9.8 t$$ Setting it equal to zero we have: $$t=\frac{v_0}{9.8}$$ $$\int \left(v_0-9.8 t\right) \, dt=-4.9 ...
1
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2answers
47 views

Derivative of certain piecewise function

I've got function $f(x) = \begin{cases} \frac{ln(1+x)}{x} &\text{for $x\not=0$} \\ 1 &\text{for $x=0$} \end{cases}$ And I need to find the derivative. (also one sided) I've found that ...
2
votes
0answers
47 views

Proof of Second Partials Test

How does one rigorously prove the second partials test without firstly assuming that $D(a,b)=AC-B^2$ that states the following: $ A=\frac {\partial^{2}f(a,b)}{\partial x^{2}},B=$$\frac ...
1
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2answers
878 views

why area under curve or riemann sum equals to definite integral

i do get that Riemann sums is sum of infinite triangles with with infinitely small length. But definite integral is completely different you are taking anti derivative of f(x) at b and subtract anti ...
2
votes
2answers
131 views

Using Stokes theorem to integrate $\vec{F}=5y \vec{\imath} −5x \vec{\jmath} +4(y−x) \vec{k}$ over a circle

Find $\oint_C \vec{F} \cdot d \vec{r}$ where $C$ is a circle of radius $2$ in the plane $x+y+z=3$, centered at $(2,4,−3)$ and oriented clockwise when viewed from the origin, if $\vec{F}=5y ...
1
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2answers
57 views

Relating $\sin(x)^{2a+1}$ on interval $(0,\pi/2)$ to a factorial form

The integral: $$\int_0^{\pi/2} \sin(x)^{2a+1}dx $$ has a closed form solution in terms of factorials: $$[(2^a)(a!)]^2 [(2a+1)!]^{-1}$$ How does this come about?
9
votes
2answers
215 views

Computing $\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$

I don't know how to compute: $$\int {\dfrac{\csc^{2014}x-2014}{\cos^{2014}x} dx}$$ I have tried substituting $t=\tan ^{2} x$ but got nothing out of it. I know there's some trick involved, but ...
1
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4answers
69 views

Finding $p'(0)$ for the polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$

The question goes as follows: Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p'(0)$ is... What I ...
2
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2answers
91 views

Integration of $1/\sin^3 x$

I need a explanation of this problem: $$ \int \frac{1}{\sin^3 x}\,dx $$ Change the variable $$ t = \tan (x/2) $$ With use of $\tan$, $\cos$, $\sin$ and $\cot$, only. So how do I ...
2
votes
0answers
88 views

asymptotic expansion of the integral for large tau

How can I proceed to resolve this integral? $$ c_1\int_{-\infty}^{\infty}{\frac{\cos\left(x\tau\right)}{\left(1 + c_{2}\,x\right)^{\alpha}}}\, \,{\rm d}x $$ where $c_1, c_2$ are positive constants, ...
2
votes
1answer
72 views

$f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite

I need to prove: $f$ is uniformly continuous on $(a, b]$ implies $\lim\limits_{x\to a^+} f(x)$ exists and finite Now, I already have a sketch for the proof: Let $\{x_n\}$, a sequence such ...
1
vote
6answers
121 views

Initial value $\left ( \frac{dy}{dt} \right )+3y=11$, $y(0)=1$

I have never done an initial value problem, and would like some help on how to start this please.
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
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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
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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
320 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
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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
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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
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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 ...