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

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1answer
20 views

Expressing limit of sum definite integral

Evaluate limit by expressing it as a definite integral. ...
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3answers
30 views

$\varepsilon$ - $\delta$ proof for $\lim_{x \to 27}2x^{2/3}=18$

Construct a careful $\varepsilon$ - $\delta$ argument to show $$\lim_{x \to 27}2x^{2/3}=18$$ From the definition of a limit $$\forall \varepsilon > 0, \space \exists \delta >0 \space : ...
2
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4answers
51 views

Derivation of the integral

Evaluate $$\large\frac{d}{dx}\int_{0}^{\large\int_0^{e^x}{\cos (s)\,\mathrm ds}}\sec(t^2)\,\mathrm dt$$ I got the answer to be $$e^x\cdot\sec(\sin^2(e^x))\cdot \cos(e^x)$$ but do not know if ...
0
votes
1answer
26 views

Consider $f(x) = \frac{2x^3-1+\sin x}{x^2-3}$. Show that $f (x) < 2x$ for most negative values of $x$.

Consider $$f(x) = \frac{2x^3-1+\sin x}{x^2-3}$$ Show that $f (x) < 2x$ for most negative values of $x$. How do I start this/ what concepts does this questions test?
2
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2answers
41 views

Homework help. From spivak calculus book

Show that $f$ is convex on an interval if and only if for all $x$ and $y$ in the interval we have $$f(tx+(1-t)y)<tf(x)+(1-t)f(y), 0<t<1$$ The only thing I know is that we have to approach ...
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0answers
15 views

Maximum volume of an open box with a square base?

A box with a square base and an open top is to be made. You have 1200cm^2 of material to make it. What is the maximum volume the box could have? Here's what I did: 1200 = x^2+4xz; where x=length of ...
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2answers
23 views

Continuous increasing bounded function, derivative

Is it true that a differentiable (and hence continuous) increasing bounded function $f:\mathbb{R} \to \mathbb{R}$ has derivative $f'$ that must go to zero as $x \to \infty$. If it is, could someone ...
3
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2answers
71 views

$\frac{1}{x^2} \int xe^x dx$ without using integration by parts

On a test i just had, i needed to solve a differential equation which lead me to having to find the result of $$ \frac{1}{x^2}\int xe^x dx $$ I then attempted to do this integral without integration ...
2
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2answers
39 views

Trouble Understanding Continuity Theorem

I am looking at Calculus on Manifolds by Michael Spivak, but there's a theorem that I don't quite understand. 1-8 Theorem. If $A \subset \mathbb{R}^n$, a function $f: A \to \mathbb{R}^m$ is ...
1
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1answer
57 views

Proof of $\lim_{x \to \infty}\tan x/x$ does not exist

There is an answer for this question as follows: If we approach to infinity with the sequence $a_{n}=n\pi$ then limit is zero, on the other hand if we approach with the sequence ...
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5answers
52 views

Composition of two functions is not commutative

I have been always shown that the composition of two functions is, in general, not commutative with a counterexample. But can you give a more general proof of this statement (that is to say, one that ...
2
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5answers
46 views

Integration by parts of $\cos(x)e^{-x}dx$

I do the integral but I end up getting the original $\cos(x)e^{-x}dx$ on both sides and canceling them out resulting in no solution. Can I get a step by step break down of how to solve?
1
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3answers
146 views

Optimization problem?

Hi I was having trouble figuring out this question. Find the point on the circle $x^2 + y^2 = 1$ in the first quadrant where the tangent line to the circle encloses with the coordinate axes a ...
1
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0answers
30 views

Derivative question with series

I am having this question here. Find the 66th derivative of $\displaystyle \cos x^3$ Yes, the cube is on the $x$. The idea is to do it with series. I got an answer which I want to verify if it is ...
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2answers
61 views

Deducing if the series converges. [duplicate]

$$\displaystyle \sum\limits_{}^{} \dfrac{1}{k(ln(k)^2)}$$ Integral test $$\int_{} \frac {1}{u^2} du = \int u^{-2} du = \frac {-1}{u} = \frac{-1}{\ln k} +c$$
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2answers
27 views

Deduce if the series converges absolutely or conditionally.

$$\sum_{k=0}^\infty (-1)^k \frac{(3^k)(k!)^2}{(2k)!}$$ I start by using the absolute convergence test. This eliminates the -1: |1|^k = 1 Then I use the ratio test. $$\left|\frac{3^{k+1} ...
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0answers
14 views

How to prove whether the series problem converges or diverges?

$Σ$ $(-1)^k$ $ \frac{k^2+3k}{k^3+k+2)}$ I use the absolute value theorem for this problem. Then I use the a limit comparison test on it. $\frac{k^2+3k}{k^3+k+2}*1/k =\frac {k(k^2+3k)}{(k^3+k+2)}→ ...
1
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2answers
47 views

Finding the exact amount of a sigma problem?

$$\begin{align*} \sum_{k=1}^\infty \frac{1}{k(k+1)} \end{align*}$$ This is a telescoping series; therefore I use partial fractions to solve. $\int_{1}^{∞} (1)/(k+1) $ = $ ((A/k)+(B)/(k+1))$ A= 1 ...
1
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4answers
55 views

without using l'hopital rule

Can someone give me please some guidance hoe to solve the following limit, without using L'Hopital rule? $$\lim\limits_{n \to \infty } \frac{n}{\ln\left(\frac{3n}{5}\right)}$$ Thanks a lot!
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2answers
21 views

Cauchy sequence in practice

Let $a_n$ be a sequence. Based on Cauchy can I say that if $|a_{n+2}-a_{n+1}| < |a_{n+1}-a_n|$ then $a_n$ converges? The reason behind this is that $a_{n+1}$ is just a small offset of $a_n$ and ...
0
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5answers
44 views

Consider the following limit: $\lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)}$

Can someone give me some guidance on where to begin with the following limit? $$ \lim_{n \to \infty } \frac{\ln(1+n)-\ln(n^{2})}{\sin(1/n)} $$ Thanks!
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5answers
44 views

Find the center and radius of polar circle equation [on hold]

Find the radius and center of the circle $$r=2\cos \theta+3\sin \theta$$
-1
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2answers
37 views

Using Taylor series to find $\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$ [on hold]

How to find the limit $$\lim_{x\to 1}\frac{2-(x+3)^{1/2}}{x-1}$$ using the Taylor series? I have done the derivatives on function following the regular steps. But all I get is zero.
1
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4answers
54 views

Trig differentiation

Prove that there is a constant C such that $$ \arcsin{\frac{1-x}{1+x}} + 2\arctan (\sqrt{x}) = C $$ for all $x$ in a certain domain. What is the largest domain on which this identity is true? What ...
0
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0answers
44 views

Integration of a polynomial

I am facing a problem in finding the integral $$\int\frac{r^2}{-C r^3 + r^2 -2 M r +Q^2}\,dr$$ Here M, Q, and C are parameteres (to be fixed later). Could anybody Please help me in finding it? I ...
2
votes
2answers
77 views

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculation of $\displaystyle \int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$ given that $ a>b>0$ $\bf{My\; Try::}$ Let $\displaystyle I = \int_{0}^{\pi}\frac{\sin^2 ...
0
votes
3answers
57 views

How to solve a differential equation?

I'm trying to solve the system $$\frac{d^4x}{dt}+4x=0\quad ,\quad\frac{d^3x}{dt}+x=0$$. However, I don't know of any method of tackling such a problem. Can anyone please provide a route to a solution? ...
0
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1answer
17 views

Sequences with intervals

I'm trying to play a bit with sequences & intervals and I've got a few questions which I'm not sure about: Let $a_n$ be a sequence and $I=(a,b)$ interval such that {$a_n|n\in N$} densed in $I$ ...
0
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1answer
16 views

About restricting variables in an integrand, and also changing the look of an integrands.

So, in the last step of, many, integrands, Wolfram chooses to restrict the $x$-values, even if I didn't specify which values $x$ can take on. Take for example: $$\int\frac{dx}{x(x^2-1)^{3/2}} = ...
0
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2answers
29 views

infimum and supremum of subsets question

Let $B \subseteq \mathbb{R}_{+}$ such that B is non-empty. consider $B^{-1} = \left \{b^{-1} : b\in B \right \}$. Show that if $B^{-1}$ is unbounded from above, then $\inf\left(B\right)=0$ How can i ...
1
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2answers
36 views

Calculating semi axes from given tilted ellipse equation

Hopefully no duplicate of Ellipse $3x^2-x+6xy-3y+5y^2=0$: what are the semi-major and semi-minor axes, displacement of centre, and angle of incline? (see below) Let the following equation $$x^2 - ...
4
votes
1answer
83 views

How to find the maximum and minimum of $\dfrac{\sin x}{x^2+1}$?

How can we find the values of $x$ that give the maximum and minimum of $$\frac{\sin{x}}{x^2+1}$$ I took a lucky guess and found that $\dfrac\pi4$ was fairly close to giving the max, but how does one ...
5
votes
2answers
78 views

Evaluation of $\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$

During my calculation I ended with the following product: $$P=\prod_{n=1}^\infty e\left(\frac{n}{n+1}\right)^{n}\sqrt{\frac{n}{n+1}}$$ I tried to express in term of series by taking the logarithm ...
0
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1answer
22 views

how to find a closed form expression for a power series

my question is how do i find a closed form expression for a function f(x) which the power series $\sum_{n=0}^\infty 2n(-7)^n x^{n+2}$ converges to and the value of x for which f(x) equals the given ...
3
votes
3answers
55 views

Is $\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$ convergent?

Does the following integral $$\int_1^\infty \frac{\log(x-1)}{x(x-1)}\,dx$$ converge? If it is convergent can we compute it?
0
votes
1answer
25 views

Counting function for the number of zeros of a continuous positive function?

Let $f(x)$ within $x\in[a,b]$ an absolute continuous function with $f(x)\geq0$ $f(x_m)=0$ for all absolute minima $x_m$ no other zeros than at $x_m$ I am trying to define a counting function for ...
3
votes
3answers
58 views

Problems with this integral $ \int \sqrt{1 + {1 \over t^2} + {2 \over t}} dt$

$$ \int \sqrt{1 + {1 \over t^2} + {2 \over t}}\,\mathrm dt$$ I tried making substitution, using $ u=1 + \dfrac{1}{ t^2} + \dfrac{2 }{ t} $, then , $dt=\dfrac{du}{-2\left({1 \over t^3 }+ {1 \over ...
0
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2answers
93 views

Evaluating $\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} dy$

I was trying to find arc-length of $x = \sqrt{y}-7y$ So basically right now I am stuck with this $$\int^{4}_{1} \sqrt{1+\left(\frac{1}{2\sqrt{y}}-7\right)^2} \,\mathrm dy$$ $$\int^{4}_{1} ...
5
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2answers
53 views

Integration $\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R$.

Give a compact form for the solution of integral: $$\frac{1}{2\pi}\int_{-\pi}^{\pi}(x-a)^ke^{-i\omega x}dx, \ \ \ \ a\in\mathbb R,k\in\mathbb N$$ any suggestions please?
0
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2answers
31 views

Write sums in factorial form. [on hold]

Show that in such a way that we can generalize this proof $$(1+2+3+4+5)+(1+2+3+4)+(1+2+3)+(1+2)+1= \binom{7}{3}$$
2
votes
1answer
24 views

Partial Derivative of a nonexistant variable?

I am wondering how I would find the partial derivative of $z = g(r, \theta) = \theta$ with respect to both $r$ and $\theta$. I realize that if you take the partial in respect to $\theta$, it is 1. I'm ...
3
votes
3answers
342 views

Find a value for “c”

For what value of $c$ is $$\lim_{x\to\infty} \left(\frac{x+c}{x-c}\right)^x = e?$$ I am unsure of how to start this question in any sense.
1
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0answers
38 views

Antiderivative of $|x − 2| + |x − 3|$ [on hold]

Find the most general antiderivatives of the following function. $$|x − 2| + |x − 3|$$ I started with showing that the antiderivative for $|u|$ is $\dfrac{u|u|}2$. How to proceed then?
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0answers
18 views

About Calculus Cost, Revenue, and Profit Functions

1.) An international firm produces a new line of product. Their production cost is PhP 250 for each small unit and PhP 350 for each large one. The firm’s production budget for the product is PhP ...
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2answers
17 views

Find a Cartesian Equation for the Plane Satisfying Those Properties

Find the Cartesian equation of the plan parallel to j and passes through the intersection of the planes described by the equations x + 2y + 3z = 4, and 2x + y + z = 2. I was able to get the ...
0
votes
1answer
106 views

Optimization question on graph

I was having trouble with this question. A triangle has one side parallel to the x-axis, two vertices on the part of the parabola $$y =3 − {x^2\over 12}$$ above the x-axis and the third vertex at ...
2
votes
3answers
61 views

Suppose that $f ' (x)$ exists and $f(x)$ has two roots $x_1$ and $x_2$. Try to prove that:

Suppose that $f'(x)$ exists and $f(x)$ has two roots $x_1$ and $x_2$. Try to prove that: there is $\xi \in (x_1,x_2)$ such that $f(\xi)+f'(\xi)=0$. We cannot use the knowedge of integration.
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0answers
14 views

Normal derivative property

I have seen in many papers that to obtain some results about PDEs is used the following argument: If $\phi=0$ in $\partial\Omega$ then $\bigtriangledown\phi=\dfrac{\partial\phi}{\partial n}n$, where ...
1
vote
1answer
229 views

Related Rates/ Optimization problem

I was having trouble figuring out this problem. A fisherman is in a boat 3 km from the nearest point A on the coast. The fisherman wishes to return to his camp C located 5 km from the point A. The ...
2
votes
1answer
25 views

Is it true that a quasiconvex, increasing and continous function, is convex?

Let $f:\mathbb R^n \to \mathbb R$ be a continuous and increasing function. Let $f$ be quasiconvex. Let $f(0)=0$. Can we say that $f(x)$ is convex ? If yes, how do we prove it ? Thank you very much ...