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

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0answers
11 views

Smallest Volume of an ellipsoid

Find values a, b, c so that the ellipsoid $\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2}=1$ passes through the point (1,1,1) and encloses the smallest volume.
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1answer
22 views

How can i visually or geometrically say that integration is just the inverse of differentiaiton?

I have just finished a course on calculus. So I started pondering on the fundamental theorem of calculus and the relation between integration and differentiation . We all know that integration is just ...
0
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0answers
10 views

Properties of a function needed for not having a zero

While studying ODE I thought of the following problem: Let $f:A\subset\mathbb{R}\to\mathbb{R}$ and $x_0\in A$ such that $f(x_0)=0$. What properties should have $f$ so as to allow us to conclude that ...
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1answer
24 views

How to show this integral (Error function)

I'm given this question. Show that $$\int_{0}^{0.25}\frac{1}{\sqrt{x}}e^{-x}dx=\int_{0}^{0.5}2e^{-u^2}du$$. As I know this integral is an error function. How to show? Can anyone give me some hints? ...
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2answers
23 views

Line integral of vector field

Compute the line integral $\int_\gamma g \cdot dx $ for an arbitrary piecewise smooth curve $\gamma$ traversing in the upper half plane from $(-a,0)$ to $(b,0)$ where $a > 0$ and $b>0$. ...
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1answer
21 views

Volume of solid by Cartesian, Cylindrical, & Spherical

I am having trouble just setting up the integrals for this problem. Find the volume of the solid bounded by $x^2 + y^2 = 1, z = 0$, $z = 6$, $y\geq 1/2$. a) Use integration with Cartesian ...
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1answer
33 views

Do there exists continuous functions on compact sets with infinite length?

Is it possible to construct a continuous function from $[0,1] \to \mathbb{R}$ whose length is infinite?
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2answers
45 views

how to convert log(x) into linear form? [on hold]

I have simple function which is non-linear like log(x) I want to convert it into linear function. Anyone could help out? Thanks
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3answers
43 views

Convergence test of an integral [on hold]

$$\int_1^\infty \left( 1- \frac{2x+1}{2(x^2+x)^{1/2}} \right) \ dx$$ Anyone can suggest hint for how to determine whether it is convergent or not? Thank you
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3answers
57 views

Compute the improper integral $\int_{2}^{+\infty}{\frac{ \sqrt{(3x^7+5x^5)}}{x^3}}dx$ [on hold]

$$\int_{2}^{+\infty}{\frac{ \sqrt{(3x^7+5x^5)}}{x^3}}dx$$ Anyone can give hint on how to solve this?
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2answers
70 views

Evaluate $\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$

My question is: Evaluate $$\int_{-\pi}^\pi \! \cos(kx)\cos^n(x) \, \mathrm{d}x$$ for $k=0,1,...,(n-1)$ and $n \in \mathbb{N}$. I've tried integration by parts but without much success. Any ...
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1answer
41 views

If a monotonic function and its inverse equals to itself, then it must be function $x$?

If $f$ is a monotonic function on [0,1] with property $f = f^{-1}$, then $f(x) = x$ for all $x$ in this interval. I don't how to determine if this statement is true or not. Any hints and how do I ...
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3answers
54 views

Finding lower/upper bounds for $\prod_{i=2}^n \log(i)$

I have a homework problem where I need to asymptotically order a set of functions, and $\prod_{i=2}^n \log(i)$ is one of them. Is there a tight upper/lower bound for this function? I've tried the ...
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0answers
55 views

The coefficients of $\frac{1}{\cos(x)}$ are even

Let's consider $G(z)=\dfrac{1}{\cos(z)}$ as an exponential generating function of the Euler numbers' sequence. How to prove that all $a_{i}$ in the expansion of$\dfrac{1}{\cos(z)}=\sum_{k=0}^{ ...
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2answers
103 views

Two convergent subsequences and their limits

$\{a_n\}$ is a sequence. I'm asked to verify the following statement: "If $\{a_{2n}\}$ and $\{a_{3n}\}$ converge then $$\lim_{n\to\infty}a_{2n}=\lim_{n\to\infty}a_{3n}$$ I think this is not true, but ...
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0answers
7 views

bound on Lagrange multipliers

Under what conditions is it possible to bound the Lagrange multipliers in a given optimiztion with constrains problem?
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4answers
79 views

How to evaluate $\lim _{n\to \infty }\:\int _{1/(n+1)}^{1/n}\:\frac{\sin\left(x\right)}{x^3}\:dx$?

We have to evaluate the following limit: $$\lim _{n\to \infty }\:\int _{\frac{1}{n+1}}^{\frac{1}{n}}\:\frac{\sin\left(x\right)}{x^3}\:dx,\:n\in \mathbb{N}$$ First step I wrote that $\int ...
2
votes
0answers
15 views

Small proximity of important points of a function

Let $a,b,c$ be coprime integers with $a^2 + b^2 \gt c^2$ and consider the function $f(x) = a^x + b^x - c^x$. It is easy to verify that there exist $r$ and $s$ such that $f(r) \ge f(x)$ for all $x$ and ...
0
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0answers
14 views

strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
1
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1answer
20 views

Convergent sums and rate of decay

True or False: If $a_n\in\ell^1,$ then $\overline{\lim}n a_n<\infty$ (i.e. $a_n=O(\frac{1}{n})$) Edit: My intuition says the answer should be positive.
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0answers
42 views

Find all differentiable functions $f$ such that $f(f(x))=f'(x)$ [duplicate]

Here is a problem I made up: Find all differentiable functions $f$ from the reals to the reals such that $f(f(x))=f'(x)$ for all real $x$.
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3answers
54 views

Problem in indefinite integral. (Exponential)

I'm given this integral to integrate. I've no idea where to start with. Perhaps someone can give me some hints or guide me. Thanks a lot. $$\int\frac{(x^3)e^{x^2}{}}{x^2+1}dx$$
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2answers
39 views

What's the relation between a fixed point and a root of a function?

A fixed point of a function $f$ should be an $x$ in the domain of $f$, such that $f(x) = x$. On the other hand, a root (or zero) of a function, should be an $x$ in the domain of $f$, where $f(x) = ...
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3answers
59 views

differentiation an integration

Let $x,y$ and $r,\theta$ in Cartesian and polar cordinate. So $$x=r\cos\theta , y=r\sin \theta$$ Therefor $dx=\cos \theta dr-r\sin\theta d\theta $ and $dy=\sin \theta dr+r\cos \theta d\theta$. ...
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0answers
19 views

How would i find the volume of a cone in the $interval [0,a]\times[0,a]\times[0,a]$ and how it's surface area? (using integration?)

whichEssentially i want to find the measure of $z^2\leq x^2+y^2$ and $z^2=x^2+y^2$. Now i know for one of them i would incorporate cilindrical coordinates: $$g(r,\phi,z)=(rcos \phi, r sin\phi,z)$$ ...
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1answer
18 views

Having trouble with discretization and boundry value problems

I have the following homework question: Consider the boundary value problem $y''(x) + 5y'(x) − (2 + x)y(x) = e^x$ on $x ∈ (0, 2)$ with boundary conditions $3y(0) + y'(0) = 5$ and $y'(2) = 7$. ...
4
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3answers
36 views

Proving statement about convergent sequence $(a_n)$ and the sequence $(\max\{a_n,a_n^2\})$

Suppose $(a_n)$ is a sequence and $\lim_{n\to\infty} a_n = a$ and let $(b_n)=(\max\{a_n,a_n^2\})$. I have to prove/disprove that: If $a>1$ then $\lim_{n\to\infty} b_n = a^2$ If $a=1$ then ...
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2answers
30 views

At what argument $x$ is the tangent to the graph $y=\frac{1}{2}x^2-\ln x$ horizontal?

At what argument $x$ is the tangent to the graph $y=\frac{1}{2}x^2-\ln x$ horizontal? Well this is a question which I found in a website. I found the Derivative to be $(x^2+1)/x$. As far as I ...
0
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2answers
18 views

How to find volume of the given solid analytically?

Here is the question - I am able to visualize the solid, but how do I find its volume? I'm unable to figure out the 2D structure that when rotated, produces this solid. Please help. Edit: The ...
2
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3answers
44 views

Question about maximizers and trig

Hi there I have a quick question about the following Consider the simple maximization problem of $$f(x,y)= \frac{x}{1+x^2+y^2}$$ It can be easily seen from analysis of critical points obtained from ...
4
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1answer
62 views

Help in finding the sum of the series

$$\sum_{n=1}^\infty \frac{1}{n^4+n^2+1}$$ I tried breaking into factors but it is not telescoping. $$\frac {1}{(n^2+n+1)(n^2-n+1)} = \frac {1}{2n} \left(\frac {1}{n^2-n+1} - \frac ...
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2answers
20 views

compute the smallest affine subspace containing $S$, where $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ is a set of vectors in $\mathbb R^3$

I've started to study convexity to enchance my optimization skills. Given a set $S=\{(1,1,1),(2,3,4),(1,2,3),(2,1,0)\}$ of vectors in $\mathbb R^3$ an exercise asks to compute the smallest affine ...
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3answers
57 views

Q: Why is this the limit?

Why is the limit ... $$\lim_{n\to \infty} {-7^n + 8^{n-2}\over 7^{n+1} + 8^{n+2}} = \frac {1}{4096}$$ I don't get it. Since the denominator has an $8^{n+2}$, isn't the limit supposed to be 0? When you ...
3
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4answers
381 views

Why is the Riemann sum less than the value of the integral?

Why is $ \frac{1}{n}\sum_{k=1}^{n}\frac{1}{1+\frac{k}{n}}\leq\int_{0}^{1}\frac{dx}{1+x}=\log 2 $? Because I think: $$\int _0^1\frac{dx}{1+x}=\frac{1}{n}\sum _{k=1}^n\frac{1}{1+\frac{k}{n}}$$ Why is ...
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5answers
401 views

Why the radius of convergence and not “areas of convergence” for power series?

My calculus is quite rusty and I'm trying to rebuild it on an intuitive basis. Currently, I am looking at power series and have trouble understanding the radius of convergence. I am comfortable with ...
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1answer
36 views

Prove that the following integrals are equivalent.

In my linear algebra course, we are looking into inner product spaces. The following came up with regards to an inner product on a subspace of the infinitely-differentiable real functions. Let ...
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1answer
27 views

$\nabla \varphi . \text{d}\mathbf{x} = \text{d}\varphi$ or $\nabla \varphi . \text{d}\mathbf{x} = 3\text{ d}\varphi$?

This might be a daft question, but I am confused by the vector identity $\nabla \varphi . \text{d}\mathbf{x} = \text{d}\varphi$, where $\varphi(\mathbf{x})$ is a scalar function, that is used in my ...
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4answers
58 views

How to find the maximum and minimum of the function $f(x) = \frac{3x}{x^2 -2x + 4}$

How would one find the maximum and minimum of such a function: $$f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto f(x) = \frac{3x}{x^2 -2x + 4}$$ I have just been introduced to functions in my ...
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1answer
13 views

Extermal curve for specific problems?

I ran into a quiz question last month. how we can find the Extermal curve for following problem. $$ \int_1^2 \frac {\dot {x}^2}{t^3} dt $$ where $x(1)=2, \ x(2)=17$
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1answer
27 views

Convergence of infinite series of function with factorial and power

Determine whether the series is convergent or divergent: $$\sum_{n=0}^\infty \frac{(3n)!+4^{n+1}}{(3n+2)!}$$ I guess we have to use comparison test for this question, but I am not sure what to use ...
1
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1answer
62 views

Proof of $\left| x\right| <1$, then $\lim_{n\to \infty } \, x^n=0$.

Struggling with the proof: If $\left| x\right| <1$, then $\lim_{n\to \infty } \, x^n=0$. The proof is given like this: Now this is how I see it, but Im not sure where I am going wrong so I ...
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0answers
14 views

Approach on solving limit equation systems and finding some f given assymptotes?

This is a "reverse" question of finding the asymptote of a function Recently, I am interested in doing some sort of modelling which involve equations of the form $$@(t)=1-f(t)$$ where $f(t)$ is ...
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0answers
20 views

Another integral equation question

Suppose that $z = \int_{- \infty}^z f (y) d y$. If $f$ were continuous, we can differentiate both sides to get $f(y)=1$. But what if $f$ does not have to be continuous, is this still true or are there ...
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1answer
77 views

If $\frac{x-1}{e^x-1} = y$ then $x=?$

I have following equation: $$\frac{x-1}{e^x-1} = y$$ I want to solve this equation such that I have the value of $x$ in the term of $y.$ i.e. inverse of the equation
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2answers
24 views

If a continuous function is strictly decreasing before a point and strictly increasing afterwards, is the point a global minimum?

I'm in the middle of a proof that a point on a function is a global minimum. Usually I'd just solve an inequality to prove by contradiction that there are no points less than the minimum. But I can't ...
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2answers
32 views

Delta epsilon proof statement logic [on hold]

In the delta epsilon proof, it says the following: For every $\delta > 0$ there is an $\epsilon > 0$ such that (some statement) What is the difference between the above statement and if we ...
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0answers
54 views

Application of the Fundamental Theorem of Calculus

I was wondering if someone could help me with a problem I'm having. I'm reading a paper 'Spatiotemporal dynamics of continuum neural fields' and on page 13 they authors derive a model for spatially ...
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3answers
19 views

How to prove expected value of uniform random variable?

I tried this: $$\int_a^b t~dt = \frac{t^2}{2}\Big]_a^b = \frac{b^2-a^2}{2} = \frac{(b+a)(b-a)}{2}$$ Isn't it supposed to be $\frac{b+a}{2}$ or something like that? Obviously if I multiply the ...
1
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1answer
19 views

Integration about x and y axes to find area

I have a problem statement that requires me to find area between the curves about x axis and about y axis. But my answers are not matching. Please find below my worked out solution - The ...
2
votes
2answers
76 views

Using exclusively the definition of limit proof that $\lim_{x \to 0} \frac{x^3-2x+x}{\sin(x)} = -1$

Using exclusively the definition of limit proof that $$ \lim_{x \to 0} \frac{x^3-2x+x}{\sin(x)} = -1 $$ I have to learn how to prove limits by the delta-epsilon definition, I know how to do basic ...