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

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0
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0answers
5 views

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

I have a homework problem where I need to asymptotically order a set of functions, and $\prod_{i=2}^n log \left (i\right )$ is one of them. Is there a tight upper/lower bound for this function? I've ...
0
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0answers
34 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}^{ ...
1
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2answers
55 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 ...
-1
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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?
4
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4answers
64 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 ...
0
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0answers
10 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
41 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
48 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
35 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
57 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 $dy=\sin \theta dr+r\cos \theta d\theta$. ...
1
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0answers
18 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: $$g(r,\phi,z)=(rcos \phi, r sin\phi,z)$$ ...
0
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1answer
17 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
35 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 ...
0
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2answers
29 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
37 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
56 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 ...
0
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2answers
19 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 ...
0
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3answers
54 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
votes
4answers
337 views

Why Riemann sum is less than value of the integral?

Why $ \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 Riemann ...
6
votes
5answers
291 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 ...
0
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1answer
35 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 ...
1
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1answer
26 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 ...
1
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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 ...
1
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1answer
12 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 $x(1)=2, \ x(2)=17$
1
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1answer
24 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 ...
0
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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 ...
0
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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 ...
4
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1answer
76 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
0
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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 ...
-2
votes
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 ...
1
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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 ...
0
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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
74 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 ...
0
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0answers
35 views

Minimizing surface area for a given volume

Math question:An open-top box with a square base is to have a volume of 4 cubic ft. Find the dimensions of the box that can be be made with the smallest amount of material. This is the only thing I ...
0
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2answers
31 views

Different results for the same equation

Why does the chart of $xy+yz+xz=-1$, a one sheeted hyperbolid, is different from the chart of $z = -\frac{1}{x+y} - \frac{xy}{x+y}$? Aren't they both the same equation?
2
votes
1answer
23 views

How do I write the generic finite difference approx of f'(x) using Lagrange interpolating polynomial approximation?

I have the following homework problem: (10 points) Differentiation Formulas by Lagrange Interpolating Polynomials. (a) Write the generic finite difference approximation to f'(x) using the Lagrange ...
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0answers
13 views

Envelope Theorem and Static Optimization

The Statement of the Problem: For fixed $r \gt 0$ and $m$, find the maximum value of $1-rx^2-y^2$ on the constraint set $x+y=m$. Find the value function $f^*(r,m)$ and compute $\frac{\partial ...
1
vote
1answer
68 views

Dimension of garden to minimize cost

Math question: A homeowner wants to build, along her driveway, a garden surrounded by a fence. If the garden is to be $5000$ square ft, and the fence along the driveway cost $6$ dollars per foot while ...
4
votes
1answer
31 views

Difficult exercise on unicity of solutions for an IVP

Suppose $f$ and $g$ are continuous and $g$ is odd and strictly increasing function. I have to prove that the IVP $$y'=f(x)g(y)$$ $$y(0)=1$$ has a unique solution if and only if $$\lim \limits_{u \to ...
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1answer
15 views

Extending a convex function

Suppose $f:(a,b) \to \mathbb R$ is twice differentiable with the property that $c_1 \leq f''(x) \leq c_2$ for every $x \in (a,b)$, where $c_1, c_2$ are positive constants. Is it possible to extend $f$ ...
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0answers
42 views

Find the values of $c$ that satisfy the Mean Value Theorem [on hold]

Find the value or values of $c$ that satisfy the equation $f'(c) = \frac{f(b)-f(a)}{b-a}$ in the conclusion of the Mean Value Theorem for the function and interval. $$f(x)= \ln(x-1), \ I = [2,6]$$ ...
0
votes
0answers
50 views

How to differentiate $y$ with logarithmic differentiation

I am asked to find the differentiate $y$ using logarithmic differentiation $$y=\frac{ x(x^5+1)^{1/2}}{(x-1)^{1/3}}?$$ I tried it 3 times and I got three different answer each time Any help
0
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3answers
79 views

Integrating $f(x) = 1/x$ from $x=a$ to $x=\infty$

Can the integration of $f(x)=1/x$ from $x=a > 0 $ to $x=\infty$ ever be finite? That is, can $\int_{x=a}^{\infty} 1/x$ be finite?
0
votes
1answer
35 views

Differential Equations: Linear or Nonlinear

In my textbook, the authors said that a differential equation is linear if it can be expressed in the form $$a_0(t)y^{(n)}+a_1(t)y^{(n-1)}+\cdots+a_n(t)y=g(t)$$ According to the definition, why the ...
-3
votes
3answers
74 views

Calculus II Function Construction [on hold]

I need help please! Construct a function that is continuous and non-negative [0,1], with the property that the area under the function on [0,1] is finite yet the arc length on [0,1] is infinite.
0
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2answers
23 views

What theorem can I use to decide if an ODE which admits separation of variables has a unique solution?

Suppose that I have the IVP : $$y' = f(x)g(y)$$ $$y(a)=b$$ It's easy to show that any solution of such an equation will satisfy the implicit formula: $$ \int \frac{1}{g(y)}dy = \int f(x) dx$$ I ...