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

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0
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0answers
7 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.
0
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0answers
5 views

Question on the application of extreme value theorem

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
45 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
votes
2answers
17 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
48 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 ...
2
votes
4answers
93 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 ...
5
votes
5answers
115 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
34 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
vote
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 ...
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4answers
54 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
vote
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
vote
1answer
23 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 ...
2
votes
1answer
60 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
13 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
19 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
votes
1answer
74 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
votes
2answers
23 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
31 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
51 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
18 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
votes
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
22 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 ...
0
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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
67 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
30 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 ...
1
vote
1answer
14 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 $f$ ...
-5
votes
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
47 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
78 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
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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 ...
2
votes
4answers
36 views

How can I find if $\sum_{n=2}^\infty {(-1)^n*4 \over (ln(n))^2} $ converges or diverges using the alternating series test?

$$4\sum_{n=2}^\infty {(-1)^n \over (\ln(n))^2} $$ If $4 \over (\ln(n))^2$ = $u_n$, then $u_n$> 0, and: $$\lim_{n\to\infty} u_n = 0 $$ But there is one more test to prove convergence which says ...
2
votes
4answers
93 views

the derivative of $\sin{x}$ is $\cos{x}$

I can prove the $\sin'{x}=\cos{x}$ by formula $\lim_{h\to 0}{\frac{f(x+h)-f(x)}{h}}$ but the proof is not known for me by the formula $\lim_{z\to x}{\frac{\sin{z}-\sin{x}}{z-x}}$? Can anyone give ...
2
votes
2answers
27 views

Finding an ODE given some of its solutions

Find $a, b, f(x)$ such that $$y''+ay'+by = f(x)$$ Is satisfied by $g_{1}=\sin x + e^x$ and $g_{2}=\sin x - e^{-x}$ What I tried to do: First, I used the fact that if $g_{1}$ and $g_{2}$ are ...
1
vote
2answers
68 views

Coefficient calculation on Fourier series !? [on hold]

in a Fourier series for function $$f(x)=\begin{cases}-1&\text{for }-\pi<x<0\\\sin x&\text{for }0<x<\pi\end{cases}$$ with $f(x)=f(x+ 2 \pi)$, is $f(x)= \dfrac{a_0}{2}+ ...
4
votes
5answers
75 views

Something wrong at $\int \frac{x^2}{x^2+2x+1}dx$

I have to calculate $$\int \frac{x^2}{x^2+2x+1}dx$$ and I obtain: $$\int \frac{x^2}{x^2+2x+1}dx=\frac{-x^2}{x+1}+2\left(x-\log\left(x+1\right)\right)$$ but I verify on wolfram and this is equal with: ...
0
votes
1answer
70 views

Can dx be equal to dy?

I am taking baby steps into calculus. While reading "Calculus made easy" by S.Thompson, I came across this confusing statement: "Suppose we make $x$ to vary, that is to say, we either alter it or ...
0
votes
4answers
39 views

Solving the following question without using lagrange multipliers

This is an 11th and 12 grade problem which doesn't require multivariable calculus but i cant see any way of doing it without using Lagrange multiplier. Find the values of x; y for which $x^2 + y^2$ ...
1
vote
1answer
15 views

Space curve torsion

Hello I am looking for anyone to maybe look over my ideas and see if they think it is correct. Say I am looking for the torsion $\tau$ of a space curve given by $r(t)=(cos(3t),sin(3t),4t)$ I know if ...
0
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0answers
24 views

Confusion about Flow Integral

I am asked to calculate the flow integral $$\int{\vec{F}\cdot\hat{T}ds}$$ of $\vec{F}=<2x,-3y>$ along the fourth quadrant path from (5,-3) to (8,0) along the curve $x^2-10x+y^2+16=0$. So I ...
1
vote
1answer
20 views

derivative of some function

I have the following function : $$f_{x}\left(x,s\left(x,z\left(x\right)\right)\right)$$ When I try to differentiate it according to $x$, I find something but I am not sure : ...
0
votes
1answer
53 views

How can I solve like this exercise

Let we have the following initial value problem : $$y'=f(x,y)=e^y$$ With the condition $y(0)=0$ Find the largest interval $|x| \le a $ makes the initial value problem has an unique solution ...
0
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0answers
22 views

Area between curves (Calc) [on hold]

How do you know whether to solve with respect to y or respect to x? I know how to do the rest of the problem, I never know which one to do though.
0
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1answer
32 views

$f:U \rightarrow \mathbb{R}$, $U$ is an open conected subset of $\mathbb{R}^n$ and $f \in C^1$ need to show that $f$ is $M$ Lipschitz on any compact

It is a more general form of the question here, only here $U$ is not a convex set but an open and connected subset of $\mathbb{R}^n$. I need to show that $f$ is $M$ Lipschitz on any compact $K \subset ...
0
votes
0answers
26 views

Getting solutions for an ODE in different intervals

I have to obtain solutions for: $$ x(x-1)y' + y = x^2 -1$$ In the intervals $(- \infty, 0), (0,1), (1,+ \infty)$. I got the solution $$f(x) = \frac {x}{x-1}(C+x+\frac{1}{x})$$, where $f$ is defined ...
0
votes
0answers
21 views

True/false statements about convergence of a sequence

Let $(a_n)$ be a sequence. True or false: If $\lim_{n\to\infty}(a_{2n}-a_n)=0$ then $(a_n)$ converges. If $(a_n)$ converges then $\lim_{n\to\infty}(a_{2n}-a_n)=0$ Intuitively, I ...