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

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1
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1answer
25 views

Partial derivatives confusion in the equation $PV=nrT$

The question is to exactly: "If the variables $P,V$ and $T$ are related by the equation $PV=nRT$ where $n$ and $R$ are constant, simplify the expression" $$ \frac{\partial V}{\partial T}\frac{\partial ...
3
votes
1answer
22 views

What is the geometrical meaning of the integral of a vector valued function?

If $f:\mathbb{R}\rightarrow\mathbb{R}$ is an integrable function. then $\int_a^b f(x)dx$ can be considered as the area between the graph and the x-axis. But what if $f:\mathbb{R}^n\rightarrow \...
1
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4answers
39 views

Prove using induction the following equation is true.

If $$(1-x^2)\frac{dy}{dx} - xy - 1 = 0$$ Using induction prove the following for any positive integer n$$(1-x^2)\frac{d^{n+2}y}{dx^{n+2}} - (2n+3)x\frac{d^{n+1}y}{dx^{n+1}} - (n+1)^2\frac{d^ny}{dx^n} ...
1
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4answers
80 views

$\int f(x)\,dx - \int f(x)\,dx$

which is true $$\int f(x)\,dx - \int f(x)\,dx = 0$$ or $$\int f(x)\,dx - \int f(x)\,dx=c\text{ ?}$$ with $c$ some arbitary constant. My intuition says that 'something' subtracted by itself is ...
-5
votes
3answers
17 views

What relationship between t and k would make the lines with Cartesian equations sx-ky+7=0 and tx+2y-3=0 perpendicular?

What relationship between t and k would make the lines with Cartesian equations sx-ky+7=0 and tx+2y-3=0 perpendicular?
4
votes
1answer
94 views

What is the derivative of $x^i$?

What would the derivative be of $x^i$? Would it simply be $ix^{(1-i)} $? I tried running the Power rule, and I got that is that right?
1
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0answers
23 views

Finding a Relation Between Two Sequences

Consider the following recurrence relation for $C_i(r)$s $$\begin{align} &C_0(r)=r-r_2 \\ &q(r+n)C_n(r)+\sum_{k=0}^{n-1}[(r+k)\alpha_{n-k}+\beta_{n-k}]C_k(r)=0, \qquad n\ge1 \end{align} \tag{...
-6
votes
1answer
20 views

The distance from a point to a plane

Im not sure how to do these questions, Ive tried it but can someone please do the step by step process so i can verify, thank you in advance (Idk if u guys can see the image or not)
0
votes
1answer
44 views

Solution to the convoluted integral equation

A have the following equation: $$f(a)=\int_0^ag(x)f(x)\,dx,$$ where $g(x)$ is a known function. Is there any solution to $f(a)$ just in terms of $g$?
-2
votes
2answers
15 views

What is the the Cartesian equation of the plane having x-, y-, and z-intercepts of 2, 5, and 3 respectively?

What is the the Cartesian equation of the plane having x-, y-, and z-intercepts of 2, 5, and 3 respectively? Im not sure how to do this
-2
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0answers
21 views

Prove the following integral identity

Does any one have any idea on how to prove the following: $$\int_0^Sf(z)\,\mathrm{d}z+a\int_0^Sf(z)\left(\int_0^zf(z_1)\,\mathrm{d}z_1\right)\,\mathrm{d}z+a^2\int_0^Sf(z)\left[\int_0^zf(z_1)\left(\...
0
votes
0answers
7 views

Deriving a minimal amount of different size squares to fill a space from a function

Suppose we have many of squares of the following sizes: $1\times 1, 2\times 9, 3\times 5$ and we want to fill a board of size $16\times 32$ with squares such that we use a minimal amount of squares ...
0
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1answer
17 views

Determine a set of parametric equations for the xz plane.

How do you determine a set of parametric equations for the $xz$ plane?
0
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4answers
40 views

False proposition then true

Please, take a look to this inequation (m is a natural number): $$2\sqrt {m+1} - 2\sqrt m \lt\ \frac{1}{\sqrt{m+1}} $$ For m = 3, the expression is false: 0.53 $\lt$ 0.5 So, the expression doesn't ...
13
votes
4answers
502 views

Integral $\int_0^\infty\frac{\tanh^2(x)}{x^2}dx$

It appears that $$\int_0^\infty\frac{\tanh^2(x)}{x^2}dx\stackrel{\color{gray}?}=\frac{14\,\zeta(3)}{\pi^2}.\tag1$$ (so far I have about $1000$ decimal digits to confirm that). After changing variable $...
0
votes
3answers
2k views

Volume of a horizontal cylinder using height of liquid

“Tanks” are cylinders with circular cross-section and axis horizontal. These cylinders are variable in size with radius and length different for each tank. We need to determine the amount of liquid ...
2
votes
3answers
23 views

Center of mass of 1 dimensional rod

In my calculus 2 course, we have been studying how to calculate the center of mass of a 1 dimensional "rod" with dimensions $[a,b]$ (so that the left end of the rod is at $x=a$ and the right end of ...
-1
votes
0answers
17 views

Growing ink-blot

A Circular ink blot grows at the rate of 2 cm^2/s. Find the rate at which the radius is increasing after 28/11 seconds. I am getting 0.177 cm/s as the answer. But original answer is given as 0.25 cm/...
1
vote
1answer
246 views

Finding angle of sector which forms a cone

To find the angle where it is in rad, am I right to say that $10*(angle\ in\ rad)=2\pi*(radius\ of\ cone)$
0
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1answer
36 views

Is taking the partial derivative of $x^n$ the same thing as the derivative of it?

Is taking the partial derivative of $x^n$, the same thing as taking the derivative of $x^n$? The derivative of $x^n$, is $nx^{n-1}$, so would the partial derivative be the same thing?
0
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5answers
66 views

Using power series representation find an approximation to $\int_{0}^{0.1} \arctan(2x)dx$

Using power series representation find an approximation to $$\int_{0}^{0.1} \arctan(2x)dx$$ This is my solution: $$\frac{d}{dx}\arctan(2x)=\ \frac{2}{1+4x^2} $$ $$\int_{0}^{0.1}\frac{d}{dx}\arctan(...
0
votes
1answer
23 views

Choice of the limits for multivariable integral

Let $A \subseteq \mathbb{R}^2$ a limited set bordered through $x=0, x=1, y=-1+x, y=1-x^2$. Rotate A around the y-axis and define this set with $B$. Calculate the integral $$\int_B y\,\mathrm{d}x\...
3
votes
1answer
46 views

How to describe a summation of $\frac{1}{2^x3^y}$ and evaluate.

I want too calculate the value of this sum: $$\sum \frac{1}{2^x3^y}$$ Where we sum up all permutations of terms involving a nonnegative integer $x$ and a nonnegative integer $y$. How can I ...
4
votes
3answers
32 views

Find all local extremums of $f(x)=x^{2}e^{-x}$ and decide if these are global extremums

As all my other questions, this one isn't homework (it's preparation for an exam). I'd like to know if I did everything correctly. In my previous task, I had a mistake in the first derivation. But ...
79
votes
19answers
4k views

Why does the series $\sum_{n=1}^\infty\frac1n$ not converge?

Can someone give a simple explanation as to why the harmonic series $$\sum_{n=1}^\infty\frac1n=\frac 1 1 + \frac 12 + \frac 13 + \cdots $$ doesn't converge, on the other hand it grows very ...
0
votes
0answers
22 views

Is it possible construct a well-defined series through integration by parts?

I have a function $$G=\int_0^S\mathrm{d}x\,f(x)g(x)\mathrm{e}^{-\int_0^x\mathrm{d}z\,g(z)}~.$$ If $f(x)$ was unity, the above integral could have been easily written as $$G=1-\mathrm{e}^{-\int_0^S\...
0
votes
1answer
31 views

Finding the velocity of a position vector

Let $\{\tilde{i}, \tilde{j}\}$ be the standard basis vectors for IR2. Define two paths in IR2 by $\tilde{v1}$(θ) = cosθ$\tilde{i}$ + sinθ$\tilde{j}$ $\tilde{v2}$(θ) = −sinθ$\tilde{i}$ + cosθ$\tilde{j}...
1
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1answer
21 views

Integration of the following trignometry

We have to find the integration of the following , I tried but got stuck , can anyone help me
1
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2answers
42 views

Equations with integrals

In a math-textbook I have, they have solved the equation $$ -c_{2}\int_{0}^{S}f(x)dx+c_{1}\int_{S}^{\infty}f(x)dx=0$$ as $$\int_{S}^{\infty}f(x)dx=\frac{c_{2}}{c_{1}+c_{2}}$$. Anyone who know how they ...
0
votes
1answer
29 views

How to distinguish between arc length and arc length parametrisation?

I am trying to understand and distinguish the difference between arc length and arc length parameterisation. The first thing, how do denote the $\text{arc length}$ and $\textit{arc length ...
1
vote
2answers
22 views

If an integrable function is orthogonal to all derivatives, then is f a constant?

Suppose that I have a function in $f \in L^1(\mathbb{R})$ such that $$\int_{\mathbb{R}}f(x)v'(x)\,dx = 0$$ for all test functions $v$ which are smooth with compact support. Can I show that $f(x)$ is ...
1
vote
1answer
32 views

Find all local extremums of $f(x)=\frac{x}{x^{2}+x+1}$ and decide if these are global extremums

Did I do everything correctly? Find all local extremums of the following function and decide if these are global extremums (i.e. maxima or rather minima of the function on its entire domain) ...
2
votes
3answers
47 views

Recursive System of Equations and one Solved Example in 2007 GATE Exam?

The solution of $\frac{a_{20}}{a_{20}+a_{20}}$ is $-39$ from the recursive system of equations: \begin{cases} a_{n+1}=-2a_n-4b_n \\ b_{n+1}=4a_n+6b_n\\ a_0=1,b_0=1 \end{cases} This is taken from $...
3
votes
2answers
33 views

Are polynomial fractions and their reductions really equal? [duplicate]

I'm reading Larson's AP Calculus textbook and in the section on limits (1.3) it suggests finding functions that "agree at all but one point" in order to evaluate limits analytically. For example, ...
0
votes
0answers
20 views

Obatin $\int_{\gamma_1}F\cdot dl =\int_{\gamma_2}F\cdot dl$

Let $F = (F_1,F_2)$ be a $C^1$ vector field such that all its components are continuously differentiable in $\Omega$. Assume that $\frac{\partial F_1}{\partial y}=\frac{\partial F_2}{\partial x}$ Let ...
3
votes
1answer
55 views

Numerical Method Sample Question via Truncation error Methods?

I have one multiple choice question: Approximation of integration $\int_0^{0.1} e^{x^2}dx $ by using simple formula of following options has lower Truncation error: Choice Part: $a)$ ...
0
votes
1answer
31 views

Diagram of a multivaribale function

I have to draw the diagram of the function: $$(x^2+y^2)^{\frac{3}{2}}=x^2-y^2$$ I transformed it with polar coordinates to: $$r=\cos^2(\varphi)-\sin^2(\varphi)$$ with $r \ge 0$ and $\cos^2 \ge sin^2$....
0
votes
1answer
28 views

Recursive formula in term of original value

$$P_1=P_0G_{0,1}A_1\\ P_m=P_{m-1}G_{m-1,m}A_m+A_0\sum_{i=0}^{m-2}P_i G_{i,m}~~\text{for}~~m\geq 2$$ Is it possible to write $P_m$ in terms of only $P_0$, i.e., without other $P_j$ terms?
0
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2answers
32 views

How to find the limit of this rational sequence?

I'm stuck on a limit problem and I can't understand how can I prove this series diverges. I know it diverges but I don't know how to prove it. The problem is $$\lim_{n\rightarrow\infty}\frac{7n^3-3n^4-...
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votes
0answers
33 views

FourerIntegrals

Trying to understand a proof about Fourer Intergrals in my book. I can't understand how they got formula (3). As you can see delta W goes to $0$ then $L$ goes to infinty. Doesn't that mean that we get ...
6
votes
3answers
100 views

Convergence of a Harmonic Continued Fraction

Does this continued fraction converge? $$\large\frac { 1 }{ 1+\frac { 1 }{ 2+\frac { 1 }{ 3+\frac { 1 }{ 4+\dots } } } } $$ ($[0;1,2,3,4, \dots]$) I tried approximating a few values but I ...
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votes
0answers
32 views

What is less value side c =? [on hold]

Consider a right triangle hypotenuse legs a and b c, it is satisfied that: $ab + bc + c = 100$ What is less (smaller) value side c (without brute force)
0
votes
2answers
46 views

Evaluate $\int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }}$

How do you solve this integral $$ \phi = \int \frac{dr}{r^2} \frac{1}{\sqrt{-(\frac{1}{r} - \frac{1}{p})^2 + \frac{\epsilon^2}{p^2} }} $$ ? Note: It appears in the Kepler problem and it should ...
1
vote
3answers
79 views

Trying to study to convergence of the series $\sum_{n=1}^{\infty} \frac{ (n!)^2 4^n }{(2n)!}$

I am trying to find out whether the following series converges: $$ \sum_{n=1}^{\infty} \frac{ (n!)^2 4^n }{(2n)!} $$ I have tried to use the Ratio Test but it gives the inconclusive case. What other ...
1
vote
1answer
50 views

Defined integral with min function.

I have to resolve an integral that I didn't see before and I can't find any examples online. So my integral is: $$\int_{0}^{\pi/2} \min(1, \tan (x)) dx $$ I have no idea what $\min(1, \tan x)$ means. ...
4
votes
3answers
343 views

Why is $\displaystyle\int_{x=-\infty}^{x=\infty} f(x) \delta(x) \, \mathrm{d}x = f(0)$?

I understand that $\delta(x)=0$ whenever $x \ne 0$ and that $\displaystyle\int_{x=-a}^{x=b} \delta(x) \, \mathrm{d}x = 1 \space$ $\forall a,b \gt 0$ and also $\displaystyle\int_{x=-\infty}^{x=\infty} ...
8
votes
5answers
125 views

Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real

Prove that the roots of the equation: $$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$ are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
0
votes
3answers
87 views

Evaluate $\int \frac{\sqrt{64x^2-256}}{x}\,dx$

$$\int \frac{\sqrt{64x^2-256}}{x}\,dx$$ Image. I've tried this problem multiple times and cant seem to find where I made a mistake. If someone could please help explain where I went wrong I would ...
-6
votes
2answers
62 views

If $A$ and $B$ are positive constants, show that $\frac{A}{x-1} + \frac{B}{x-2}$ has a solution on $(1,2)$ [on hold]

If $A$ and $B$ are positive constants, show that $$\frac{A}{x-1} + \frac{B}{x-2}$$ has a solution on $(1,2)$. Please, support your answers with rigorous proof.
0
votes
0answers
37 views

Calculus Limit Comparison test [on hold]

Limit Comparison Test Problem The three series $\sum A_n$, $\sum B_n$, and $\sum C_n$ have terms $$A_n=\frac1{n^9}, B_n=\frac1{n^4}, C_n=\frac1n.$$ Use the Limit Comparison Test to compare the ...