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

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11 views

Finding roots of a fractional exponential equation.

If we consider a polynomial equation its easy to find the number of roots associated with the expression by applying Descartes Rule. This method, however, doesn't work with non integer exponents. ...
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2answers
77 views

Why do we bother with $u$-substitution?

This question has bothered me ever since I learned $u$-substitution (A note here: I have no formal education at this level, so I may definitely have missed something). The method is presented as an ...
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4answers
56 views

Let $f$ be a continuous function on $[0,1]$.Then how to evaluate $\lim_{n\rightarrow\infty}\int_0^1 x^nf(x)dx$

Let $f$ be a continuous function on $[0,1]$.Then how to evaluate $\lim_{n\rightarrow\infty}\int_0^1 x^nf(x)dx$. Thanks for your kind help.
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3answers
33 views

How to solve integration of $\int x(x^2+k^2)^{-1/2} \, dx$?

As said in title, how do you solve integral $\int x(x^2+k^2)^{-1/2}\,dx$ where $k$ is some constant?
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2answers
40 views

Please review my proof.

I am working on a problem from Spivak 13.7 which states: Prove: $$m_i'' + m_i' \leq m_i$$ Where: $$m_i'' = \inf \{f(x): t_{i-1} \leq x \leq t_i\}$$ $$m_i' = \inf \{g(x): t_{i-1} \leq x \leq ...
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1answer
28 views

Computing ${\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right)$

Let $t \in \mathbb{R}$. Is the following elementary calculation correct? $$ {\mathrm{d} \over \mathrm{d}t}\left(e^{it}\right) = \underbrace{{\mathrm{d} \over \mathrm{d}t}\left(it\right) \cdot ...
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1answer
33 views

How do the steps of this definite integral work?

Sorry if this is a really basic question but I can't seem to get my head around the steps involved in this integration at all. My equation to be integrated is as follows: ${ds \over s}=\mu dt$ ...
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1answer
38 views

Suppose $f(x)\in L_1$ - Prove that $\lim_{n\rightarrow\infty}\int_0^\infty f(x)\cos(nx)dx = 0$

Assuming knowledge of the cyclic behavior of $cos(x)$, integration by parts, and $\int_0^{\infty} f<\infty$ is enough here? Consider \begin{align} & \int_0^\infty f(x)\cos(nx)dx = ...
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1answer
39 views

Computing $\int_{|z|=1} {e^z \over z}\ dz$

Goal: Let $\gamma$ be the unit circle. Then I aim to compute $$ \int_{|z|=1} {e^z \over z}\ dz = \int_{\gamma} {e^z \over z}\ dz $$ Attempt: Consider that $\gamma$ is a closed curve. Let $a = 0$. ...
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1answer
42 views

Are there integrals you can't solve without inverse hyperbolic substitution?

Are there any integrals that can't be solved with only trig substitution? An integral that requires you to use a hyperbolic or inverse hyperbolic substitution?
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2answers
43 views

Problem with Floor Function

I have $f_k(x)=kx-\lfloor kx \rfloor$, where $k\in \mathbb N$ and $x\in\{0,1\}$ and $x\in \mathbb Q$. When I plug in some numbers it seems obvious that $$ f_k(x)+f_k(1-x)=1 $$ for example $k=28, ...
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2answers
50 views

Does a solution to the differential equation $y'=y$ exist?

What is the solution to this differential equation : $$f'(x)=f(x)$$ I'm very interested in this because if it have a solution this means that the slope of that function at a point $a_0$ is the height ...
5
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7answers
281 views

Interesting calculus problem advice

Can someone suggest a really hard calculus problem that can be solved with the knowledge of a high school student ? I would really like to work my brains on something interesting . Thanks a lot !
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2answers
46 views

Indefinite Integral Confusion

Solve the indefinite integral $$\int\frac{x^3−11x^2+x+2}{x^4−2x^3}\text{d}x.$$ My answer was $\frac{1}{2 x^2}+\frac{1}{x}-4\log(2-x)+5\log(x)+ C$ and I put that into my webwork and it was wrong. Then ...
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2answers
269 views

What is the value of that integral?

The Maple code int(exp(-z^2*sin(z)^2), z = 0 .. infinity, numeric, epsilon = 0.1e-1) outputs $2.835068335 $. However, I am not sure if the answer is correct. ...
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1answer
44 views

Proving something about $|f(x)|$ when the lim of $f(x)/x^2$ is known

I've been trying to crack this issue for 2 days and I got pretty much nothing Given that $f$ is a continuous function and the following limits exists and are finite: $$ (1) ...
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4answers
147 views

How to prove that $\frac{d}{dx} \left(\sin^2(x)+\cos^2(x)\right)=0$

I have seen in a mathematics.stackexchange.com thread that to prove that $\sin^2x+\cos^2x=1$ one have to prove that the derivative of $\left(\sin^2(x)+\cos^2(x)\right)$ is $0$. But how can we prove ...
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2answers
21 views

Definite integral (mixture of functions)

I have problem with this integral and I generally don't know how to approach it: $$\int_{-2}^2 (x^4+4x+\cos(x))\cdot \arctan\left(\frac{x}{2}\right)dx$$ I know that I probably have to make some ...
0
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0answers
28 views

Is the relation correct?

Accoeding to my notes: $\sum_{n=1} ^{m} z^n=z+z^2+....=\frac{1}{1-z}(z(1-z)+z^2(1-z)+....+z^m(1-z))=\frac{z(1-z^m)}{1-z}, |z|<1$,so $\lim_{m \to +\infty} z^m=0$,therefore $\sum_{n=1}^{+\infty} ...
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1answer
24 views

Solid of revolution vs $\iiint$

In calc 1, I learned about rotating a curve around an axis, say $y=x^2$ around the y-axis. In calc 3, I learned about the shape of 3D objects in the context of $\iiint$ triple integrals . These ...
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0answers
21 views

Convergence of series (putnam training) [on hold]

Does the series $\sum_{n=1}^{\infty} \frac {|\sin n|}{n} $ converge?
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1answer
24 views

Determine the values of real parameters …

If you have an idea, please, do not leave the page, just write it, I will be very thankful. We have the function $$f:R\setminus \{-1 \}\to{R}$$ ...
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0answers
31 views

To show a function differentiable

Let $A \in \mathbb{R}^n$ be a fixed vector and $T : \mathbb{R}^n \rightarrow \mathbb{R}^n$ a linear transformation . Define $f : \mathbb{R}^n \rightarrow \mathbb{R}$ by $$f(x) = \langle ...
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0answers
14 views

Find the values of the parameters for which the function admits an oblique asymptote…

can you please help me solve this exercise: Find the values of real parameters $a$ and $b$ so that the function $$\color{maroon}{f(x)={(ax^3+bx^2)}^{1/ 3}}$$ admits an oblique asymptote: ...
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0answers
28 views

Prove continuity of $\frac{x^3y+2xy^3}{x^2+y^2}$ using the definition

$f(0,0)=0$ and $f(x,y)=\dfrac{x^3y+2xy^3}{x^2+y^2}$ when $(x,y) \neq (0,0)$. Is $f$ continuous at $(0,0)$? I went to polar coordinates, $$ f(x,y)=g(r,t)=r^4(\cos^3t \sin t+2\cos t\sin^3t)/r^2=r^2 ...
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0answers
13 views

How to convert vector field from cartesian to spherical

I have a vector field $A ( r) = \omega \times r$, where $r=(x,y,z)^T$ and now I want to express this field in cylindrical coordinates. How do I do this?
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1answer
23 views

Separable Differential Equation

The question is: $$t^5\frac{\mathrm{d}y}{\mathrm{d}t} + y^5 = 0$$ The next step says $\frac{1}{y^5}\frac{\mathrm{d}y}{\mathrm{d}t} + \frac{1}{t^5} = 0$ i understand this. However it then says: ...
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3answers
31 views

Finding a tangent to an ellipse parallel to a given line

Problem: Find the lines that are tangent to the ellipse $x^2 + 4y^2 = 8$ and parallel to $x +2y = 6$. I tried to find the derivative of $x^2 + 4y^2 = 8$ and I got: $$\frac{dx}{dy} = -\frac{x}{2y}.$$ ...
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0answers
29 views

Minimise sum of squares [duplicate]

For real numbers $x_{1},..,x_{n}$, minimise $x_{1}^2+..+x_{n}^2$ subsject to the condition $x_{1}+..+x_{n}=2$. This has cropped up in a stats question.
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1answer
40 views

Given one solution, can a second solution always be found?

Let's consider a second order ODE: $$y''+p(x)y'+q(x)y=f(x)$$ A common procedure is to find linearly independent solutions $y_1,y_2$ to the homogenous ODE, and then apply the technique of variation ...
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3answers
42 views

How can one determine whether the following series converges or diverges [duplicate]

$$ \sum_{n = 2}^{\infty}\frac{(-1)^{n}}{\sqrt{n} + (-1)^{n}} $$ Wolfram Alpha returns nothing useful, except that the ratio test was inconclusive.
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0answers
27 views

How to find $\nabla$ in spherical coordinates

I want to derive(!) just a few components like the $\hat{e}_r$ component of the divergence operator in spherical coordinates and the $\hat{e}_{\phi}$ component of the curl operator in spherical ...
4
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1answer
88 views

a doubt with the series $ \sum_{n=0}^{\infty}e^{-nx} $

I know that the series is equal to $$ \sum_{n=0}^{\infty}e^{-nx}= \frac{1}{1-e^{-x}}$$ However, if I expand each exponential term into a Taylor series I get $$ \sum_{n=0}^{\infty}e^{-nx}= ...
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3answers
24 views

Find all points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the x-axis

Find all the points on the curve $y=2x+x^{-1}$ which have a tangent parallel to the $x$-axis.
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3answers
40 views

How to find $\int\sqrt{(26x-x^2)}dx $

How do I find $\int \sqrt{(26x-x^2)} dx $ Is this an integration by parts question? Thanks, --Nick
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2answers
45 views

Is there a difference between limit and “two-sided limit”?

If we have a function of real-variable, then we can talk about one-sided limits $\lim\limits_{x\to a^+} f(x)$ and $\lim\limits_{x\to a^-} f(x)$ and, of course, also about the limit $\lim\limits_{x\to ...
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2answers
52 views

How to Simplify Sin/tan problem.

I am trying to simplify $\displaystyle\frac{\sin^2}{\tan^2}$ but I don't know how to go about it. Any help is appreciated.
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1answer
59 views

Does the series $\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ converge?

$\sum\limits_{n=1}^\infty\frac{\sin(n)n!}{n^n}$ Please let me know how you did it. Thank you.
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2answers
77 views

Maximum among $1, 2^{1/2}, 3^{1/3}, 4^{1/4},…$

What is maximum value among $1, 2^{1/2}, 3^{1/3}, 4^{1/4},....$ ? My approach: let $f(x)=x^{1/x}$ then I found out the derivative of $f$. Since $f(x)$ is maximum where $f'(x)=0$ and $f''(x)<0$ ...
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2answers
19 views

maximized profit w/ a cost & demand function

I'm having trouble with this problem: If $C(x) = 14000 + 500x − 4.8x^2 + 0.004x^3$ is the cost function and $p(x) = 4100 − 9x$ is the demand function, find the production level that will ...
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1answer
29 views

Easy calculus question

Let $D = \{ (x,y) : 0 \leq x \leq 1, \; \; x^2 \leq y \leq x, \; \; 0 \leq z \leq x \} $ and suppose $f(x,y,z) = x + y $. Want: $\int_D f $ IS this the correct integral? $$ \int_{0}^1 ...
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1answer
29 views

Minimize total area of a square and triangle made of 13m long wire

I'm a little bit confused about this problem. I've gotten the first part, but I can't get the second! ...
2
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1answer
37 views

Another integral calculus question from Apostol

Am stuck on another question from Apostol "Calculus" Volume 1 (Section 5.11, Question 22). The question reads: Determine a pair of numbers $a$ and $b$ such that $$ \int_0^1 (ax+b)(x^2+3x+2)^{-2} dx ...
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2answers
82 views

Equality of integrals: $ \int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \int_{0}^{1} \frac {1}{1+x^2} \, \mathrm{d}x $

In Street-Fighting Mathematics (page 16), Prof. Sanjoy Mahajan states that $$ \displaystyle\int_{0}^{\infty} \frac {1}{1+x^2} \, \mathrm{d}x = 2 \cdot \displaystyle\int_{0}^{1} \frac {1}{1+x^2} \, ...
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2answers
26 views

Showing that ${d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)}$

I'm trying to show that $$ {d \over dz}\log\left[ z - a \over z - b \right] = {1 \over (z - a)} - {1 \over (z - b)} $$ However my attempt yields that $$ {d \over dz}\log\left[ z - a \over z - b ...
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0answers
15 views

Finding the distance from a parabola (ballistic trajectory) to a point (for use in collision detection)

I need to have some form of collision detection / prevention for an object moving along a ballistic trajectory and a second stationary object on the same plane plane. The ballistic trajectory is ...
0
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0answers
12 views

Domain of function of form $f(x)=\frac{g(x)}{k(x)}$

I just want to know did we have a rule to find the domain of function in form of $f(x)=\frac{g(x)}{k(x)}$ .I know $k(x)\ne 0$ . but in general do we have any rule to compute domain of function like ...
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3answers
36 views

For solid volumes, why does the Integral behave as a summation?

When you take a definite integral, you can think about calculating the area under the curve (via Riemann rectangle slices approximation) Now, when you take the volume of a 3D object, you sum the ...
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1answer
30 views

Convergence when integrating a not-quite power series

This is a question where I don't have serious doubts about the truth of the statements; it's more about how to prove things rigorously. Consider $\displaystyle\frac{1-x^t}{1-x}$ where $t$ may be a ...
2
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3answers
65 views

$\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$

Evaluate $\int_0^\frac{\pi}{2}\cos ^2x\log(\tan x)dx.$ Sidenote:Via mathalpha I know that answer is $-\pi/4$ but do not know how to derive that.