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

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1answer
15 views

Evaluate the limit $\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$

I have this assignment Evaluate the limit: 5.$\displaystyle\lim_{x \to 0^{+}} x \cdot \ln(x)$ I don't think we are allowed to use L'Hopital, but I can't imagine how else.
0
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3answers
34 views

How to solve for an unknown upper bound in a summation

I've always wondered: if you have something like $$ \sum\limits_{k=0}^x k $$ And you want to find x such that the value of the summation is the closest to a constant c, how would you proceed? And ...
0
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2answers
21 views

By applying the second version of the Fundamental Theorem of Calculus find the integral:

The second version of the Fundamental Theorem of Calculus states that if $F'(x)=f(x)$ then $\int_{a}^{b} f(x) dx = F(b)-F(a)$. I need to use this to find a) $\int_{-2}^{-1} \frac{1}{x^3} dx $ and b) ...
0
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4answers
41 views

Proving an equation is a fuction

Prove that the equation $y^3 + 3xy -5x^3 + 1 = 0$ defines $y$ as a function of $x$ for all $x$ in the real numbers.
3
votes
1answer
45 views

Integrate $e^{-\frac{y^2}{2}}\left(\frac{1}{y^2}+1\right)$

I'm trying to find $$\displaystyle \int{e^{-\frac{y^2}{2}}} \left(\frac{1}{y^2}+1\right)dy$$ I tried using integration by parts and some substitutions, but nothing seem to work. The answer is ...
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3answers
38 views

Weird integration question

let = e(x) = $$\text{e(x) =}\int\frac{e^x}{e^x+e^{-x}}$$ and f(x) = $$\text{f(x) =}\int\frac{e^{-x}}{e^x+e^{-x}}$$ The question wants.... a) calculate e(x) + f(x) b) calculate e(x) - f(x) c) Use ...
0
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1answer
33 views

Integrating $x\cdot{\cosh(x^2)}$

How do integrate $x\cdot{\cosh(x^2)}$? Do i just use integration by parts? I know that integration by parts is $\int{u\cdot{\mathrm{d}v}} = uv - \int{v\cdot{\mathrm{d}u}}$ Making ...
0
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1answer
21 views

Calculus and Lagrange Multiplier

If I have the function $u(x_1,x_2) = (x_1-a_1)^{1/2}(x_2-a_2)^{1/2}$ If I take the derivative of the above with respect to $x_1$, would it equal $1/2 (x_1)^{-1/2} (x_2-a_2)^{1/2}$? And wrt to ...
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0answers
11 views

Setting up a volume-finding calculation

I'm asked to find the volume inside the sphere $x^2+y^2+z^2=25$ and outside the cylinder $x^2+y^2=1$. I approached the volume $V$ in the following way: ...
4
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1answer
35 views

Area and integration question, is this area under the curve?

Find the exact area between $x$ and the graph $f(x)=(x-1)(x-2)(x-3)$. $$f(x) = x^3-6x^2+11x-6$$ I found that this is an odd shaped positive polynomial with a maxima between 1 and 2 and minima ...
-2
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1answer
59 views

$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x = -1.38104$

$$\int_0^1\frac{\tanh ^{-1}(x)\log(x)}{(1-x) x (x+1)} \operatorname d \!x = -1.38104$$ When I look at it, I have no idea how to work on it. any hits! Thank you
0
votes
1answer
10 views

invertibility, derivative, and difference quotient

Suppose that $f$ is an invertible differentiable function, that the domain of $f^{-1}$ contains an interval around $a$, and that $f^{-1}$ is continuous at $a$ and that $f^{-1}$ is continuous at $a$. ...
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4answers
24 views

Derivative of $y=\tan(3)e^x$,

If, $y=\tan(3)e^x$, wouldn't the derivative be $y\;'=\sec^2(3)e^x \times e^x$? The outer function times the inner function, using the chain rule? The answer key gives the derivative as $y=e^x \tan ...
3
votes
1answer
17 views

If $f(1) = 3$ and $\int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt\;\forall x,y \in \mathbb{R^{+}}\;,$ Then $f(e) =

Let $f:\mathbb{R^{+}}\rightarrow \mathbb{R}$ be a differentiable function with $f(1) = 3$ and satisfying:: $\displaystyle \int_{1}^{xy}f(t)dt = y\int_{1}^{x}f(t)dt+x\int_{1}^{y}f(t)dt\;\forall ...
4
votes
1answer
79 views

L'Hopital's Rule, Factorials, and Derivatives

I have the following limit $\displaystyle\lim_{n\to\infty}\frac{e^n}{n!}$. Now if I try to solve this using this using L'hopital's rule, I won't be able to since I can't take the derivative of $n!$. ...
3
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0answers
25 views

How to find the Maclaurin series for the integral of $e^{x^2}$?

I am trying to find the Maclaurin series for the integral of $e^{x^2}$? What I done so far is that the Maclaurin series for $e^{x^2}$ is $$e^{x^2}=\sum_{n=0}^{\infty}\frac{x^{2n}}{n!}$$ So would ...
1
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2answers
20 views

Even function divided by Odd function is an Odd function PROOF?

An Even function divided by Odd function is an Odd function,that is a fact. However is there a means to prove this?
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2answers
31 views

Struggling to find the second derivative of this function's first derivative

So I've found the first derivative of this function but now I have to find the second derivative. I've tried everything but I cannot seem to get it. Here's the original function: $x = a sec(θ)$, $y = ...
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2answers
20 views

Related Rates- Expanding Circle

I just wanted to see if I did this correctly. Only asking for B. So the question ask : The area of a circle increases at a rate of 1cm^2/s. a. How fast is the radius changing when the radius is ...
0
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0answers
8 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
0
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1answer
15 views

Calculate duration of task

Say I have some task to process 100 days of data, and it takes 5 hrs to process a day. But each day that it takes to process it a new day of data comes in. So for the initial set of data it takes: 5 ...
0
votes
2answers
27 views

How to prove functions are odd and even

Show that any function f on [-a,a] where a is a positive constant, can be written as the sum of an even and an odd function?
0
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1answer
21 views

how to show this manipulation in the integral

Let we have: $$G(t)=y_1(t)\int y_2(s)ds$$ when we take the limits as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds$$ then is it possible to write it as: $$G(t)=y_1(t)\int^t_{t_0} y_2(s)ds=\int^t_{t_0} ...
0
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0answers
20 views

Simplifying Gamma functions yet having a complication while graphing when the function was able to be graphed previous to simplification?

According to the Euler's duplication formula: $$ \Gamma(z) \Gamma(z+\frac{1}{2}) = 2^{1-2z} \sqrt{\pi} \Gamma(2z) \therefore $$ $$ \Gamma(2z) = \frac{\Gamma(z) \Gamma(z+\frac{1}{2}) ...
4
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0answers
30 views

Integral involving hyperfactorial

I'm trying to prove that: $$ \int_0^1 \ln\left(K(x)\right)\space dx =-\zeta'(-1)=\ln(A)-\frac{1}{12} $$ Where $A$ is Glaisher Kinkelin's constant and $K(x)$ is a generalization of the hyperfactorial ...
2
votes
3answers
45 views

Find derivative of $f(x)=\frac{1}{\sqrt{x+2}}+2x$ by definition

Use the definition of a derivative to find the derivative of: $$f(x)=\frac{1}{\sqrt{x+2}}+2x$$ my work: $$\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ ...
0
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1answer
16 views

Derivative of complex conjugate

In general, two different mathematical operations need not commute. Let f(x,y) be a complex valued function, taking in two real-valued inputs x and y. Then under what circumstances is the partial ...
0
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1answer
11 views

Volume of a region after some transformation.

Consider $$R=\{(x,y,z):x^2+y^2\leq 1,0\leq z\leq 2\}$$ and the transform $$T:(x,y,z)\to(x,y+\tan \alpha z,z)$$ where $0<\alpha<\pi$ Then what is the volume of $T(R)$? I tried myself but I ...
2
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0answers
21 views

Reference request regarding calculus exam

I'm currently a first year computer science student and I'm deeply interested in calculus . That being said, what we studied so far consists of: Cantor sets, sequences and a brief introduction to ...
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3answers
11 views

Integrate the differential equation of a simple rate equation

Could somebody please show me how to integrate the following: $dA/dt = -kA$ I'm told that the answer is: $A(t) = A(0)e^-kt$ but I do not know why. Could you be explicit in your answer and explain ...
0
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1answer
13 views

how to find the interval at which a derivative function is increasing

Alright, so here's the deal. I need to find the interval of this derivative function: f(x)= −5x2+12x−7 So far, I've gotten that the derivative is this: ...
1
vote
1answer
14 views

Summation of arithmetic-geometric series of higher order

There is a closed formula for the summation of arithmetic-geometric series: $\sum_{x=1}^{+\infty }(ax+b)r^x=\frac{(a+b)r-br^2}{(1-r)^2}$ when $-1<r<1$ But consider: $\sum_{x=1}^{+\infty ...
0
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1answer
17 views

How do I find the relative coordinates of a picture of a plane in 3d space.

I have a box, with corners $A$ through $H$, as depicted above. I'll consider $B$ the origin of a coordinate system, with the $x$ axis in the direction through $C$, the $y$ axis through $A$ and the ...
0
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1answer
23 views

What are the rules being used to compute $\lim\limits_{x\rightarrow \frac{\pi}{2}} (1-\cos x)^{\tan x}$?

I am given $\lim\limits_{x\rightarrow \frac{\pi}{2}} \frac{\ln(1-\cos x)}{\cos x} = -1$ So, $(1-\cos x)^{\tan x} = e^{(\tan x) \ln(1-\cos x)}$ and as $x\rightarrow \frac{\pi}{2}$, we have: $(\tan ...
7
votes
3answers
49 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
votes
1answer
32 views

Is there any way to solve this inequality by hand?

I have had this one equality problem I've been stuck on for a while. Could anyone share on what methods I would have to use to solve this by hand? No need for full answer, just a hint is okay. ...
0
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2answers
14 views

What is $1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$?

I want to express $$1 + \sum_{k=1}^{\infty} \frac{(it)^k}{k!}a^{2k+1}$$ in terms of standard functions (exp, cos, sin, etc.), but I just don't see what this function is. Does anybody here have an ...
0
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1answer
20 views

Given that $x(\theta)=5\cos \theta,y(\theta)=5\sin \theta,z(\theta)=\theta$

Given that $x(\theta)=5\cos \theta,y(\theta)=5\sin \theta,z(\theta)=\theta$ $L(\theta)$ is the arclength at the point $P(x(\theta),y(\theta),z(\theta))$ and $D(\theta)$ is the distance from origin to ...
0
votes
1answer
17 views

Rate of Change with Derivatives

We just started with rate of change while using derivatives and I am stuck on a question, hope you can help. A particle moves on a vertical line so that its altitude at time $t$ is $y=t^3−12·t+3$, ...
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0answers
21 views

Maturity and Proficiency in calculus, linear algebra for successful research

Will the high level maturity and proficiency in basic calculus, linear algebra (both calculation and theorem aspects) be required or recommended as an important factor to be successful in mathematical ...
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0answers
24 views

Proving $\int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)} dx < \frac{1}{e}$

While i was playing around with very weird functions and came across this: $$ \int_{\frac{\pi}{4}}^{\frac{3\pi}{4}}{e^{\cos(x)}\cot(x)}dx \approx 0.3676932086...\approx \frac{1}{e} - ...
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0answers
24 views

Area of a circle shall equal the area of a square [on hold]

How can I, using bolzanos theorem, discuss the equal areas of a circle and a square? How can this be shown in a graph? Would be really grateful if any could help me! :)
0
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1answer
10 views

Show that the z-coordinate of the center of mass is 2/3 and so independent of the parameter a.

For a>0, consider the family of solids bounded below by the paraboloid z=a(x^2+y^2) and above by the plane z=1. If the solids all have constant mass density 1 gm/cm^3, show that the z-coordinate of ...
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2answers
70 views

Why “$\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ does not exist” is not an acceptable answer?

Find the limits: $\lim\limits_{x\rightarrow \infty} \frac{x+\sin x}{x}$ Since the numerator and denominator tends to infinity as $x$ tends to infinity, then applying Lhopital's rule: ...
3
votes
1answer
14 views

Writing the integral $ \int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$ in simpler form?

I was wondering if $\int_{t}^{\infty}(\frac{1}{4\pi s^{3}})^{1/2}\frac{r(|x|-r)}{|x|}e^{-\frac{(|x|-r)^{2}}{4s}}ds$ can be written more simply, where $x,r\in \mathbb{R}$ ? wolfram alpha doesn't ...
1
vote
2answers
50 views

integral of ∫xdx where x is constant [on hold]

I am trying to convince my friend that : ∫xdx where x is constant ⇨The result of integration is zero. This is important because by believing that the result is non zero, he believes that one can ...
1
vote
1answer
18 views

Double integral and integration by parts

Let $f:[0,b]\to[0,d]$ be a continuous bijection. If $h:[0,d]\to \mathbb{R}$ is a Riemann integrable function, how to prove that $$\int_{0}^b\left(\int_{0}^{f(x)}h(y)dy\right)dx = \int_0^d ...
0
votes
0answers
7 views

Looking for an alternative solution for optimal control problem

Let's say we have the following function ; $\intop_{0}^{\infty}\int_{0}^{N}V\left(C(t,\tau\right)dtd\tau$ and we want to maximise it according to the following constraint ; ...
1
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0answers
24 views

Does there exist such function?

Fix an integer value $k\geq 1$. Let $[0,1]$ the unit interval and let $s\in [0,1]$. Does there exist a function $f$ (which depends on $k$ of course but not on $s$) such that $$\int_s^1 \left( ...
0
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0answers
24 views

how can I give an elementary proof of Maximum Modulus Theorem for polynomials?

how can I give an elementary proof of Maximum Modulus Theorem for polynomials? I got proof, but not elementary. This question in this book Conway.