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

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0answers
49 views

A problem in limits

I am solving problems in Limits and stuck in this problem. Problem Let $f:R\rightarrow R$ be a positive decreasing function with $$\large{\lim \limits_{x\to \infty }\frac{f(\frac{x-x^{3}}{6} ...
0
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2answers
72 views

How to solve $3\sin^3x-5\sin x\cos x+2\cos^2 x=0$?

Solve $3\sin^3x-5\sin x\cos x+2\cos^2 x=0$. It should use simple identities, but no identity I used helped me. There has to be a trick but I don't seem to find it. I could really use any kind of help. ...
1
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3answers
205 views

Determine whether this limit statement is true or false and why

I am having difficulties determining whether or not the following statement is true or false but more specifically, why it would be so. If $\lim\limits_{x \to 2} f(x) = f(2)$, then $\lim\limits_{h ...
5
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2answers
103 views

Directional derivative

The governor Ralph has trouble on the bright side of Mercury. The temperature in the wall of the vessel, when it is in the position $(x, y, z)$ is given by $T(x, y, z)=e^{-x^2-2y^2-3z^2}$, where $x$, ...
1
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1answer
50 views

Finding optimum value for a function

Suppose you have $f\colon\mathbb{R}\to\mathbb{R}$, If we can rewrite f as: $f(x)=Kp(x)^\alpha q(x)^\beta$, where, $p,q$ functions, $k$ constant and $$K'=(p(x)+q(x))'=0,$$ then a candidate for a ...
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1answer
68 views

Help with an Inverse Integral

Evaluate $$\int^{1/{\sqrt{3}}}_{-1/{\sqrt{3}}} \frac{x^4}{1-x^4}\cos^{-1}\frac{2x}{1+x^2} \mathrm{d}x\\= \frac{\pi}{a}\ln(b+\sqrt{c}) +\frac{\pi^{d}}{e} - \frac{\pi}{\sqrt{f}}$$ Then Find ...
1
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5answers
64 views

Integration of $(\tan x\sec x)^3$

How do you integrate the following function? I've been struggling with this one for quite a while now. Any help would be very much appreciated. $$\int(\tan(x)\sec(x))^3dx$$
3
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1answer
839 views

limits and continuity: irrational and rational piecewise function

I have noticed similar topics, but people seem to solving them with sequences which I have not learned yet. I need to prove that the function: $$f(x)=\begin{cases} x, & \text{ if $x$ is an ...
0
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2answers
28 views

Integral with logarithmic and exponential function

I have a problem with integration of $$\int \frac{1}{2e^tt^2}+\frac{\ln t}{2e^t}\,\mathrm{d}t$$ It seems that the integral has simple solution in terms of elementary functions, but I have no idea ...
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6answers
542 views

Evaluate $\int_0^{\infty} {\sin(\tan(x)) \over x}dx$

I tried to solve it the Feynman way and defined: $$I(a):=\int_0^{\infty} {\sin(\tan(a \cdot x)) \over x} \ dx$$ And look what happens when one substitutes $u=ax$ $(a>0)$: $$I(a)=\int_0^{\infty} ...
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1answer
34 views

Definite integration (Weierstrass)

By using the substitution $u=\tan\frac{1}{2}x$, prove that $\int_{0}^{\frac{\pi}{2}}\frac{1}{5-4\ cosx}dx=\frac{2}{3}\tan ^{-1}(3)$. Hence find $\int_{0}^{\frac{\pi}{2}}\frac{\cos x}{5-4\cos x}dx $. ...
6
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3answers
1k views

Can I solve limit of natural log without using L'Hospital Rule

I want to find the following limit: $$\lim_{x\to 2} \dfrac{\ln(x+3)-\ln(5)}{(x-2)}$$ but can we find the limit without using L'Hospitals Rule? It is an indeterminate form but I do not know how to ...
7
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3answers
329 views

Why it is absolutely mistaken to cancel out differentials?

In many of my physics courses, (don't worry, this is a mathematics question!) My teachers cancel out differentials, and every time, they say: "If a mathematician saw me canceling out this ...
6
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8answers
184 views

Find $\int\limits_{0}^{2\pi} \frac{1}{5-3\cos(x)} \,\,dx$

I have to find $$\int_0^{2\pi} \frac{1}{5-3\cos(x)} \,\,dx$$ I tried to do it by substitution $t = \tan(\frac{x}{2})$ Then we have that $$\cos(x) = \frac{1-t^2}{1+t^2} \quad dx = ...
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3answers
80 views

How to determine the limit of $f(\mathbf{x}) = \frac{2x^6y}{x^8+y^4+5x^4y^2}$ as $\mathbf{x} \rightarrow \mathbf{0}$?

How do I determine the limit of $f(\mathbf{x}) = \frac{2x^6y}{x^8+y^4+5x^4y^2}$ as $\mathbf{x} \rightarrow \mathbf{0}$, where $f:\mathbb{R^2}-\{(0,0)\}\rightarrow \mathbb{R}$? Following the ...
2
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1answer
44 views

Unit of value calculated by using integral

I have a really simple question that is driving me crazy! I have a program which can log power and the result is a log file which contains two arrays, one with time values that show when a power value ...
0
votes
1answer
66 views

Solving limit Lim ar r goes to zero $ 1/e^{1/r^{2}}.r $

What is $$\begin{equation*} \frac{1}{e^{1/r^{2}}\cdot r} \end{equation*}$$ as $r\to 0?$ I tried using L'hopital's rule once, then I used the series for the exponential and the limit turned out to ...
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3answers
46 views

General Formula of the $n$th Derivative for $f(x) = xe^{2x}$

Find the general formula for the nth derivative of $f(x)=xe^2x$ in the form: $$ f^{(n)}=A(n)e^{2x}+B(n)xe^{2x} $$ I've evaluated the first five derivatives in that for and for $A(n)$ have found ...
0
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2answers
30 views

Showing that $\lim_{t \rightarrow \infty}-e^{-t}t^x=0$

I'm trying to prove that $\Gamma(x+1) = x \Gamma(x)$ and after doing integration by parts on $\Gamma(x+1)$ I'm left with a term $-t^xe^{-t}$ that I need to evaluate at the limit as $t \rightarrow ...
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1answer
42 views

How to deal with limits of expressions that have a $(-1)^n$?

For example, I need to find the limit of $$\frac{n^3+(-1)^n n^2}{\sqrt{n^6+1}}$$ as $n$ tends to infinity. The given answer is 1. I have trouble getting the answer because I don't know how to take ...
2
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1answer
61 views

Maclaurin series for a function

Provided I have the function \begin{equation*} f(x)=(1+x)^{1/x}, \end{equation*} and I want to calculate a 3rd order Maclaurin series, how can that be done without taking direct derivatives (as ...
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1answer
60 views

Verification that $\int x\sin x=\sin x- x\cos x + C$ by differentiating both sides of the equation

The original question is: Confirm that the formulae stated below are correct by differentiating both sides: $\int x\sin x=\sin x-x\cos x+C$ Where does the cancellation occur, and what is the ...
1
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1answer
26 views

Interest for a period

I am trying to turn my code into a maths equation if that's possible. To work out when the interest for a period is over X amount. ...
0
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2answers
717 views

Can arc length be negative?

I have a quick question that has been bothering me since I took my calc 2 final earlier. The question dealt with arc length and was pretty much plug and chug using the the arc length formula. My ...
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1answer
24 views

General Sequence Equation

I want to find the general equation to express the following: *1,1,-1,-1,1,1,...* Thanks for the help in advance! :)
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1answer
30 views

About existence of derivative

Please give me details that either derivative exist or not at end points of any interval? Thanks to everybody who will help me.
3
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2answers
200 views

Change of variables for heat equation

How to make a change of variables to turn the equation $$\frac{\partial{u}}{\partial{t}}=D\frac{\partial^2{u}}{\partial{x}^2}+cu$$ back to the heat equation? Where can I read about change of ...
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1answer
40 views

Power Series Representation of $x^3/(2-x)^3$

I don't need an answer, as this was a question I got wrong on a problem set, but could someone explain this? So, we have to represent f(x)= $x^3$/$(2-x)^3$ My professor writes consider g(x) = ...
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0answers
24 views

Function of variation equals integral of absolute value [duplicate]

Exercise 11, chapter 6 from second edition of Baby Rudin: Suppose $g$ is Riemann integrable on $[a,b]$, put $$\int_a^x g(t)dt,$$ and define $g^+(t)=\max(g(t),0)$, $g^-(t)=-\min(g(t),0)$. Prove ...
0
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2answers
44 views

Where is the point that has the shortest distance to the orgin?

I just finished an exam, it has the following question: Where is the point on the plane $3x + 5y + z = 18$ has the shortest distance to $(0,0,0)$? I found this question similar: Find the point on the ...
6
votes
1answer
45 views

First exercise of Guillemin-Pollack. [closed]

If $k < l$ we can consider $\mathbb{R}^k$ to be the subset $\{(a_1, \dots, a_k, 0, \dots, 0)\}$ in $\mathbb{R}^l$. Show that smooth functions on $\mathbb{R}^k$, considered as a subset of ...
2
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0answers
204 views

Strict inequality in proof of First Mean Value Theorem for Integrals

Let $M=\sup|f(x)|$ and $m=\inf|f(x)|$ $x \in [a,b]$ The first mean value theoremn for Riemann integrals says: If $f$ continuous (and in this case we will asume non-constant, constant is trivial) ...
2
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1answer
30 views

How to find the volume of a solid of revolution?

The region bounded by $x^2-y=0$ and $x+y=0$ is rotated around $y=1$ I solved for: $$y = x^2 $$ and $$y = -x$$ And I then also solved for x for both of those. I set up the integral like so: $$ V = ...
2
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3answers
140 views

Definite integral using $u$-substitution.

Evaluate the definite integral: $$\int_{1/3}^{\sqrt{2}/3} \dfrac{1}{x\sqrt{9x^2-1}}{dx}$$ So, I am to use $u$-substitution, and immediately, it would appear that perhaps the integral may be some ...
4
votes
2answers
89 views

integral of logarithm and rational function

i'm wondering how can i evaluate this integral using real methods: \begin{equation*} \int_{0}^{\infty}\frac{\log x}{1+x^{2}}dx. \end{equation*} I tried using mclaurin series of $\log x$ but really ...
1
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1answer
38 views

Find directional derivative - simple

The directional derivative of $f(x,y)$ at $(1,2)$ in the direction of $\vec a =\vec i + \vec j$ is $2\sqrt{2}$. We also know that the directional derivative of $f(x,y)$ at $(1,2)$ in the direction of ...
1
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1answer
35 views

Area between two curves in terms of x

I am given two equations and a graph. The equations are $$x=-y^3+4y+9$$ $$x=y^2-5y$$ The problem shows a graph with a shaded region, and I am only to find the area above $y=-1$. I want to set up ...
2
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6answers
71 views

How do I find the limit of this function as $x\to1^-$?

I need to find $$\lim_{x\to1^-}\frac{\sqrt{1-x^2}}{\sqrt{1-x^3}}.$$ I tried l'Hôpitals, however it seems like no matter how many times you differentiate, it will still be in the indeterminate form. Is ...
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2answers
146 views

To prove that no such function can be continuous. [closed]

Suppose $f: [a,b] \to R$ is two to one. that is, for each $y$ in $R$, $f^{-1}({y})$ is empty or contains exactly two points. How to prove that no such function can be continuous.
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4answers
69 views

How often is $x\to\infty$ used to denote ($x\to +\infty$ or $x\to -\infty$)?

How often is $x\to\infty$ used to denote ($x\to +\infty$ or $x\to -\infty$)? Both my textbook and my teacher use $x\to\infty$ as above, so e.g. it's false for us that ...
4
votes
5answers
108 views

Why $\int_0^af(x)dx=\int_0^af(a-x)dx$?

Is it true that: \begin{equation*} \int_0^af(x)dx=\int_0^af(a-x)dx? \end{equation*} Because I know that: \begin{equation*} ...
6
votes
4answers
136 views

<s>Solve</s> Evaluate $\lim_{x\to\infty}\frac{\sqrt{x-1} - \sqrt{x-2}}{\sqrt{x-2} - \sqrt{x-3}}$

I'm trying to solve evaluate this limit $$\lim_{x\to\infty}\frac{\sqrt{x-1} - \sqrt{x-2}}{\sqrt{x-2} - \sqrt{x-3}}.$$ I've tried to rationalize the denominator but this is what I've got ...
2
votes
1answer
118 views

Fourier transform of Gaussian? [duplicate]

For the Fourier transform defined as $$\frac {1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(x) e^{-i\alpha x}\,dx$$ I know there is simple formula for the Fourier transformation and inverse transformation ...
3
votes
0answers
58 views

detemine a limit involving log function

I just came up with the following limit $$ \lim_{n\to \infty} \frac{\displaystyle\log_a\Bigg(\sum_{\substack{k\in \mathbb{N}\\k\leq n~(1-\frac{1}{a})}}\binom{n}{k} (a-1)^k\Bigg)}{n} \qquad \text{ ...
2
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2answers
87 views

Proving $\frac{x}{x-\lfloor \sin x \rfloor}$ has no limit as $x\to 0$ using the definition of limit

I need to show that $$\lim_{x\to 0} \left(\frac{x}{x-\lfloor \sin x \rfloor} \right )$$ doesn't exist using the definition of limit (its negation). I fail to choose $\varepsilon$ and $x$ correctly ...
1
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1answer
80 views

Calculate the Taylor series of $f(x) =\ln( 1 -x +x^2) $ and the domain of convergence

I just stuck at the following exercise: Show that the function f has a Taylor series and calculate it, with $x_0 = 0$. $$ f(x) = \ln{(1-x+x^2)}$$ Because I already know the Taylor series from ...
7
votes
1answer
170 views

Calculating $\int_0^{\infty } \frac{\log (v+1)}{\sqrt{(v+1)^2+1} \sqrt{(v+1)^2+4 \sqrt{(v+1)^2+1} (v+1)+4}} \, dv$

What tools would you recommend me for this one? $$\int_0^{\infty } \frac{\log (v+1)}{\sqrt{(v+1)^2+1} \sqrt{(v+1)^2+4 \sqrt{(v+1)^2+1} (v+1)+4}} \, dv$$ It's related to Calculate in closed form ...
4
votes
2answers
65 views

compute improper integrals using integration by parts

Compute \begin{equation*} \int_0^\infty \frac{\sin^4(x)}{x^2}~dx\text{ and }\int_0^\infty \frac{\sin (ax) \cos (bx)}{x}~dx. \end{equation*} For the first integral I tried letting $u = \sin ^4 x$ ...
2
votes
5answers
119 views

How can I find the indefinite integral of $\int\sin^3x \cos^3x dx $?

I am looking to evaluate the indefinite integral \begin{equation*} \int\sin^3x \cos^3x dx. \end{equation*} I'm not sure if I started this right but I broke the terms up like this: ...
0
votes
1answer
78 views

Boundary layer problem

This question is taken from Bender & Orszag "perturbation methods" $y' = (1 + X^{-2}/100)y^2 - 2y + 1$ ,$y(1)=1$ first we can see that if we set $\epsilon=100x^{2}$ we can translate the above to ...