For basic questions about limits, derivatives, integrals, and applications, mainly of one-variable functions.
0
votes
1answer
48 views
Roots of $x^{2}+e^{0.1x}-1$
I saw an exercise that asks to prove that $f(x):=x^{2}+e^{0.1x}-1$
have a root $r<0$.
The solution stated that $f''(x)=2+(0.1)^{2}e^{0.1x}>0$ hence there
is a maximum of two roots, since $0$ is ...
1
vote
1answer
38 views
Find the area of the portion of the surface of the sphere $x^2+y^2+z^2=4x$ that is cut off by a nappe of the cone $y^2+z^2=x^2$.
I'm having trouble with a Calculus question where I am supposed to use the following formula to calculate the area of a surface:
Suppose that $f$ and its first-order partial derivatives are ...
1
vote
1answer
34 views
Evaluating the definite integral $\int^{\pi/2}_0 \frac{x+\sin x}{1+\cos x}\,\mathrm dx$
Problem :
$$\int^{\pi/2}_0 \frac{x+\sin x}{1+\cos x}\,\mathrm dx \tag{i}$$
My approach :
$$\frac{x+\sin x}{1+\cos x}\,\mathrm dx= \left ( \frac{x}{2\cos^2(x/2)} + \tan(x/2) \right )\,\mathrm dx$$
...
1
vote
2answers
74 views
What is the derivative of $\frac{x}{2-\tan x}$?
I am too stupid to figure this out so I won't even try anymore
$$y = \frac{x}{2 - \tan x}$$
I am sure this will take someone about four seconds to solve, but I spent about ten minutes looking at it ...
1
vote
2answers
22 views
General way to find out whether a curve is positively oriented
I have a general question, that deals with the question how I am able to find out whether a particular curve(in $\mathbb{C}$) is positively oriented? Take e.g. $ y(t)=a+re^{it}$. Obviously this one ...
3
votes
5answers
56 views
prove that $ \lim_{x \to 0} \frac{e^{1/x}}{x}$ does not exist.
By Substitution of y = $\frac {1}{x}$ i have managed to show that
$\lim_{x \to 0^+}\frac{e^{1/x}}{x} = \infty $
but i can't find a way to show that
$\lim_{x \to 0^-}\frac{e^{1/x}}{x} = 0 $
I've ...
-1
votes
0answers
30 views
Math exercise using plans complex?
This is a very short question.In what line of the plan Cw is the circle $|z|=1$ mirrored using the function $w=\sqrt {(z+1)} $ ?
0
votes
3answers
24 views
Derivative of a quotient
I am trying to find
$$y = \frac{x^2 + 4x + 3}{ \sqrt{x}}$$
I am reviewing this so I am suppose to do it without the quotient rule, just using what I know about 16 years of algebra and the power ...
2
votes
0answers
40 views
Integration by Parts for PDEs
I'm reading a paper on PDEs in preparation for some research.
In it, integrals like this appear repreatedly:
$$ I(x,t) = \int_{|y|>1} e^{-i\alpha xy} \frac{e^{i\beta y^2}}{(1+y^2)^m} dy. $$
Here ...
1
vote
2answers
31 views
Inverse fourier transform of exponentially decaying function in the frequency domain
I want to take the inverse Fourier transform of the following function:
$$ \hat{f}(\omega) = \begin{cases}e^{-r \sqrt{\omega}} & \text{for } \omega > 0 \\ 0 & ...
0
votes
3answers
43 views
Summing a series - Calculus 1.
I'm learning Calculus 1 at the collage,
and the semester's end is close, which bring with it the exams period.
So I pretty much understand all the topics,
except for a series summing.
I don't know ...
1
vote
1answer
27 views
Function Projection: Orthogonal Polynomials
I am currently reading a paper called "Numerical Quadrature" by Timothy J. Giese (2008) which describes the numerical quadrature technique in detail. At one point (just before equation 19) it states ...
0
votes
1answer
46 views
Trajectory of a projectile.
From the definition of a parabola can we prove that the trajectory of a projectile is parabolic? And can this be proved by calculus?
0
votes
0answers
47 views
Obtaining the cardioid by mirroring the square root function in a line
In what line of the plane $C_{W}$ is the cardioid $$p= 2 (1 + \cos\theta)$$ mirrored, from the branch of the function $$w=\sqrt{Z}$$ which takes positive values in $X>0$ and $Y=0$.
Seriously this ...
-5
votes
1answer
37 views
Calculus homework problem help! [closed]
Evaluate the following limits:
limit of x^2/x^3 as x approaches infinity $\lim\limits_{x\to\infty} \frac{x^2}{x^3}$
limit of (x^3-x^2+x)/(1000x^2-1) as x approaches infinity ...
0
votes
0answers
20 views
Diagonalizing a system of differential equations
I need to solve this system of differential equations:
$\dot{A}=g_1AC+g_2A^2+g_3C^2$
$\dot{B}=g_1BC+g_2C^2+g_3B^2$
$\dot{C}=\frac{g_1}{2}(AB+C^2)+g_2AC+g_3BC$
Probably I can try to write it in the ...
6
votes
1answer
52 views
Does the integrability of $\log(f(x))$ imply $f(x)$ is bounded?
Let $f(x):(a,b)\rightarrow \mathbb{R}$. Suppose $\int_a^b\log{f(x)}\,\mathrm{d}x<\infty$, can we claim that $0<f(x)<M<\infty$ a.e.. Why and why not?
1
vote
0answers
43 views
Is this a valid application of “separation of variables”?
I asked this question over at Physics SE. I am not satisfied with the answer. At the heart of the question is this mathematical concern:
Can I invoke "separation of variables" to go from this:
...
1
vote
0answers
29 views
Representing a diverged integration in term of non-elementary functions
I know that the following integration diverges
$$
I = \int_0^\infty {\frac{{\sqrt {{{(x + a)}^2} + {b^2}} }}{x}{e^{ - x}}dx},
$$
where $a$ and $b$ are constants. But are there any ways to represent ...
0
votes
2answers
54 views
How to find this mass?
Let $R$ be the region in the first quadrant of the plane bounded by the lemniscates of the following equations:
$\rho^2=4\cos(2\theta)$,
$\rho^2=9\cos(2\theta)$,
$\rho^2=4\sin(2\theta)$, and
...
4
votes
1answer
53 views
Can an apparently intractable integral be made tractable in this way?
Say there are two integrals:
$$F(\beta)=\int_{b_0}^{\beta}f(b)\: db$$
$$H(\rho)=\int_{r_0}^{\rho}h(r)\: dr$$
where $F(\beta)$ is (apparently) intractable while $H(\rho)$ is tractable (expressible in ...
2
votes
2answers
51 views
Indefinite Integration : $\int \frac{dx}{(x+\sqrt{(x^2-1)})^2}$
Problem :
Solve : $\int \frac{dx}{(x+\sqrt{(x^2-1)})^2}$......(i)
I tried :
Let $x =\sec\theta$ therefore , (i) will become after some simplification
$$\int ...
3
votes
5answers
74 views
Indefinite integration : $\int \frac{1+x-x^2}{\sqrt{(1-x^2)^3}}$
Problem :
Solve : $\int \frac{1+x-x^2}{\sqrt{(1-x^2)^3}}$
I tried :
$\frac{1-x^2}{\sqrt{(1-x^2)^3}} + \frac{x}{\sqrt{(1-x^2)^3}}$
But its not working....Please guide how to proceed... thanks..
2
votes
6answers
77 views
Help with differentiation of natural logarithm
Find $\;\dfrac{dy}{dx}\;$ given $y=\frac{\ln(8x)}{8x}$.
The answer is $\;\dfrac{1-\ln(8x)}{8x^2}\;$.
Can you show the process of how this is worked?
Thanks.
3
votes
3answers
34 views
Finding the derivative of a relational problem
I am self studying some calculus and I have gotten really stuck! I thought I had the right idea but I keep getting the answer totally wrong. I am sure I am missing something important. Here is the ...
6
votes
1answer
81 views
Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?
How could one prove that
$$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$
is such that $x+{1\over2}-{1\over3} = \gamma$ ?
I am having problems just calculating ...
3
votes
1answer
45 views
prime notation clarification
When I first learned calculus, I was taught that $'$ for derivatives was only a valid notation when used with function notation: $f'(x)$ or $g'(x)$, or when used with the coordinate variable $y$, as ...
1
vote
3answers
42 views
Help with implicit differentiation: $e^{9x}= \sin(x+9y)$
Find $\;\dfrac{dy}{dx}\;$ given $\;e^{9x}= \sin(x+9y)$
the answer is $\;\displaystyle\frac{e^{9x}}{\cos(x+9y)}- \frac{1}{9}$.
Can you show the process of how this is worked?
thanks.
3
votes
3answers
52 views
How to determine if a function is decreasing, constant or increasing (in a given interval) if its derivative function has no zeroes?
Let us suppose you have a certain function $f(x)$ and you want to find out in which intervals this function is decreasing, constant or increasing. I know you need to follow these steps:
Find out ...
2
votes
1answer
38 views
Analysis of $\sum_{k=1}^{\infty}\frac{x^2}{(1+x^2)^k}$
Given the sum $\sum_{k=1}^{\infty}\frac{x^2}{(1+x^2)^k}$, I have two questions. (Sorry for that, it's for my exam preparations):
Prove that $\sum_{k=1}^{\infty}\frac{x^2}{(1+x^2)^k}$ converges ...
3
votes
2answers
38 views
Relation between differentiable,continuous and integrable functions.
I have been doing lots of calculus these days and i want to confirm with you guys my understanding of an important concept of calculus.
Basically, in the initial phase,students assume that ...
1
vote
1answer
50 views
limit of expression with first and second derivatives
I got the expression: $$\frac{X'(t) X''[t]+Z'(t) Z''(t)}{\sqrt{X'(t)^2+Z'(t)^2}}$$
How do I find the limit when I get to a point with $X'(t)=Z'(t)=0$, it's like everytime it happens, the limit is ...
1
vote
4answers
76 views
How can evaluate $\lim_{x\to0}\frac{3x^2}{\tan(x)\sin(x)}$?
I know this
$$\lim_{x\to0}\frac{3x^2}{\tan(x)\sin(x)}$$
But I have no idea how make a result different of:
$$\lim_{x\to0}\frac{3x}{\tan(x)}$$
I would like understand this calculation without using ...
1
vote
3answers
65 views
How to show convergence or divergence of a series when the ratio test is inconclusive?
Use the ratio or the root test to show convergence or divergence of the following series. If inconclusive, use another test:
$$\sum_{n=1}^{\infty}\frac{n!}{n^{n}}$$
So my first instinct was ...
1
vote
1answer
32 views
Simpsons rule & Lagrange?
What is the relation between Lagrange interpolation and Simpson's rule to integrate some function with some points $x_0,f(x_0)$; ... $x_n, f(x_n)$ ?
2
votes
0answers
33 views
Problem solving Integration prob
I have a problem with solving this integration problem:
$\int\limits_0^b x^{n-1}e'(t(x))dx$
I tried to solve it through partial integration by setting:
$f'(x) = e'(t(x)), f(x) = e(t(x)), g(x) = ...
3
votes
9answers
152 views
Finding the definite integral $\int_0^1 \log x\,\mathrm dx$
$$\int_{0}^1 \log x \,\mathrm dx$$
How to solve this? I am having problems with the limit $0$ to $1$. Because $\log 0$ is undefined.
2
votes
2answers
36 views
Binomial expansion of expression with numerator and denominator both linear equations of x
How can we expand the following by the binomial expansion, upto the term including $x^3$? That'll be 4 terms.
This the expression to be expanded: $\sqrt{2+x\over1-x}$
I understand how to do the ...
5
votes
1answer
71 views
Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$?
Find the limit $\displaystyle\lim_{n\rightarrow\infty}{(1+1/n)^{n^2}e^{-n}}$?
I found the limit as $e^{-1/2}$ using l'Hospital rule. I guess I made a mistake. Because the limit seems to be 1. Also, ...
1
vote
2answers
44 views
Area between $y=e^{-x}$ and $y=e^{-x}\sin x$
Let $x_n$ denote the $x$-coordinate of the $n$th point of contact between the curves $y=e^{-x}$ and $y=e^{-x}\sin x$, with $0<x_1<x_2<\cdots$, and let $A_n$ denote the area of the region ...
0
votes
3answers
65 views
Evaluating the limit $\lim\limits_{x\rightarrow 0}\frac{\int_{0}^{\sin{x^2}}e^{(t^2)}dt}{x^2}$
I am to find the $$\lim_{x\rightarrow 0}\frac{\int_{0}^{\sin{x^2}}e^{(t^2)}dt}{x^2}$$ and I have no idea how to approach this. As its widely known, there's no way to integrate $e^{(x^2)}$, how should ...
0
votes
1answer
90 views
I don't understand this notation… - Series with ln
I found this notation in my book
$$
\sum\limits_{i=1}^n \ln^n3
$$
and I don't know how to interpret it.
Is it
$$
\sum\limits_{i=1}^n \ln((1^n)\cdot3)\;?
$$
And btw, how to check if this series ...
1
vote
1answer
44 views
Texture mapping and conformal transformation
I'm a 3D programmer with background in physics, interested to better know how texture mapping can be made using conformal maps for simple surfaces.
I want to texture map a paraboloid:
...
1
vote
2answers
83 views
Defining infinitesimals
Can such definition of infinitesimals hold?
$$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$
And, if the above definiton works, then obviously
...
4
votes
2answers
64 views
If $p(2x+1)=p(x^2)$ for all $x\in\mathbb{R}$, then $p\equiv\text{const.}$
Let $p\in \Bbb{R}[x]$ (polynomial) with $\deg(p)=n$. Suppose that $p(2x+1)=p(x^2)$ for all $x\in\mathbb{R}$. Prove that $p\equiv\text{const.}$
4
votes
3answers
72 views
Determine if $\sum^{\infty}_{n=1}\int^{\frac{\sin n}{n}}_0\frac{\sin x}{x} \, dx $ is converges or diverges.
Determine if $\sum^{\infty}_{n=1}\int^{\frac{\sin n}{n}}_0\frac{\sin x}{x} \, dx $ is converges or diverges.
Could someone please show me how to do it? And especialy showing why ...
-1
votes
1answer
23 views
How to calculate equation of normal to the curve which is parallel to another line
Find the equation of the normal to the curve y=3x^2-2x-1 which is parallel to the line y=x-3 .
Hi I'm having trouble figuring out when to use which gradients as initially the gradient you get is one ...
1
vote
2answers
107 views
Possible proof(?!): $\int_1^\infty \frac{1}{x} dx$ converges
I'm probably wrong, as I am not a calculus professor. But I hear that the
$$\int_1^\infty \frac{1}{x} dx$$
does NOT converge. because $\ln |x|$ approaches infinity but:
We know that the $\lim_{x \to ...
2
votes
2answers
51 views
Need help applying the root test
I'm not sure if I am doing something wrong, or not... I've got an answer but it doesn't look right to me.
Given the following series, determine if it is convergent or divergent using the root or ...
5
votes
0answers
51 views
Is there a bijection $f:\mathbb R^+\to\mathbb R^+$ s.t. $f'(x)=f^{-1}(x)$? [duplicate]
Is there a bijection $f:\mathbb R^+\to\mathbb R^+$ s.t. $f'(x)=f^{-1}(x)$?
And if there is, can I prove uniqueness? This problem troubles me.
