Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.
It has two major branches, differential calculus and integral calculus; the former concerns instantaneous rates of change, and the slopes of curves, while integral calculus concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence of infinite sequences and infinite series to a well-defined limit.Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Today, calculus has widespread uses in science, engineering, and economics.In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus (plural calculi) is a Latin word, meaning originally "small pebble" (this meaning is kept in medicine – see Calculus (medicine)). Because such pebbles were used for counting (or measuring) a distance travelled by transportation devices in use in ancient Rome, the meaning of the word has evolved and today usually means a method of computation. It is therefore used for naming specific methods of calculation and related theories, such as propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus.
My attempt:
I have proved (i), it is continuous since ##\lim_{(x,y)\rightarrow (0,0)}=f(0,0)##
I also have shown the partial derivative exists for (ii), where ##f_x=0## and ##f_y=0##
I have a problem with the directional derivative. Taking u = <a, b> , I got:
$$Du =\frac{\sqrt[3] y}{3 \sqrt[3]...
a) The Euler-Lagrange equation is of the form ## \frac{d}{dx}(\frac{\partial F}{\partial y'})-\frac{\partial F}{\partial y}=0, y(a)=A, y(b)=B ##.
Let ## F(x, y, y')=(y'^2+w^2y^2+2y(a \sin(wx)+b \sinh(wx))) ##.
Then ## \frac{\partial F}{\partial y'}=2y' ## and ## \frac{\partial F}{\partial...
Epsilontic – Limits and Continuity
I remember that I had some difficulties moving from school mathematics to university mathematics. From what I read on PF through the years, I think I’m not the only one who struggled at that point. We mainly learned algorithms at school, i.e. how things are...
Hi, PF
Trigonometric Integrals
"The method of substitution is often useful for evaluating trigonometric integrals" (Calculus, R. Adams and Christopher Essex, 7th ed)
Integral of cosecant...
I'll start with a characterization of the Green's function as a fundamental solution to a differential operator. This theorem is given in Ordinary Differential Equations by Andersson and Böiers.
##E(t,\tau)## is known as the fundamental solution to the differential operator ##L(t,D)##, also...
I just wanted to know if my solution to part (b) is correct. Here's what I did:
I took the partial derivative with respect to x and y, which gave me respectively.
Then I computed the partial derivatives at (-3,4) which gave me 3/125 for partial derivative wrt x and -4/125 for partial derivative...
I was recently recommended this book and told it was a standard textbook at an upper undergraduate level or lower graduate level. Well that's certainly above my level, but specifically what would be the prerequisites?
I've no formal math training but self taught calculus at a level somewhere...
(a)
$$\frac{ds}{dt}=|r'(t)|$$
$$=\sqrt{(x(t))^2+(y(t))^2+(z(t))^2}$$
$$=\frac{2}{9}+\frac{7}{6}t^4$$
$$s=\int_0^t |r'(a)|da=\frac{2}{9}t+\frac{7}{30}t^5$$
Then I think I need to rearrange the equation so ##t## is the subject, but how?
Thanks
Edit: wait, I realize my mistake. Let me redo
a) Observe that ## \frac{\partial}{\partial z}F(y, z)=y^{n-1}\cdot \frac{2z}{2\sqrt{y^2+z^2}}=\frac{zy^{n-1}}{\sqrt{y^2+z^2}} ##.
This means ## G(y, z)=\frac{z^2\cdot y^{n-1}}{\sqrt{y^2+z^2}}-y^{n-1}\cdot \sqrt{y^2+z^2}=\frac{z^2\cdot...
Proof:
(i) Consider the second-order linear differential equation ## \frac{d^2u}{dx^2}+\frac{fu}{2}=0, f=f(x) ##.
Then ## u''+\frac{f}{2}u=0\implies r^2+\frac{f}{2}=0 ##, so ## r=\pm \sqrt{\frac{f}{2}}i ##.
This implies ## u_{1}=c_{1}cos(\sqrt{\frac{f}{2}}x) ## and ##...
Here I want to address of the question if it is possible to make a sum over an uncontable set and discuss integration rules involving uncountably infinite constants.
I will provide introduction in very condensed form to get quicker to the essense.
Conservative part
First of all, let us...
Hi, PF
First I will quote it; next the doubts and my attempt:
"In mathematics, trigonometric substitution is the replacement of trigonometric functions for other expresions. In calculus, trigonometric substitution is a technique for evaluating integrals. (...)
Case I: Integrands containing...
I'm reading Ordinary Differential Equations by Andersson and Böiers, although this is more related to multivariable calculus. There is a Lemma regarding Lipschitz continuity which I have a question about. Below ##\pmb{f}:\mathbf{R}^{n+1}\to \mathbf{R}^n ## is a vector-valued function defined by...
In my opinion , if it can be shown that this is a monotonically bounded sequence, one can confirm that there is a limit.
First,we know $$ \frac{1-x^{4n}}{1+x^{2}}dx=(1-x^{2}) (1+x^{2}) ^{n-1}=(1-x^{4}) ^{n-1}(1+x^{2}).$$
According to the integral median theorem,we can get $$a_n=(2- \sqrt{3} )...
See my insights article for those interested in an unconventional approach to doing Precalculus at an accelerated pace and beginning Calculus.
It is different from the usual way that a precalculus is done text in that it covers in the US what is called Algebra 1, Geometry, Algebra 2, and...
Let's see how messy it gets...
##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
So, first of all, why and how are we taking the derivative of the vector r or s as d/dt if t is not a parameter of the equations?
Second question is what is the difference between d/dt(r) and d(theta)/dt(r) and also between d/dt(s) and d(theta)/dt(s)? Like, both of these appear at the bottom of...
I have tried to use the natural Taylor expansion of ln(t+1)
and working with long division but I get the result 1/t+1/2+t/12−t^2/24+⋯
instead of 1/t+1/2-t/12+t^2/24+... that is the right result
I have tried to do this several time but still don't works. Is there a miscalculation in my long...
As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/
show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
Note that ## \frac{\partial F}{\partial x}=\frac{2x}{2\sqrt{x^2+y'^2}}=\frac{x}{\sqrt{x^2+y'^2}}, \frac{\partial F}{\partial y}=0, \frac{\partial F}{\partial y'}=\frac{2y'}{2\sqrt{x^2+y'^2}}=\frac{y'}{\sqrt{x^2+y'^2}} ##.
Now we have ## \frac{dF}{dx}=\frac{\partial F}{\partial x}+\frac{\partial...
My interest is only on the highlighted part, i can clearly see that they made use of chain rule i.e
by letting ##u=1+x^2## we shall have ##du=2x dx## from there the integration bit and working to solution is straightforward. I always look at such questions as being 'convenient' questions.
Now...
I am on differential equations today...refreshing.
Ok, this is a pretty easier area to me...just wanted to clarify that the constant may be manipulated i.e dependant on approach. Consider,
Ok I have,
##\dfrac{dy}{6y^2}= x dx##
on integration,
##-\dfrac{1}{6y} + k = \dfrac{x^2}{2}##...
let ##k \in\mathbb{N},## Show that there is ##i\in\mathbb{N} ##s.t ##\ \left(1-\frac{1}{k}\right)^{i}-\left(1-\frac{2}{k}\right)^{i}\geq \frac{1}{4} ##
I tried to use Bernoulli's inequality and related inequality for the left and right expression but i the expression smaller than 1/4 for any i...
We have ##L(v^2 + 2v\epsilon + \epsilon^2)##. Then, the book proceeds to mention that we need to expand this in powers of ##\epsilon## and then neglect the terms above first order, we obtain:
##L(v^2) + \frac{\partial L}{\partial v^2}2v\epsilon## (This is what I don't get).
We know taylor is...
Hi, PF
I am going to reproduce the introduction of the textbook; then the Theorem:
The method of substitution cannot be forced to work. There is no substitution that will do much good with the integral ##\int{x(2+x^7)^{1/5}}\,dx##, for instance. However, the integral...
I'll be starting Apostol's calculus book in a little less than two weeks, as I'm finishing up an easier textbook. I looked ahead and I see that there is a four part introduction, and I was wondering if I could just skip that. I briefly skimmed it, and it just looks like a review of summation...
Place hemisphere in xyz coordinates so that the centre of the corresponding sphere is at the origin.
Then notice that the centre of mass must be at some point on the z axis ( because the 4 sphere segments when cutting along the the xz and xy planes are of equal volume)
y2 + x2 = r2
We want two...
I have to write taylor expansion of f(x)=arctan(x) around at x=+∞.
My first idea was to set z=1/x
and in this case z→0
Thus I can expand f(z)= arctan(1/z) near 0
so I obtain 1/z-1/3(z^3)
Then I try to reverse the substitution but this is incorrect .I discovered after that...
Hi, PF, I hope the doubts are going to be vanished in a short while:
This is the limit of Riemann Sum
##\displaystyle\lim_{n\rightarrow{\infty}}\displaystyle\frac{1}{n}\displaystyle\sum_{j=1}^{n}\cos\Big(\displaystyle\frac{j\pi}{2n}\Big)##
And this is the definition of the limit of the General...
My first point of reference is:
https://math.stackexchange.com/questions/154968/is-there-really-no-way-to-integrate-e-x2
I have really taken time to understand how they arrived at ##dx dy=dA=r dθ dr## wow! I had earlier on gone round circles! ...i now get it that one is supposed to use partial...
Hi, PF
There goes the solved example, the doubt, and the attempt:
Example 8 Solve the integral equation ##f(x)=2+\displaystyle\int_4^x\,f(t)dt##.
Solution Differentiate the integral equation ##f'(x)=3f(x)##, the DE for exponential growth, having solution ##f(x)=Ce^{3x}##. Now put ##x=4## into...
Welcome to this month's math challenge thread!
Rules:
1. You may use google to look for anything except the actual problems themselves (or very close relatives).
2. Do not cite theorems that trivialize the problem you're solving.
3. Have fun!
1. (solved by @AndreasC) I start watching a...
Hi,PF
The book is "Calculus" 7th ed, by Robert A. Adams and Christopher Essex. It is about an explained example of the first conclusion of the Fundamental Theorem of Calculus, at Chapter 5.5.
I will only quote the step I have doubt about:
Example 7 Find the derivatives of the following...
Hi,
unfortunately I have problems with the task d and e, the complete task is as follows:
I tried to form the derivative of the equation ##f(x)##, but unfortunately I have problems with the second part, which is why I only got the following.
$$\frac{d f(x)}{dx}=f_0 g(x) \ exp\biggl(...
My take:
$$\int_{x^2}^{2x} \sin t \, dt$$
using the fundamental theorem of calculus we shall have,
$$\int_{x^2}^{2x} \sin t \, dt=-2x \sin x^2 +2 \sin 2x$$
I also wanted to check my answer, i did this by,
$$\int [-2x \sin x^2 +2 \sin 2x] dx$$
for the integration of the first part i.e...
Hi, there. I am reading this thesis. On page 146, it reads that
I do not know how to calculate the limits when they are viewed as distributions. I am trying to integrate a test function with the limits. So I try (##Q## is defined as ##Q>0##) $$\lim_ {r\rightarrow \infty} \int_{0}^\infty dQ...
I remember taking Physics in high school, so I guess it is possible, but it's been so long ago, I can't remember. It just seems that Calculus is indispensable when teaching Physics topics, except for a few like heat expansion or geometric optics. I would imagine that there is a lot of Δ this &...
part (a)
The number of Bees per Wildflower plant.
part (b)
##\dfrac{dB}{dF}= \dfrac{dB}{dt} ⋅\dfrac{dt}{dF}####\dfrac{dB}{dF}=\left[\dfrac{2-3\sin 3t}{5e^{0.1t}}\right]##
##\dfrac{dB}{dF} (t=4)= 0.4839##part (c)
For values of ##t>12## The number of Bees per wildflower plants reduces...
Part (a) no problem...chain rule
##\dfrac{dy}{dx}= (2x+3)⋅ e^{x^2+3} =0##
##x=-1.5##
For part b,
We need to determine and check if ##\dfrac{d^2y}{dx^2}>0##
...
##\dfrac{d^2y}{dx^2}=e^{x^2+3x} [(2x+3)^2+2)]##
Now any value of ## x## will always give us, ##\dfrac{d^2y}{dx^2}>0##
The other...
Hi, PF
The aim is to prove how the approach from the left and right sides of the ##x##x axis eventually renders a vertical asymptote for the function ##\frac{1}{x}##. I've been searching in the textbook "Calculus", 7th edition, by Robert A. Adams and Christopher Essex, but I haven't found...
Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
By the Euler's equation of the functional, we have
## J(\mathrm u)=\int ((\mathrm{u})^{2}+e^{\mathrm{u}}) \, dx ##.
Then ## J(\mathrm{u}+\epsilon\eta)=\int ((\mathrm{u}'+\epsilon\eta')^{2}+e^{\mathrm{u}+\epsilon\eta}) \, dx=\int...
For exercise 3 (2),
,
The solution for finding the eigenvector is,
However, I am very confused how they got from the first matrix on the left to the one below and what allows them to do that. Can someone please explain in simple terms what happened here?
Many Thanks!
Hi. I'm self-studying functions which relate to calculus. Let me post what I feel I know and what I'm not grasping yet. Please correct any mistakes I'm making.
I'm just talking real numbers: A function is a rule that takes an input number and sends it to another number. We can describe it...
Why not use these number systems, in place of the real number system, when these allow us to divide by infinity exactly?
According to these, division by infinity equals exactly zero! No need for calculus limits, which only can say it approaches zero when tending towards infinity...