Calculus Definition and 1000 Threads

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 &...

For part (a), Using partial fractions (repeated factor), i have... ##7e^x -8 = A(e^x-2)+B## ##A=7## ##-2A+B=-8, ⇒B=6## $$\int {\frac{7e^x-8}{(e^x-2)^2}}dx=\int \left[{\frac{7}{e^x-2}}+{\frac{6}{(e^x-2)^2}}\right]dx$$ ##u=e^x-2## ##du=e^x dx## ##dx=\dfrac{du}{e^x}## ... also ##u=e^x-2##...

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...

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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...

TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift? Say you were given the equation : How would you find : with a calculator that can only add, subtract, multiply, divide Is there a general formula?

I'm struggling with section a. This is my calculation: The expression remains depend on the variable t, while in the answer is a concrete number:

Hi, so I know how to find domain but how about range in this problem? I don't understand the way he did it? I always get answers wrong when it comes to range.

EXAMPLE 4 Find the area of the region ##R## lying above the line ##y=1## and below the curve ##y=5/(x^2+1)##. Solution The region ##R## is shaded in Figure 5.24. To find the intersections of ##y=1## and ##y=5/(x^2+1)##, we must solve these equations simultaneously: ##1=\frac{5}{x^2+1}## so...

I'm given the wavefunction and I need to find the normalization constant A. I believe that means to solve the integral The question does give some standard results for the Gaussian function, also multiplied by x to some different powers in the integrand, but I can't seem to get it into...

Hi, With respect to the techniques mentioned in point 2 and 3: Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques. Below excerpt shows problems 4 and 5 referenced in the above 2 points:

Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks

(a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##. I also tried to calculate the limit directly (not using geometrical argument at...

I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...

I'm talking about the x^(-4/3) how does it equal 0 when x approch infinite?? so I can use L'Hopital's Rule

Hello, I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?

For this, I first try to work out where function is increasing My working is ##f'(x) = 12x^3 - 12x^2 - 24x## For increasing, ##12x(x^2 - x - 2) > 0## ##12x > 0## and ##(x - 2)(x + 1) > 0## ##x > 0## and ##x > 2## and ##x > -1## However, how do I combine those facts into a single domain...

Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...

For this, Does someone please know why we are allowed to take limits of both side [boxed in orange]? Also for the thing boxed in pink, could we not divide by -h if ##h > 0##? Many thanks!

For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!

When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...

For this problem, Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)? Many thanks!