What is Calculus: Definition and 1000 Discussions

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.

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  1. S

    Existence of directional derivative

    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]...
  2. S

    Find f(x,y) given partial derivative and initial condition

    My attempt: $$\frac{\partial f}{\partial x}=-\sin y + \frac{1}{1-xy}$$ $$\int \partial f=\int (-\sin y+\frac{1}{1-xy})\partial x$$ $$f=-x~\sin y-\frac{1}{y} \ln |1-xy|+c$$ Using ##f(0, y)=2 \sin y + y^3##: $$c=2 \sin y + y^3$$ So: $$f(x,y)=-x~\sin y-\frac{1}{y} \ln |1-xy|+2 \sin y + y^3$$ Is...
  3. M

    Determine whether ## S[y] ## has a maximum or a minimum

    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...
  4. fresh_42

    Insights Epsilontic – Limits and Continuity

    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...
  5. mcastillo356

    B Integral of cosecant function: understanding different approaches

    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...
  6. P

    How to get general solution via Green's function?

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  7. S

    Multivariable calculus problem involving partial derivatives along a surface

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  8. J

    Introduction to Modern Astrophysics -- What are the prerequisites?

    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...
  9. S

    Unit tangent vector and curvature with arc length parameterization

    (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
  10. M

    Calculus ## G(y, z)=z\frac{\partial}{\partial z}F(y, z)-F(y, z) ##?

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  11. M

    Proofs about the second-order linear differential equation?

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  12. A

    A Summing over continuum and uncountable numerocities

    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...
  13. mcastillo356

    B Trigonometric substitution, a case I'd like to share

    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...
  14. P

    I Lipschitz continuity of vector-valued function

    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...
  15. KungPeng Zhou

    A question about definite integrals and series limits

    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} )...
  16. KungPeng Zhou

    How Do You Solve Calculus Homework Problem Six Using LaTeX?

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  17. bhobba

    Insights Beginner's Guide to Precalculus, Calculus and Infinitesimals

    [url="https://www.physicsforums.com/insights/precalculus-calculus-and-infinitesimals/"]Continue reading...
  18. bhobba

    Insights Precalculus, Calculus and Infinitesimals

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  19. chwala

    Find the derivative of the given function

    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}##...
  20. Anonymous001

    I Difference between d/dt and d(theta)/dt? Why is it dr or ds/dt?

    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...
  21. L

    I Taylor series of 1/ln(t+1) at t=0

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  22. I

    Show that the given function is decreasing

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  23. M

    Can anyone please verify/confirm these derivatives?

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  24. chwala

    Solve the given differential equation

    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...
  25. chwala

    Solve the given differential equation

    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}##...
  26. I

    Let k∈N, Show that there is i∈N s.t (1−(1/k))^i − (1−(2/k))^i ≥ 1/4

    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...
  27. G

    Show that the Taylor series for this Lagrangian is the following...

    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...
  28. mcastillo356

    I Substitution in a definite integral

    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...
  29. A

    Calculus Can I skip the introduction in Apostol's calculus?

    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...
  30. akrill

    Finding the Centre of Mass of a Hemisphere

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  31. L

    I Taylor expansion of f(x)=arctan(x) at infinity

    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...
  32. mcastillo356

    B I don't recognize this limit of Riemann sum

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  33. chwala

    I Integration of ##e^{-x^2}## with respect to ##x##

    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...
  34. mcastillo356

    I Solve Int. Eq.: Exponential Growth Diff. Eq.

    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...
  35. Infrared

    Challenge Math Challenge - July 2023

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  36. mcastillo356

    Calculus Confusion over Calculus Book example footnote

    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...
  37. L

    Linear first-order differential equation with an initial condition

    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(...
  38. chwala

    Differentiate the given integral

    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...
  39. Haorong Wu

    I Calculate limits as distributions

    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...
  40. S

    How is Physics taught without Calculus?

    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 &...
  41. chwala

    Solve the given problem that involves integration

    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##...
  42. chwala

    Rate of Change: Bees in Wildflower Meadow (a-c)

    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...
  43. chwala

    Show that the graph is convex for all values of ##x##

    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...
  44. mcastillo356

    B Vertical asymptote with an epsilon-delta proof?

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  45. Silvia2023

    For this Partial Derivative -- Why are different results obtained?

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  46. M

    Is this the correct way to find the Euler equation (strong form)?

    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...
  47. bhobba

    Insights What Are Infinitesimals – Simple Version

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  48. C

    Find Eigenvalues & Eigenvectors for Exercise 3 (2), Explained!

    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!
  49. R

    B Functions which relate to calculus: Questions about Notation

    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...
  50. jaketodd

    B Dividing by infinity, exactly, finally!

    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...
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