Calculus Definition and 1000 Threads

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. 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(...
  2. 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...
  3. 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...
  4. 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 &...
  5. 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##...
  6. 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...
  7. 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...
  8. mcastillo356

    B Vertical asymptote with an epsilon-delta proof?

    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...
  9. Silvia2023

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

    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...
  10. 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...
  11. bhobba

    Insights What Are Infinitesimals – Simple Version

    Continue reading...
  12. M

    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!
  13. 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...
  14. 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...
  15. PeaceMartian

    How to find integrals of parent functions without any horizontal/vertical shift?

    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?
  16. I

    Calculus Problem: Blowing Up a Spherical Balloon

    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:
  17. Mohmmad Maaitah

    How to find range inside square root

    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.
  18. mcastillo356

    B Why is this definite integral a single number?

    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...
  19. casparov

    Help Solve for the normalization constant of this QM integral

    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...
  20. 1

    Integration Substitution Techniques for quadratic expressions under square roots

    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:
  21. S

    Solving this definite integral using integration by parts

    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
  22. S

    Radii of stacked circles inside the graph of y = |x|^1.5

    (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...
  23. carlsondesign

    I What is the official name for a Field Series in mathematics/physics?

    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...
  24. Mohmmad Maaitah

    L'Hopital's Rule case: How does x^(-4/3) equal 0 when x approches infinity?

    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
  25. P

    I Tensor Calculus (Einstein notation)

    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?
  26. M

    Finding where this function is increasing or decreasing

    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...
  27. A

    Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

    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...
  28. M

    Why Can We Take Limits of Both Sides? [Answered]

    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!
  29. M

    Proof of ##M^n## (matrix multiplication problem)

    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!
  30. bhobba

    B How can hyperreal numbers make infinitesimals logically sound in calculus?

    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...
  31. M

    Why is continuity necessary before applying the Extreme Value Theorem?

    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!
  32. M

    Why Is the Chain Rule Not Used in Differentiating h(x) = 3f(x) + 8g(x)?

    For part(a), The solution is, However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)## Many thanks!
  33. Sailor Al

    I Can Air Resist a Piston? A Serious Discussion

  34. M

    Intro Math Wanting to study Functions from scratch for my Calculus preparation

    Does Precalculus by James Stewart teaches function from scratch ?
  35. Argonaut

    Maximizing Range/Time in Air of an Airplane: Solving with Calculus

    Is my solution correct? (I only have answers to odd-numbered exercises.) Is it a good solution or have I overcomplicated things? (a) The forward force provided by the engine balances the air resistance force, so ##F_{engine}=F_{air} = \alpha v^2 + \beta /v{^2}##. Let ##W_{engine}## be the...
  36. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    We were taught that in cylindrical coodrinates, the position vector can be expressed as And then we can write the line element by differentiating to get . We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
  37. bhobba

    Unlock the Power of Calculus: Algebra 1 to Boaz for Students

    Here is an interesting book a student could do after after Algebra 1, or even integrate into an Algebra 1 course: https://www.amazon.com/dp/B077VV95N3/?tag=pfamazon01-20 And a website: https://www.calculussolution.com/ Several topics become easier, such as logarithms, when you know a...
  38. YAYA12345

    I Integral Bee Preparation -- Trouble with this beautiful integral

    While I was preparing for an integrals contest, I had a doubt about the following integral, I tried several substitutions but nothing worked.I would appreciate your support for this beautiful integral. $$ \int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...) \ \mathrm{d}x$$
  39. MatinSAR

    Vector Calculus in 1D: ± to Show Magnitude?

    [mentor's note - moved from one of the homework help forums] Homework Statement:: It's a question. Relevant Equations:: Vector calculus. Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector? ± can show possible directions in one dimension and...
  40. WMDhamnekar

    Stochastic Calculus: Conditional Expectation

    Are my following answers correct?
  41. WMDhamnekar

    Stochastic Calculus: Conditional Expectation

  42. Demystifier

    A Found a new formula of Dirac calculus

    I have found a new formula in Dirac calculus. The formula is elementary, so probably I'm not the first who found it. Yet, I have never seen it before. As many other formulas in Dirac calculus, it is not rigorous in the sense of functional analysis. Rather, it is a formal equality, which is only...
  43. A

    Calculus Calculus book between Stewart & Spivak levels

    Hi, Are there calculus books that lie between Stewart (or Thomas) level and Spivak (Courant/Apostol) level? Thanks.
  44. mcastillo356

    I Express the limit as a definite integral

    Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt: (i) General Riemann Sums Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
  45. T

    Calculus Multivariable calculus PDF books

    Multivariable calculus is a branch of mathematics that extends the concepts of single-variable calculus to functions of multiple variables. In this subject, vectors and partial derivatives are introduced to represent and manipulate multi-dimensional data. The gradient of a function represents...
  46. binbagsss

    I Using the Chain Rule for Vector Calculus: A Tutorial

    This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example ## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ## and not ## 2 \bf{v} \cdot \nabla \bf{v} ## Is there an obvious way to see / think of this...
  47. mcastillo356

    I The Basic Area Problem (introduction to the topic of integrals)

    Hi PF There goes the quote: The Basic Area Problem In this section we are going to consider how to find the area of the region ##R## lying under the graph ##y=f(x)## of a nonnegative-valued, continous function ##f##, above the ##x##-axis and between the vertical lines ##x=a## and ##x=b##, where...
  48. R

    I Multivariable function optimization inconsistency

    Mentor note: For LaTeX here at this site, don't use single $ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane. I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
  49. V

    Is it possible to find the range of this function?

    I can get the domain, but getting the range seems impossible. Domain $$x-5=0$$ $$x =5$$ $$\therefore x \in (- \infty ,5) \cup (5, + \infty)$$ Range  I can simplify the function to the form below, but I don't know how to go from there. $$ f(x)= x + 5 + \frac {1}{x-5}$$  
  50. V

    Limit question to be done without using derivatives

    I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...
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