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. TheAnt

    Calculus Any suggestions for good beginner calculus textbooks.

    Hi, i am looking for a good textbook for someone who knows nothing about calculus.
  2. H

    I Prove Pappus's centroid theorems without calculus

    Pappus's centroid theorems were discovered 17 centuries ago, when calculus wasn't invented yet. How are these theorems proved without using calculus? "The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C...
  3. JasMath33

    A Is This Proof Correct? Ask & Discuss Here

    I was reading this book yesterday and looking at this proof/justification. I was thinking it is possibly incorrect, but wanted to get some other opinions. Here is the example they gave in the book with the work attached.
  4. JasMath33

    Proving Contractive Homework: Tips & Solutions

    Homework Statement I worked on this question and I made it so far, and now I am stuck on how to finish it. Here is the problem and below I will explain what I attempted. Homework EquationsThe Attempt at a Solution I know looking at the last part about using previous homework, I want to...
  5. MrDickinson

    I Can someone me simplify this expression....

    lim_(h->0^-) (e^(x+h)/((x+h)^2-1)-e^(x+h)/(x^2-1))/h = -(2 e^x x)/(x^2-1)^2 I know how to differentiate the expression using the quotient rule; however, I want to use the limit definition of a derivative to practice it more.This desire to practice led me into a trap! Now I just can't simplify...
  6. B

    MHB What is the relationship between P and G(t) in Calculus with Exponential?

    For P=P_0\times e^{G(t)} and G'(t) = a+bt Show G(0)=0 I get G(t) = \int a+bt ~dt = at+\frac{1}{2} b t^2+C therefore G(0)=C
  7. JasMath33

    Lipschitz Q: Show w/ Example & Derivative

    Homework Statement Homework EquationsThe Attempt at a Solution I know I will just have to show this by one example. I thought about using f(x) = x2 but I am not sure if this satisfies the last part dealing with the absolute value of the derivative. It is just the last part on which I am stuck.
  8. JasMath33

    Using the Intermediate Value Theorem to Find Fixed Points

    Homework Statement Homework EquationsThe Attempt at a Solution I started looking at this problem and I think I am going to have to use the intermediate value theorem for this proof, but I am not quite sure. I started looking at possible examples of these functions, but I know this is not good...
  9. U

    What do I need to master before I start calculus?

    I want to start learning calculus I. But, I don't know what courses I need to finish before starting precalculus. any help would be appreciated.
  10. MichPod

    Calculus Need calculus of variations book for a laymen

    While trying to study textbooks on analytical mechanics or QFT I realized that I simply cannot operate with variations of functions in the same way I can operate with derivatives and integrals. I have never learned calculus of variations in university and, frankly, I am not much interested in...
  11. JasMath33

    FDQR ( Freshman Dream Quotient Rule)

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  12. JasMath33

    Intermediate Value Property for Discontinuous Functions

    Homework Statement ' Here is the given problem Homework EquationsThe Attempt at a Solution a. For part a, I felt it was not continuous because of the sin(1/x) as it gets closer to 0, the graph switches between 1 and -1. Then I felt it might be continuous, therefore I am not sure. b. For part...
  13. JasMath33

    Help Solve Calculus Limit Proof Homework Statement

    Homework Statement I am posting this for another student who I noticed did not have the proof in the problem. Here is what she said. Let's try and help her out. I have been working on the problem below and I am stuck. I am stuck primarily because of the part where is says x=0. If x-0, it...
  14. Shafia Zahin

    Calculus A good book on Calculus to clear the concepts

    Hi, I want to know about some books that can help me to clear my concepts of calculus.I know the basics of differentiation and integration,but don't have clear idea about it.Well,I would like to buy such a book that has explained the rules and methods of calculus properly.Also,I want to know...
  15. mr.tea

    Calculus Question/Advice regarding multivariable calculus

    Hi, I want to re-learn multivariable calculus, after I have learned it, not in the best possible way... and feel bad about it. I have seen the recommendations here about Hubbard/Shifrin/Fleming/Edwards. I have also seen the books by Munkres/Spivak/Apostol. I didn't really like Hubbard's book...
  16. A

    Hi guys, introducing myself

    My name is Amit Gupta and I am a teacher of physics and calculus, I have experienced in physics last several years, I love to solve physics equations, Anyone who have problems in physics and calculus subjects, I will provide you a right solution through problems practice. We are working towards...
  17. N

    MHB Using "calculus" in a sentence

    In English grammar, we typically refer to specific branches of mathematics without a prefixed article; e.g., solve using algebra, trig., etc.. By contrast, we often prefix calculus with the definite article, "the". For instance, "solve via the calculus". Can anyone explain this exception to...
  18. M

    Convergence of series (Theoretical Question)

    1. The problem: Ive been all afternoon struggling with this doubt. Its a bit more teoric than the rest of the exercices i did and i just can't seem to get around it so here it goes ...
  19. S

    Calculate y^(p+q) using the Leibniz formula

    Homework Statement Statement:[/B] Let y = u(x)v(x). a) Find y' , y'', and y''' b) The general formula for yn, the n-th derivative, is called Leibniz’ formula: it uses the same coefficients as the binomial theorem , and looks like https://i.gyazo.com/53728964c6b3ef142fd70f600c29e037.png Use...
  20. X

    I Is this general solution correct?

    Question : General solution of dy/dt = -ay + b My solution : dy/dt = -a(y-b/a) (dy/dt)/(y-(b/a)) = -a Integrating both sides : ln | y-(b/a) | = -at + C e(-at+C) = y-(b/a) Ce(-at) = y-(b/a)
  21. samgrace

    I Cauchy's Integral Test for Convergence

    Hello, I am want to prove that: $$ \sum_{1}^{\infty} \frac{1}{n^{2} + 1} < \frac{1}{2} + \frac{1}{4}\pi $$ Cauchy's Convergence Integral If a function decreases as n tends to get large, say f(x), we can obtain decreasing functions of x, such that: $$ f(\nu - 1) \geqslant f(x) \geqslant...
  22. KF33

    I What is the relationship between contractive and Lipschitz functions?

    I was looking at this definition of a contractive function and the only difference I saw between it and a Lipschitz function was the b and M. I am just wondering how you look at the connections between them.
  23. KF33

    I Continuous Functions with Piecewise Functions

    I have been working on this exercise 5 and kind of stuck how to start the problems. I would think to start with a graph, but I feel this is wrong. I am just stuck on a and b.
  24. A

    Simple harmonic motion (with calculus)

    Homework Statement An object with mass m undergoes simple harmonic motion, following 2 perpendicular directions, described by the equations: x=a cos (wt), a>0, y=b cos (2wt), b>0 a) find the equation of the trajectory b) find the speed at any given time (so having t as a variable) c) the...
  25. K

    Work done raising an object. Calculus needed.

    Homework Statement A 50-meter rope weighing 2 N/m supports a piano weighing 600 N. Find the work done in lifting the piano 25 meters 2. Relevant equation. None of the calculus equation, techniques covers this type of question with mass/length involved.The Attempt at a Solution I know the...
  26. R

    Mechanics/Kinematic Calculus

    A particle P moves on the x-axis. At time t seconds the velocity of P is V m/s in the direction of x increasing?o V is given by V={8t-3/2t²} for 0≤t≤4 Also V={16-2t} for t>4 Why does the acceleration have to equal zero to get the greatest speed of P in interval 0≤t≤4
  27. F

    B Vector Calculus Identity proof?

    The following identity is found in a book on Turbulence: Can someone provide a proof of this identity? It isn't listed in the list of vector calculus identities on Wiki. Thanks
  28. Greg Bernhardt

    Teaching statistics before calculus?

    Here is a short (and old) TED talk where a mathematics professor suggests we teach stats and probability in depth before teaching Calculus because it's math that is more relevant to a wider range of people. Have we got our math curriculum wrong? Thoughts?
  29. RubiksMelia

    Constant of Integration when calculating displacement

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  30. S

    B Question about the math involved in E-field calculations

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  31. P

    Intro Physics Which book to read the summer before starting Physics?

    I'm starting my 1st year of Physics at university in September. Although I've learned a lot of single variable Calculus and various topics of Physics this year, I'd like to get a general overview of the topics touched in a 1st Physics course at uni. I just wonder if there's an equivalent to...
  32. ShayanJ

    A Neumann boundary conditions in calculus of variations

    In calculus of variations, extremizing functionals is usually done with Dirichlet boundary conditions. But how will the calculations go on if Neumann boundary conditions are given? Can someone give a reference where this is discussed thoroughly? I searched but found nothing! Thanks
  33. C

    Integration by Parts: Solving Integrals with √(1+x^2) and x

    Homework Statement [/B] Homework Equations ∫ f(x) g'(x) dx = f(x) g(x) - ∫ f '(x) g(x) dx f(x)=√(1+x^2) f '(x)=x * 1/√(1+x^2) g'(x)=1 g(x)=x The Attempt at a Solution ∫ √(1+x^2) * 1 dx =x * √(1+x^2) - ∫ x^2 * 1/√(1+x^2) dx Further integration just makes the result look further from what...
  34. A

    MHB Business Calculus Project

    /files/483341/math134_project1.docx /files/483341/Technical_Worksheet_for_project1.docx If the links doesn't work, here is a copy. Project on Math 134 Business Calculus, Summer 2015-2016 Project Due: 07 June 2016 Overview: The company you are working for has decided to put on a banquet to...
  35. G

    B How to Read the Index of a Calculus Textbook?

    What does "ff" mean (please see image below)?
  36. G

    I Question on Calculus of variations formalism

    Hi all, I had a quick question regarding the formalism behind calculus of variations. In one-dimensional standard calc, we consider functions f:\mathbb{R}\to \mathbb{R} and define their derivatives using the conventional definition with the limit of the quotient of the change in the function...
  37. H

    Vector calculus identity using index and comma notation

    Homework Statement Use index and comma notation to show: \begin{equation*}\text{div }(\text{curl } \underline{\bf{v}}) = 0\end{equation*} Homework Equations \begin{align*} & \text{(1) div } \underline{\bf{v}} = v_{i,i} \\ & \text{(2) curl } \underline{\bf{v}} = \epsilon_{ijk} v_{j,i}...
  38. T

    Function f(x,y,z) of three variables becomes z = g(x,y)

    Hello everyone, I have a theoretical calculus question. I am working on a exercise where you have to consider f(x,y,z) and express the variable z as a function of x and y on a certain level surface around a certain (x0,y0,z0). I found out that the condition for this to be able is that the...
  39. M

    A Tensor Calculus and Divergence

    Hi PF! I have a question on the dyadic product and the divergence of a tensor. I've never formally leaned this, although I'm sure it's published somewhere, but this is how I understand the operators. Can someone tell me if this is right or wrong? Let's say I have some vector ##\vec{V} = v_x i +...
  40. A

    How do integrals actually work?

    Homework Statement Calculate the electrical energy required to assemble a spherical volume of radius R and charge Q, homogeneous density ρ the answer is (3/5)Q/R the textbook says you have to build the volume integral one layer of sphere at a time, I'll get back to that later. I like...
  41. P

    Calculus Assignment Help: Temperature & Daylight Equations Explained

    Homework Statement [/B] 1. When a certain object is placed in an oven at 540°C, its temperature T(t) rises according to the equation T(t) = 540(1 – e^–0.1t), where t is the elapsed time (in minutes). What is the temperature after 10 minutes and how quickly is it rising at this time? I have...
  42. S

    Line integral convert to polar coordinates

    Homework Statement I need to find the work done by the force field: $$\vec{F}=(5x-8y\sqrt{x^2+y^2})\vec{i}+(4x+10y\sqrt{x^2+y^2})\vec{j}+z\vec{k}$$ moving a particle from a to b along a path given by: $$\vec{r}=\frac{1}{2}\cos(t)\vec{i}+\frac{1}{2}\sin(t)\vec{j}+4\arctan(t)\vec{k}$$ The Attempt...
  43. P

    Solving Calculus Doubts: My Attempt

    The problem, and my attempt to solve it is here: https://meocloud.pt/link/bf620660-aaf0-48ad-abfe-ac49f7ea0843/calculus_doubt.pdf/
  44. nfcfox

    Calculating Bungee Jumping Acceleration: Physics and Calculus Problem Solutions

    Homework Statement So this involves a little physics but it is to all be done using calculus (They give us everything we need). Everything is in terms of feet so acceleration due to gravity is 32 ft/(s^2) they tell us the mass of the jumper is 5 slugs. They then tell us that the Force due to...
  45. NatFex

    I Sum of Probability Density Function > 1?

    I have a Stats exam on Wednesday and while I thought I was quite well-versed, I've gone back over to the very basics only to find myself confused at what should be introductory. Suppose I have a continuous random variable modeled by a probability density function: $$f(x)=2x$$ Obviously the...
  46. S

    MHB Propositional Calculus: Is Algebra's Problem-Solver? Example

    Is there to every problem in Algebra a corresponding problem in propositional calculus?? Give an example
  47. Eclair_de_XII

    Courses Would you need some background before calculus physics?

    Let's say this person has no background in the subject whatsoever; he has not taken a single physics class in his entire life. How would he fare in calculus-based physics, in your opinion? I'm sure I've already made this topic, and asked this question before, but I can't seem to find it on this...
  48. A

    Bead sliding on a wire - calculus of variations

    I am asked to find the shape of a wire that will maximize the speed a sliding bead when it reaches the end point(Similar to the brachistochrone problem expect that the speed is to be maximized and not time minimized). But shouldn't the speed at the end be independent of the shape of the wire...
  49. G

    Algebraic equation -- six degree polynomial

    Homework Statement Solve the equation for r,r>0,r<R. \frac{-2\pi R^3}{3}-\frac{8\pi r^2\sqrt{R^2-r^2}}{3}+2r^2R+\frac{2\pi}{3}R^2\sqrt{R^2-r^2}=\frac{2\pi R^3}{3} 2. The attempt at a solution After factoring, -2R(2\pi R^2-3r^2)=2\pi\sqrt{R^2-r^2}(4r^2-R^2). After squaring, 64\pi^2...
  50. R

    Calculus Hubbard and Hubbard for multivariate calculus

    I've taken the single and multivariate calculus classes at my school, (college class offered in my high school for accelerated students). I'm currently a junior, but in the summer of my senior year, I plan to read a book on proofs, brush up on math glossed over by the american education system...
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