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. CollinsArg

    Hard Double Integral Homework: Solve & Understand

    Homework Statement I'm given the integral show in the adjunct picture, in the same one there is my attempt at a solution. Homework Equations x = r.cos(Θ) y = r.sin(Θ) dA = r.dr.dΘ The Attempt at a Solution [/B] I tried to do it in polar coordinates, so I substituted x=r.cos(Θ) y=r.sin(Θ) in...
  2. L

    Integration of even powers of sine and cosine

    Homework Statement Homework Equations cos2x = (1+cos2x)/2 sin2x = (1-cos2x)/2 The Attempt at a Solution I believe you would use the double angle formula repeatedly but that is very tedious; is there a more concise way to solve the problem?
  3. EastWindBreaks

    Why is Theta 2 Independent in Solving for Theta 3 in a 4-Bar Mechanism?

    Homework Statement Homework EquationsThe Attempt at a Solution it seems like because theta 2 is independent, therefor, you can solve theta 3 by just using one equation from the system of equation? on a previous problem where its a 4 bar mechanism( which it didn't specify that theta 2 is...
  4. D

    MHB Time Scale Calculus Research - Discuss with Fellow Researchers

    Anyone here do research with time scales (differential equations(dynamic equations) combining both the continuous and discrete). I know its more of a new topic from Hilger, but I think it is a new wave of modeling that will be prevalent in the future. If anyone is interested on the topic or...
  5. M

    Calculus 3 help -- Is the gradient of a plane the normal to the plane?

    Homework Statement Is the gradient of a plane, the normal to the plane? If so, why? Homework Equations No idea, just a question that popped up in my head eon of plane: n(x-x1)+n(y-y1)+n(z-z1) The Attempt at a Solution I found the partial derivative of each, and got the normal.[/B]
  6. S

    I Help please with biocalculus question involving differentiation

    Hi, I was just wondering how one would arrive at the answers to these questions. I have the solution for parts a and b, but not for part c. Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus...
  7. M

    How to find the area of a triangular region using Green's Theorem

    Homework Statement You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree. Homework Equations Line integral of Pdx + Qdy = Double integral of...
  8. Adgorn

    I Understanding the Taylor Expansion of a Translated Function

    I recently found out the rule regarding the Taylor expansion of a translated function: ##f(x+h)=f(x)+f′(x)⋅h+\frac 1 2 h^ 2 \cdot f′′(x)+⋯+\frac 1 {n!}h^n \cdot f^n(x)+...## But why exactly is this the case? The normal Taylor expansion tells us that ##f(x)=f(a)+f'(a)(x-a)+\frac 1...
  9. M

    Calculus 3 help -- The equation of a plane and finding a point on that plane

    Homework Statement Why is that we can set two variables zero in an equation of a plane to find a point on that plane? What is the proof for this? Homework EquationsThe Attempt at a Solution
  10. EastWindBreaks

    What is the direction of P relative to A?

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  11. EastWindBreaks

    Which side should I put constant C on?

    Homework Statement the second solution is the correct, I know you can put C on both sides and it simplifed to C2 on one side, but why can't you put C2 on the right side? Homework EquationsThe Attempt at a Solution
  12. M

    Center of Mass of a Sphere with uniform density

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  13. M

    Proving the equation for the height of a cylinder

    Homework Statement Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed. Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the...
  14. M

    Question about Finding a Force with line integrals

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  15. M

    Question about Vector Fields and Line Integrals

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  16. M

    MHB Why Is Problem #4 in Calculus 3 Series So Challenging?

    i have attached the problem set. I have done the first three problems but number 4 is very difficult. Can someone help me out? Thanks [Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]
  17. S

    B Volumes and Hyper Volumes Related Special Relativity

    How can volumes and hypervolumes be related to Einstein's theory of special relativity and to quantum mechanics? Also, can volumes and hypervolumes of objects be used for modeling how different scenarios can change over time? Oh yeah, and hi my name is Sasha Jaffarove!
  18. N

    Proving Fermat's principle without calculus?

    According to me this topic must be raised and discussed how fermat did it without calculus.What problems he faced since calculus was developed afterwards by Newton leibniz. http://aapt.scitation.org/doi/10.1119/1.1514235 Moderator's edit: File substituted by link due to potential copyright...
  19. M

    Intro Physics Textbook recommendations for calculus based physics course

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  20. S

    Prove Continuity of f(x+y) = f(x) + f(y)

    Homework Statement f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ. Homework Equations lim{x->a}f(x) = f(a) The Attempt at a Solution I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)
  21. allanwinters

    Basically solved, Last coordinate does not match?

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  22. HorseRidingTic

    B Question about calculus my dad made me think about

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  23. Math Amateur

    MHB Another Question On B&S, Theorem 7.3.5 - Fundamental Theorem Of Calculus ...

    I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 7: The Riemann Integral ... I need help in fully understanding yet another aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as...
  24. Math Amateur

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  25. Delta31415

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  26. S

    Algebra Recommended books for linear algebra and multi-variable calculus

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  27. Moayd Shagaf

    I Difference Between d3x and triple Integral

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  28. W

    Solve Integral Equation: xe-axcos(x)dx from 0 to ∞

    Homework Statement Solve from x = 0 to x = ∞, ∫xe-axcos(x)dx Homework EquationsThe Attempt at a Solution I have a solution for the integral ∫e-axcos(x)dx at the same limits, and I feel that the result might be of use, but have no idea how to manipulate the integral above such that I can use...
  29. Marco Masi

    Courses Best free online calculus course for physics self-learners?

    I'm wondering if and how to establish an online community of people who would like to self-teach physics from the bottom-up without direct involvement in academic institutions. The idea is to structure it in a flipped teaching classroom format where students first follow a video course and/or...
  30. C

    Limits in multivariable calculus

    1. Find if the limit exist: sin (x^3 + y^3) / (x + y) (x,y)-> (0,0) So I am starting solving this by using polar coordinates form and I get to lim= sin r^3 ( cos^3θ + sin^3θ) / r ( cosθ + sinθ) = lim r^2 ( cos^2Θ + sin^2Θ) My question is ok so far and how...
  31. G

    How can I calculate the derivative of this function?

    Homework Statement Let f(x) be the function whose graph is shown below (I'll upload the image) Determine f'(a) for a = 1,2,4,7. f'(1) = f'(2) = f'(4) = f'(7) = Use one decimal. Homework Equations f(x+h)-f(x)/h The Attempt at a Solution Hi everybody I was trying to do this function...
  32. W

    MHB Some Geometry Some Calculus Some Trigonometry

    Show that y≈∆φ×secφ in the jpeg attached. or ∆y = sec φ A and B are points on curved surface, two lines are extended through origin to a line that is tangent to the circle, these points are A' and B', change in Angle will bring a change in length between A' and B'. I need to know how is this...
  33. OcaliptusP

    I Derivation of pi using calculus

    I tried to derivate pi using calculus but i just found a quite different result. Can you spot my wrong please?First i started with equlation of a circle which is: $$x^2+y^2=r^2$$ I am assuming circle's center stands on the center of origin. To reach pi we shoud consider the situation that...
  34. jlmccart03

    Courses Calculus 3 -- looking for ways to help me understand

    So I am in calculus 3 this year and have passed both calc 1 and 2 with a B and C+ respectively. I could have gotten a better grade but was lazy. I was lazy by using calculators and not actually learning the arithmetic and algebra. Now one serious issue I have is Trig. I can never remember trig...
  35. L

    Linear and Non-linear Equations (QM)

    1. Problem Recall that we defined linear equations as those whose solutions can be superposed to find more solutions. Which of the following differential/integral equations are linear equations for the function u(x,t)? Below, a and b are constants, c is the speed of light, and f(x,t) is an...
  36. D

    Vector Geometry: Quadrangular Pyramid with Inner and Cross Products

    Homework Statement A quadrangular pyramid OABCD with square ABCD as the bottom. OA = 1, AB = 2, BC = 2 Also, OA perpendicular to AB, OA perpendicular to AD. Question 1 : Find the inner product \overrightarrow {OA}.\overrightarrow {OB} and the size of the cross product |\overrightarrow...
  37. G

    Help with this differential calculus

    <Moderator's note: Moved from a technical forum and therefore no template.> Hi everybody I've been trying to solve this problem all the afternoon but I haven't been able to do it, I've written what I think the answers are even though I don't know if they're correct, so I've come here in order...
  38. L

    If a constant number h of fish are harvested from a fishery

    Hi! Can anyone help me? If a constant number h of fish are harvested from a fishery per unit time, then a model for the population P(t) of the fishery at time t is given by: dP/dt = P(5-P) - h, P(0) = P0. a. Solve for the IVP if h = 4. b. Determine the value of P0 such that the fish...
  39. S

    Finding the Derivative of y=sqrt(x+sqrt(x+sqrt(x)))

    Homework Statement This is a chain rule problem that I can't seem to get right no matter what I do. It wants me to find the derivative of y=sqrt(x+sqrt(x+sqrt(x))) Homework Equations dy/dx=(dy/du)*(du/dx) d/dx sqrtx=1/(2sqrtx) d/dx x=1 (f(x)+g(x))'=f'(x)+g'(x) The Attempt at a Solution My...
  40. D

    Find the limit using Riemann sum

    Homework Statement i want to find limit value using riemann sum \lim_{n\to\infty}\sum_{i = 1}^{2n} f(a+\frac{(b-a)k}{n})\cdot\frac{(b-a)}{n}= \int_a^b f(x)dx<br> question : <br> \lim_{h \to \infty} =\frac{1}{2n+1}+\frac{1}{2n+3}+...+\frac{1}{2n+(2n-1)}<br> Homework EquationsThe Attempt at a...
  41. M

    Engineering Fluid Mechanics book that's pure integration/vector calculus

    Hello. The textbook that was assigned to my class is not a good fit for my professor. The textbook simplifies the basic force equations so that there are no general equations with integrals or vector calculus calculations. Can anybody recommend a Fluid Mechanics book or books that deal only...
  42. X

    I How much would time pass between watching the sun set ....?

    How much would time pass between watching the sun set from ground level and then watching it set again from the top of a sky scraper? I heard once that this could be done using one of the towers of the World Trade Center. So I assume one could also do this using the Sears / Willis Tower in...
  43. D

    Trying to find this double integral using polar coordinates

    Homework Statement question : find the value of \iint_D \frac{x}{(x^2 + y^2)}dxdy domain : 0≤x≤1,x2≤y≤x Homework Equations The Attempt at a Solution so here, i tried to draw it first and i got that the domain is region in first quadrant bounded by y=x2 and y=x and i decided to convert...
  44. A

    B Quick question about calculus (derivatives)

    I thought Differentiation is all about understanding it in a graph. Every time I solve a question on differentiation I visualise it as a graph so it's more logical. After all, that IS what the whole topic is about, right? Or am I just wrong? But when you look at these questions...
  45. B

    Calculus What are some good resources for studying Spivak's Calculus?

    I'm currently struggling a lot in self-studying Spivak's Calculus; I've tried numerous online resources including math stackexchange, I'm currently about half-way through How To Prove It as many people have suggested. I'm only in high school, so the calculus class I'm taking barely coincides...
  46. V

    Force due to two thin charged rods acting on each other

    Homework Statement I have 2 thin rods of length L in the axial plane(x axis), they're of the same uniform linear charge distribution and are separated by a distance a . Homework Equations $$E = \frac{kq} {R^2}$$ $$F = qE$$ The Attempt at a Solution [/B] Let 0 be one at one end of the two rod...
  47. A

    Solving the heat equation using FFCT (Finite Fourier Cosine Trans)

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  48. A

    Solving partial differential equation with Laplace

    Homework Statement am trying to solve this PDE (as in the attached picture) https://i.imgur.com/JDSY4HA.jpg also my attempt is included, but i stopped in step, can you help me with it? appreciated, Homework EquationsThe Attempt at a Solution my attempt is the same as in the attached picture...
  49. A

    Schools Taking Calculus 1 over the summer -- question?

    I was talking to my friend and she told me how she was taking Calculus 1 over the summer in 5 weeks. Everyone else averaging a 30 percent as their overall grade. The teacher was planning to raise the grade. So if you get a 30%, you'll get a C. However, my question is that is it okay to pass...
  50. S

    I Calculus of variations

    when inducing that the cycloid is the least time-taking course between the two points in the two dimension, we have to use calculus of variations. Then is it possible to induce the parameter of the least time-taking course between two points in the three dimension?
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