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. M

    B Self learn calculus for UK A-levels?

    Hi, I am currently about to begin self studying for UK maths a-levels, however I am also wanting to gain a solid understanding of calculus. I know that calculus is covered in a-levels, but, the books for a-levels seem to be not as dense or as good as the US books I believe. My question is...
  2. Saqib Ali

    Calculus How do you choose problem sets in Courant's calculus texts?

    I'm going to use Courant's volume 1 and 2. I don't want to try every problem in the book, so how do I go about choosing problems to know if I understand the material?
  3. M

    Can somebody tell me what this topic is?

    Homework Statement Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I can't find notes on it.It think it has something to do with limits. Many thanks.
  4. A

    I Calculation Of the energy Of beta decay in tritium

    Hi! I hace been trying to calculate how many energy in form of beta radiation is emitted in one if those "Tritiglows" sold in Amazon. I did the following math (imatge) and got a really high energy. How is that posible? Where did I failed...
  5. jlmccart03

    I Overcoming Struggles in Calculus 3

    I am currently nearing the end of my Calculus 3 course and have been struggling all semester. First there is some background information. I passed Calc 1 and 2 with a B and C respectively. Over the summer I worked on my skills and felt prepared. Unfortunately my section was chosen for IBL...
  6. Adgorn

    I Differentials of order 2 or bigger that are equal to 0

    So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...
  7. lfdahl

    MHB Calculus inequality challenge prove ∫10f(x)/f(x+1/2)dx≥1

    Let $f$ be a positive and continuous function on the real line which satisfies $f(x + 1) = f(x)$ for all numbers $x$. Prove \[\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})}dx \geq 1.\]
  8. R

    Proving the convergence of series

    Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...
  9. Alexander350

    B Solving a differential equation with a unit vector in it

    I need to solve: \dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s} However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
  10. Peter Alexander

    Critical Points of a Parameter Dependent Integral

    1. The problem statement, all variables, and given/known data Find and categorize extremes of the following function: $$F(y)=\int_{y}^{y^{2}}\frac{1}{\ln^{2}x}dx$$ for ##y>1##. Homework Equations $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...
  11. D

    Finding the volume surrounded by a curve using polar coordinate

    Homework Statement I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space. And the three questions related to each otherA.) Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z . The equation of the...
  12. A

    Calculus Feynman's High School Calculus Book

    Hi everybody. I enjoy looking at other people's handwritten notebooks, as well as what textbooks they learned math and physics from. This evening I came across this article about how Feynman learned calculus in high school by studying Calculus for the Practical Man by Thompson. He kept very...
  13. K

    I Need a help in solving an equation (probably differentiation

    I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
  14. O

    Gaussian type integral (but not a standard form)

    When working a proof, I reached an expression similar to this: $$\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$$ I've tried the following: 1. I tried squaring and combining and converting to polar coordinates, like one would solve a standard Gaussian. However...
  15. lichenguy

    How Do You Solve a Dimensional Analysis Problem for a Block in Viscous Fluid?

    Homework Statement A block of mass ##m = 1.00 kg## is being dragged through some viscous fluid by an external force ##F = 10.0 N##. The resistive force can be written as ##R = -bv##, where ##v## is the speed and ##b = 4.00 kg/s## is a phenomenological constant. You may ignore gravity (we...
  16. R

    Calculus Will this be enough preparation for Spivak's Calculus book?

    Reading through a bit the book seems nothing like what I learned in uni (Stewarts calculus) Will "Book of Proof" By Hammack --> "Basic Mathematics" by Lang be enough preparation for Calculus by Spivak? Thanks.
  17. peadar2211

    Determine the stability of a fixed point of oscillations

    Homework Statement I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...
  18. 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...
  19. 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?
  20. 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...
  21. 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...
  22. 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]
  23. 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...
  24. 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...
  25. 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...
  26. 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
  27. EastWindBreaks

    What is the direction of P relative to A?

    Homework Statement Homework Equations The Attempt at a Solution does Vpa has the same direction as Vp? i thought Vp is tangent to the circular path that point P creates( perpendicular to vector Rpa) but from the figure, Vp doesn't seem to be tangent to the path, but Vpa does...
  28. 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
  29. M

    Center of Mass of a Sphere with uniform density

    Homework Statement Find the z -coordinate of the center of mass of the first octant of a sphere of radius R centered at the origin. Assume that the sphere has a uniform density. Homework Equations Mass = Integral of the density function Center of mass for z = Integral of density * z divided...
  30. 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...
  31. M

    Question about Finding a Force with line integrals

    Homework Statement [/B] F =< 2x, e^y + z cos y,sin y > (a) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along a straight path. (b) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along the curved path given by C : r(t)...
  32. M

    Question about Vector Fields and Line Integrals

    Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...
  33. 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.]
  34. 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!
  35. 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...
  36. M

    Intro Physics Textbook recommendations for calculus based physics course

    Hello! So I need some textbook recommendations for calculus based physics course. I have taken an algebra/ trigonometry physics course last semester, but now I'm taking a calculus based. I have a textbook which the professor told us will follow our course. The book is Fundamentals of Physics...
  37. 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)
  38. allanwinters

    Basically solved, Last coordinate does not match?

    Homework Statement Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically. Homework Equations (x,y,z) = (x0,y0,z0) +s(a1.a2,b3) + t(b1,b2,b3) The Attempt at a Solution...
  39. HorseRidingTic

    B Question about calculus my dad made me think about

    This is a question I'v got about calculus after doing my bachelors in engineering degree. So you can integrate an acceleration graph to get velocity, and integrate a velocity graph to get distance. Integrating a graph can be done by easily finding the area under the graph. This applies to all...
  40. 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...
  41. Math Amateur

    MHB Fundamental Theorem Of Calculus (Second Form) - B&S Theorem 7.3.5 .... ....

    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 an aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as follows: In...
  42. Delta31415

    Calculus Integration help please -- involves sinh(x), e^x and roots

    Homework Statement [/B] $$\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx$$ or http://www.HostMath.com/Show.aspx?Code=\int \left ( \frac{-1}{2*\sinh(x)*\sqrt{1-e^{2x}})} \right ) dx Homework Equations the sinh identity, which is (e^x-e^-x)/2 The Attempt at a Solution Tried...
  43. S

    Algebra Recommended books for linear algebra and multi-variable calculus

    hey everyone just started university and the jump i feel is huge from a level and was just wondering if you guys knew of any books that had linear algebra and/or several variable calculus in them but displayed and explained stuff in a clear simple way? or if anyone has any websites that do the...
  44. Moayd Shagaf

    I Difference Between d3x and triple Integral

    So My Question Is Simple, But It confuse me too much! What Is Difference between the notation ∫d3x and ∫∫∫dxdydz ?
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. OcaliptusP

    I Where Did I Go Wrong in Deriving 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...
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