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. K

    Absolute convergence of series

    Homework Statement Hello, I need some feedback on whether this reasons is correct. consider the series Examine the series for absolute convergence. Homework EquationsThe Attempt at a Solution How I have solved this, using the limit comparison test: we have: introducing we have that...
  2. K

    Integral help (substitution and boundaries)

    Homework Statement Hello, I´ve added an image where the task is shown. I am wondering about the integral bounds here. When doing the substitution, shouldn't the bounds be inf and 0? When using that u = ln x, i.e. u(0) =inf, u(1) = 0. Homework EquationsThe Attempt at a Solution
  3. K

    Sum of Infinite Series | Calculate the Sum of a Geometric Series

    Homework Statement Find the sum of the series Homework EquationsThe Attempt at a Solution Not sure exactly where to start. If I move 3 outside the sum I'm left with 3*sigma(1/n*4^n), which I can rewrite to 3*sigma((1/n)*(1/4)^n), which party looks like a geometric series..Any tips?
  4. K

    Find the expression for the sum of this power series

    Homework Statement Hello, I need to find an expression for the sum of the given power series The Attempt at a Solution I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite , which makes the expression even more similar to the...
  5. A

    MHB Solve f(x,y,z): Min & Max Values Subject to Constraint x^2+2y^2+6z^2=81

    Afternoon, I have been working on this problem for awhile now but have been stuck on a certain point, and once I set everything equal to each other I end up with the same thing for example: x=x, y=y, & z=z Find the minimum and maximum values of the function f(x,y,z)=3x+2y+4z subject to the...
  6. topologyfreak

    Studying Should I start with a basic physics book without calculus?

    Hi, I want to start learning physics as I have never studied it in my life but have now taken an interest towards it. I have quite literally zero knowledge on physics, so assume I haven’t taken middle school or high school courses on physics, as I haven’t. I’m 17 currently, but that shouldn’t...
  7. A

    MHB Solve Lagrange Multipliers Problem w/ e^(9x) - Find (a,b)

    I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers: Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible. I have found the gradient for both when...
  8. komarxian

    Multivariable calculus problem

    Homework Statement Find the points on the surface xy^2z^3=2 that are closest to the origin Homework EquationsThe Attempt at a Solution x,y,z=/= 0, as when x,y,z = 0 it is untrue. Right?? Otherwise, I am very unsure as to how to approach this problem. Should I be taking partial derivatives to...
  9. Hawkingo

    Help in solving an inexact differential equation

    Homework Statement The question is to solve the inexact equation by turning it into exact.the equation is ##( x + y + 4 ) d x + ( - x + y + 6 ) d y = 0## Where "x" and "y" are variable. 2. Homework Equations [/B] 1.(x+y+4)=m and (-x+y+6)=n 2.Integrating Factor =##\frac { 1 } { x ^ { 2 } + y...
  10. Delta2

    Insights A Numerical Insight for the Fundamental Theorem of Calculus - Comments

    Greg Bernhardt submitted a new blog post A Numerical Insight for the Fundamental Theorem of Calculus Continue reading the Original Blog Post.
  11. T

    Multivariable Calculus, plane sketching

    How do I know where to put the axes for the equation 4x^2 - 9y^2 = z when graphing in 3d?
  12. B

    Sanity check: Using the Jones calculus for superposition

    Homework Statement Suppose light is prepared in a coherent superposition of linear horizontal polarization and linear vertical polarization. What is the resulting polarization according to Jones calculus if it passes through: a linear polarizer at a 45-degree angle (0 degrees would be...
  13. T

    Vector Calculus: Change of Variables problem

    Homework Statement Let D be the triangle with vertices (0,0), (1,0) and (0,1). Evaluate: ∫∫exp((y-x)/(y+x))dxdy for D by making the substitutions u=y-x and v=y+x Homework EquationsThe Attempt at a Solution So first I found an equation for y and x respectively: y=(u+v)/2 and x=(v-u)/2 Then...
  14. jonathanm111

    Vector Calculus, setting up surface area integral.

    The question goes like: find the SA of the portion S of the cone z^2 =x^2 +y^2 where z>=0 contained within the cylinder y^2+z^2<=49 this is my attempt using the formula for SA, I could switch to parametric eqns, but even then I'd have hard time setting up limits of integration.
  15. J

    Definite Integral of Product/Composite Function Given Graph

    Homework Statement Given the graph of f(x) shown below, find the value of the integral. Photo attached. Homework Equations [/B] ∫23 5x·f(x2)dx The Attempt at a Solution [/B] I tried integration by parts to simplify the problem, but finding the integral of the composite function (f(x2))...
  16. Javier Lopez

    Elliptic integral of the third order for magnet calculus

    I have the first and second orders that I use in a magnetic simulator, but i need the thirth also to do also with magnetic cylinders accordingly paper: Do anybody have it in any code? I should pass to C++
  17. K

    Taylor polynomial, approximative solution of this equation

    Homework Statement The equation 4x = (1/3)*cos(3x) has a solution on the interval [0,1]. Find an approximative solution by replacing the right hand side with a Taylor polynomial of degree 2 around 0. Homework EquationsThe Attempt at a Solution So as I understand the task we should find a...
  18. O

    MHB Calculus I - Relative Growth Rate help

    Here is a picture of the problem: I am firmly stuck on part b). I know that relative growth rate is defined at f'(x)/f(x), but I cannot seem to come up with the right answer. I've tried everything I could conceivably think of. I do not understand why my answer is incorrect, unless I'm...
  19. YoungPhysicist

    I Area divided by linear function

    The original problem for anyone that can read Chinese: https://zerojudge.tw/ShowProblem?problemid=b221 The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
  20. gibberingmouther

    I Multivariate Calculus for a Game World

    Okay, I never took calculus 3 so I don't know how to write/solve an integral for a power I made up. I want to flesh out my game world with math because it's cool and I can. Like Chester wrote some of the physics for the Bizarro universe. Applying physics and math to a fictitious universe is...
  21. topologyfreak

    How to learn physics if I know calculus but zero physics?

    Hi, I have decided I like to learn some physics as I know absolutely zero physics and also including chemistry or biology, and I’d like to learn a bit of all 3, to the high school level perhaps. It seems that from, what I’ve read if I want to learn chemistry, I’ll first have to know physics, and...
  22. CaptainAmerica17

    B Spivak's Calculus as a Prerequisite for General Topology

    High school student here... Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...
  23. K

    Using the Integral Test to Show Sum is Less Than pi/2

    Homework Statement Use the integral test to show that the sum of the series gif.latex ##\sum_{n=1}^\infty \dfrac{1}{1+n^2}## is smaller than pi/2. Homework EquationsThe Attempt at a Solution I know that the series converges, and the integral converges to pi/4. As far as I´ve understood...
  24. Y

    MHB Calculus 2 (Power Series) when the limit is zero by root test

    Hi guys! Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay? In general, when we are solving power...
  25. M

    Tuple Relational Calculus in a DBMS

    <Moderator's note: Moved from a technical forum and thus no template.> I'm solving a problem using tuple relational calculus ( TRC) in DBMS. Problem Find the name of all the employees who work for XYZ Bank Corporation. Solution ( as given in my book) I don't understand why book is using...
  26. K

    Show that this equation only has one solution

    Homework Statement Show that the equation arctan(x) = x^2 have at least one solution. Then argue why the equation only has one, positive solution r. Homework EquationsThe Attempt at a Solution I know how to prove that the equation has at least one solution (IVT), but I do not know how to show...
  27. K

    Volume of revolution, region bounded by two functions

    Homework Statement Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T. Homework EquationsThe Attempt at a Solution...
  28. solarmidnightrose

    B Understanding Kinematics in Calculus for High School Students

    hi, I'm a high-school student that is just beginning to learn calculus. in calculus we are learning how to apply integration and diffrentiaiton methods regarding kinematics. there is this certain phrase i do not really understand in our textbook: e.g."in the second second" how am i meant to...
  29. M

    How do I find a plane that contains two given lines?

    Homework Statement a. Find a point at where these lines intersect b. Find the equation of a plane that contains the two lines. Homework Equations r[/B] = <1,3,0> + t<3,-3,2> r = <4,0,2> + s<-3,3,0> The Attempt at a Solution I correctly found the point of intersection to be...
  30. P

    Calculus 2 for Engineers: Riemann sums

    Homework Statement a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1. b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4. The Attempt at a Solution I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...
  31. K

    Volume of Revolution: Find V with Shell Method

    Homework Statement A container with height 4.5 is created by rotating the curve y = 0.5x^2 0 \leq x \leq 3 around x = -3 and putting a plane bottom in the box. Find the volume V of the box. Homework EquationsThe Attempt at a Solution I want to solve this by using the shell method. I have...
  32. I

    Should I Use College Physics for Self-Studying: Algebra or Calculus Based?

    Hi, my name is "Bob" I have a particular question about a specific book. Preface... It has been a while since I formally studied Physics. The type of Physics book I remember using last was I believe, Algebra-based. This was in high school. Based on my degree, I studied Algebra and Trigonometry...
  33. jonathanm111

    Vector Calculus (non conservative vector fields

    the question: My attempt: The partial derivatives did not match so i simply tried to find f(x,y) I got the set of equations on the right but that's about it.
  34. Jazzyrohan

    Calculus Resources for learning multivariable calculus

    I have recently started studying multivariable calculus and I cannot quite visualise the concepts.Problem solving is not a problem but I want a true understanding of the concepts.Which book or online resources are great at developing visualisation in this course?
  35. K

    Maximum value of this integral

    Homework Statement Find a > 0 so the integral int(exp(-ax)*cosx)dx from 0 to inf get as high value as possible. The Attempt at a Solution My way of solving this is to plot the integrand, i.e. exp(-ax)*cosx and check for different values of a. The larger a is, the smaller the area under the...
  36. S

    French student look after pre-calculus books

    Hi there , well my questions is in the title but i have to say some things : I have multiples mathematiques and physics courses calc-based and so i want to review the basic . I already search around this forum and others and i find some books who looks popular here there are : - Pre-Calculus...
  37. K

    Find Smallest b to Make Invertible Function: Homework Solution

    Homework Statement Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible. Homework EquationsThe Attempt at a Solution I know that the function has to be only increasing/decreasing, and I think it is needed to find the derivative of the function. I do...
  38. G

    Graph of Position Seems to be Wrong

    Homework Statement By relative velocity, we mean velocity with respect to a specified coordinate system. (The term velocity, alone, is understood to be relative to the observer’s coordinate system.) (a) A point is observed to have velocity Va relative to coordinate system A.What is its...
  39. archaic

    B Is Understanding 'dx' Just a Heuristic in Calculus?

    Hello! Let ##I## be an interval of size ##L##, suppose we divide it into bits of ##dx## then ##L=dx+dx+... =\alpha.dx## Since ##dx## is by definition infinitesimally small is it correct to imply that for each ##x## there's a corresponding ##dx## hence, here, ##\alpha## would be, theoretically...
  40. Adgorn

    I Spivak's Calculus: clarification on Conic Sections appendix

    Hello everyone. This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus: https://ibb.co/kAKyVU https://ibb.co/jXVSPp https://ibb.co/kwRdVU I'm pretty...
  41. Adgorn

    Spivak: Conic Sections appendix, problem 1

    Homework Statement "Consider a cylinder with a generator perpendicular to the horizontal plane; the only requirement for a point ##(x,y,z)## to lie on this cylinder is that ##(x,y## lies on a circle: ##x^2+y^2=C^2##. Show that the intersection of a plane with this cylinder can be described by...
  42. sams

    I Chain Rule of Multivariable Calculus

    I am confused when I should use the ∂ notation and the d notation. For example, on http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx, in Case 1, the author wrote dz/dt while in Case 2, the author wrote ∂z/∂t. Could anyone please explain to me when I should use the ∂ notation and the...
  43. K

    Derivation, absolute value problem

    Homework Statement Find k so that y - 36x = k is a normal to the curve y = 1 / abs(x-2). Homework EquationsThe Attempt at a Solution My problem is regarding the absolute value. I know that the tangent to the curve must be (-1/36). In the solutions manual, it is said that by knowing the sign...
  44. T

    What does “Media Upgrade” mean in Thomas' Calculus textbook?

    What is the difference between: THOMAS' CALCULUS, media UPGRADE (11TH EDITION) [ISBN-13 9780321489876] and THOMAS' CALCULUS (11TH EDITION) [ISBN-13 9780321185587]? could you provide detailed answer please? what does the title-addition "Media Upgrade" mean and what changes does it make on the...
  45. S

    MHB Can you help with these calculus questions?

    Section 1 Indices and Logarithms 1) Solve Ln(e6x−4+7)=7 for x. The answer (correct to 3 decimal places) is x = Marks : 1 Section 2 Functions 2) Let f(x) = −4x+72x−2, find f-1(x). Answer: (Correct to 3 decimal places) f-1(x) = (-2 x + ) / ( x + ). Section 3 Roots of Equations 3) Solve...
  46. P

    Projectile Motion: Initial Velocity Derivation

    Hey guys, I am working on a school project and have fired some water rockets at different angles. I am now trying to compare theoretical results vs experimental results. But I am struggling to produce an accurate mathematical model. I have attached my attempt to this post. It is quite long and...
  47. Shivam

    Derive the Volume of a Sphere using Calculus

    Homework Statement Derive the volume of sphere using Calculus. i saw videos on this topic on youtube, but i want to do it by the method of integrating a circle at angle θ (Theta) . i am posting a photo where i explained every thing i did but i couldn't know what i am doing wrong. Homework...
  48. Vanrichten

    How Does the Chain Rule Apply to Newton's Second Law in Calculus?

    Hello, I'm new to the language of calculus. I am learning about Newtons second law and I'm trying to understand it's forms in calculus. Excuse my notation, just trying to keep it as simple as possible. F=m * dV/dt I know V= dx/dt My textbook says you can 'apply chain rule' to obtain the...
  49. Jozefina Gramatikova

    Vector calculus- region-density-mass

    Homework Statement https://www.physicsforums.com/attachments/229290 Homework Equations The Attempt at a Solution I am not sure what equation to use for the volume[/B]
  50. G

    Courses How to prepare for Calculus Based Physics 2?

    hello, Some background I transfer to SUNY Albany in Fall 2017 and took the following courses Organic chemistry 1: C+ Calculus 2: B- A PHY 140 Physics I: Mechanics (3): E (like F) An introduction to the fundamentals of physics: Classical Mechanics. Topics include the concepts of force, energy...
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