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.

View More On Wikipedia.org
Recent contents Top users
  1. C

    Improper integral with spherical coordinates

    Homework Statement I have a question. I have a function f(x,y,z) which is a continuous positive function in D = {(x,y,z); x^2 + y^2 +z^2<=1}. And let r = sqrt(x^2 + y^2 + z^2). I have to check whether the following jntegral is convergent. x^2y^2z^2/r^(17/2) * f(x,y,z)dV. Homework Equations...
  2. manjuvenamma

    I Is it possible to find the limit of (1+1/x)^x as x approaches -infinity?

    Is it possible to find the limit of (1+1/x)^x as x approaches minus infinity using only the fact that it is e if x approaches infinity?
  3. R

    Line integral problems in Apostol calculus

    Homework Statement A two dimensional force field f is give by the equation f(x,y)=cxyi+x^6 y^2j, where c is a positive constant. This force acts on a particle which must move from (0,0) to the line x=1 along a curve of the form y=ax^b where a>0 and b>0 Homework Equations Find a value of a(in...
  4. danielhep

    Optimization w/ Constraint Question (Multivariable Calculus)

    Homework Statement Find any maxima/minima on f(x,y) = x2+2y2 on the unit circle, centered at the origin. Homework Equations grad f = λgrad g constraint: 1=x2+y2 The Attempt at a Solution grad f = 2xi+4yj grad g = 2xi+2yj 2x=λ2x 2y=λ4y How do I solve this? I don't see any way to get numbers...
  5. F

    Using dx/dy to find a y-parallel tangent

    Homework Statement The curve ##C## has equation $$y=x^2+0.2sin(x+y)$$ Show that ##C## has no tangent(no point where ##dy/dx=∞##), that is parallel to the y axis. Attempt $$1=2x\frac{dx}{dy}+0.2cos(x+y)(1+\frac{dx}{dy})$$ For a tangent to be parallel to y-axis, $$\frac{dx}{dy}=0$$...
  6. R

    Is there any way to calculate this integral?

    I have done it by the parametric form of σ, but if I change σ to implicit form that is G(x,y,z)=x^2+y*2+z^2-R^2=0 I don't know how continue. The theory is: where Rxy is the projection of σ in plane xy so it's the circumference x^2+y^2=R^2
  7. S

    Calculus Need help preparing for Courant's Calculus

    Hello, Just to give you some background info, I'm a software developer looking to make a career switch to Mechanical Engineering (At least I think so). My interests lie in design, complex systems, maglev, rockets and new forms of space propulsion. Before I really commit myself to University, I...
  8. S

    B Some help understanding integrals and calculus in general

    So in differential calculus we have the concept of the derivative and I can see why someone would want a derivative (to get rates of change). In integral calculus, there's the idea of a definite integral, which is defined as the area under the curve. Why would Newton or anyone be looking at the...
  9. R

    Calculus Reviewing Multivariable calculus to skip in college

    Hi, I'm asking for a friend who will be majoring in chemical engineering. We have already taken Calculus I, II, and III under a course offered by a local community college. Admittedly, it was taught from stewart's series of calculus books, and we did exactly zero proofs in the class, and all...
  10. Bunny-chan

    Epsilon distance between two terms

    Homework Statement How close is x to x_0 (x_0 \neq 0) so that 2. Homework Equations The Attempt at a Solution I tried to use absolute value properties:- \epsilon \lt \frac{\sqrt{x_0^2+1}}{x_0^3} - \frac{\sqrt{x^2+1}}{x^3} \lt \epsilonBy adding in the three sides, we...
  11. harpazo

    MHB How can self-study lead to long-term learning compared to classroom instruction?

    As I study calculus 3, I often revisit calculus 1 and 2. The following application is from single variable calculus, partucularly calculus 1, called RELATED RATES. I have not seen a related rates problem since the 2015. I am a bit rusty with the set up. Can someone help me set it up? I can take...
  12. harpazo

    MHB How do you solve this calculus 2 integration problem?

    I am currently studying calculus 3. However, once in a while, I like to review single variable calculus. I decided to tackle the following calculus 2 integration problem: ∫ tan^3 x dx Solution: Separate into two tangent functions. (tan^2 x)(tan x) Rewrite tan^2 x as sec^2 x - 1 using its...
  13. C

    Integral with transformations and bounded by x + y + z = 1

    Homework Statement I have a question. I need to know the integral dxdydz/(y+z) where x>=0, y>=0, z>=0.Homework Equations It is bounded by x + y + z = 1. The transformations I need to use are x=u(1-v), y=uv(1-w), z=uvw. The Attempt at a Solution y+z = uv. J = uv(v-v^2+uv) So I get the integral...
  14. EEristavi

    Basic Calculus: Differentiation usage

    Problem: How fast is the area of a rectangle changing if one side is I0 cm long and is increasing at a rate of 2 cm/s and the other side is 8 cm long and is decreasing at a rate of 3 cm/s?I have 2 approach and I want to know which is correct, why and what am I missing
  15. phasacs

    Why isn't Kinetic Energy always equal to Potential Energy?

    K= ∫mvdv = ∫m dx/dt dv = ∫m dx/dv dv/dt dv = ∫m dv/dt dx = ∫Fdx = U => K=U, why isn't this true? If it is, wouldn't that mean that Kinetic Energy is always equal to Potential Energy?
  16. Avatrin

    I Motivating definitions in calculus on manifolds

    Hi I am a person who always have had a hard time picking up new definitions. Once I do, the rest kinda falls into place. In the case of abstract algebra, Stillwell's Elements of Algebra saved me. However, in the case of Spivak's Calculus on Manifolds, I get demotivated when I get to concepts...
  17. Bunny-chan

    Determining a trigonometric limit

    Homework Statement Calculate the following limit: Homework EquationsThe Attempt at a Solution I don't know how to proceed with this. I've tried to multiply by the conjugate, and to simplify the expression (x+\pi) to u, but I wasn't very sucessful. To what kind of algebric device I could...
  18. shihab-kol

    Simple proof of Snell's law without calculus

    Well, I have checked out the ones with calculus but I was just wondering if there was one without calculus I tried it but could not do it I think Fermat's principle can be used to do it but I am not being successful So, anyone please help
  19. Bunny-chan

    Book demonstration about trigonometric relations

    Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution
  20. Quantum Velocity

    What calculus is needed for understanding classical physics

    Hey guy, I'm a just new in physics and i want to self-teaching it. But i don't know what calculus i need for classical mechanic. Pleas help! And if you suggest me a book can you pleas help to send me a link (and the book must be in pdf pleas) Thank
  21. Eclair_de_XII

    Courses What topics in Calculus IV are typically in a PDE course?

    Additionally, what topics from that same course are relevant to probability? I ask because I'm afraid I might forget some of the topics from my calculus series after one semester of disuse. I mean, I know I should probably brush up on my calculus skills in preparation for any math class that...
  22. H

    Calculating the equations of motion for particle in parabola

    I made the problem up myself, so there might very well not be a rational answer that I like! Homework Statement A point-particle is released at height h0 is released into a parabola. The position of the particle is given by (x, y) and the acceleration due to gravity is g. All forms of friction...
  23. D

    A Dirac Delta and Residue Calculus

    I'm an undergraduate student, so I understand that it may be difficult to provide an answer that I can understand, but I have experience using both the Dirac delta function and residue calculus in a classroom setting, so I'm at least familiar with how they're applied. Whether you're integrating...
  24. jlmccart03

    Alternating Series Estimation Theorem

    Homework Statement Using the power series for ln(x + 1) and the Estimation Theorem for the Alternating Series, we conclude that the least number of terms in the series needed to approximate ln 2 with error < 3/1000 is: (i) 333 (ii) 534 (iii) 100 (iv) 9 (v) 201 Homework Equations ln(x+1) =...
  25. cathal84

    Factorize Problem Homework: x^3 - 2x^2 -15x +36

    Homework Statement Hello, pretty back to basics with this one. How this came about was I am finding the eigenvalues for a given matrix and after forming the characteristic polynomial of the matrix i get this. x^3 - 2x^2 -15x +36Homework Equations Using the rational root theorem i came to the...
  26. R

    Integral form of Particular solution question

    Homework Statement I'm fine with the first part. Part b) is causing me trouble http://imgur.com/xA9CG5G Homework EquationsThe Attempt at a Solution I tried subbing in the solution y1 into the given equation, but I'm not sure how to differentiate this, i thought of using integration by parts...
  27. D

    Area between 2 curves, Volume around X and Y, Centroid

    g(x)= √(19x) = upper curve f(x)= 0.2x^2 = lower curve Firstly, I found the point of intersection, which would later give the upper values for x and y. x=7.802 y=12.174 Then I found the area under g(x) and took away the area under f(x) to get the area between the curves. 31.67 units^2 This is...
  28. kupid

    MHB Calculus & Diff. Eqn: Beginner Qs on Function, Derivative & Gradient

    I have some beginner doubts about Calculus and Differential equations . Is a function always a curve ? Doesn't a function already has a slope ? d/dx of a function gives the gradient of the curve between two points ? The derivative ,d/dx ,The gradient , is the rate of change of a...
  29. ccaatt

    Courses Do I need Physics and Calculus in HS for uni?

    I am very clueless when it comes to prepping for university and things like that, and as a grade 11 going into grade 12 soon, I unfortunately have to figure it out one day or another. So, I have a few quick questions, would I need Physics 30 or Calculus/Math 31 in order to become a family...
  30. W

    I Derivative of 4^x: My Exam & Answer Explained

    On my exam, we had to find the derivative of 4^x. This is what I did Y=4^x lny=xln4 y=e^xln4 and then finding the derivative for that I got, (xe^(xln4))/4 My professor said that it was wrong and even after I told her what I did to get the answer. She told me the answer was (4^x)ln4 . Which I...
  31. P

    I Multivariable Calculus Project: Spacetime/Black Holes

    I am a student currently taking both Multivariable Calculus and Differential Equations. Instead of a final exam my teacher assigned a final project for Multivariable, and I chose to do something with Spacetime/Black holes. Within the scope of <100 hours of work, is there anything I can do with...
  32. G

    Proper usage of Einstein sum notation

    Homework Statement I'm dealing with some pretty complex derivatives of a kernel function; long story short, there's a lot of summations going on, so I'm trying to write it down using the Einstein notation, for shortness and hopefully reduction of errors (also for the sake of a paper in which I...
  33. S

    Calculus Good textbooks for really learning calculus?

    I'm going to take Calc I in the fall and Calc II and III later on and I want to actually understand the stuff intuitively instead of just trying to memorize formulas and then having trouble with the applications, like optimization. I have James Stewart's Essential Calculus Early Transcendentals...
  34. D

    What Path Should a Mass Take for Shortest Descent Time Under Gravity?

    Guys I have the following homework problem to solve: There are 2 given points in a plane. If we take a point-like object with mass m and take it to the "higher" point what path should it go on to reach the other point in the shortest possible time. Only gravitational force affects our point-like...
  35. U

    B Integrating to find surface area/volume of hemisphere

    To find the surface area of a hemisphere of radius ##R##, we can do so by summing up rings of height ##Rd\theta## (arc length) and radius ##r=Rcos(\theta)##. So the surface area is then ##S=\int_0^{\frac{\pi}{2}}2\pi (Rcos(\theta))Rd\theta=2\pi R^2\int_0^{\frac{\pi}{2}}cos(\theta)d\theta=2\pi...
  36. Harshna

    Jobs in Physics/Mathematics that Actually do Calculus

    Hi, I am finding it a bit difficult to find a job where pure maths is actually done and looking for some ideas or examples from people who have done this please. I am in my last semester of a Graduate Diploma of Science (Applied Data Science) and am considering what to study for my Masters/pHD...
  37. J

    A Is the pole in this integrand integrable?

    I am trying to numerically integrate the following complicated expression: $$\frac{-2\exp{\frac{-4m(u^2+v^2+vw+w^2+u(v+w))}{\hbar^2\beta}-\frac{\hbar\beta(16\epsilon^2-8m\epsilon(-uv+uw+vw+w^2-4(u+w)\xi...
  38. CrunchBerries

    Testing Calculus II Exam points question

    I wrote a Calc II midterm today. Things went quite well, except for one question. I left it blank, as I did not have sufficient knowledge to write anything that made any sense. The question was worth 10% of the exam mark. This question was about a part of a section we covered, but this part had...
  39. T

    Derivation: Entropy of Vaporisation using Redlich-Kwong EoS

    Homework Statement For some reason it is not letting me add the image here, here is the link to the question: http://imgur.com/a/3DLWM The part I'm stuck on is the last part. Basically, the question is to obtain the following equation for the entropy of vaporisation using the Redlich-Kwong...
  40. Eclair_de_XII

    Math How often do actuaries use calculus in their field of work?

    Just curious.
  41. Puff Cube

    Object moving upwards by constant force away from planet

    Homework Statement Suppose there is an object that is a distance ##r_0## from the center of a planet that is nearby (the object is outside the surface of the planet). Let ## r ## represent the distance from the object to the planet's center. Let ## t ## represent time. The object, which is...
  42. patric44

    Calculus Is Calculus (6th edition) by Swokowaki a good book?

    hi guys i want a good book for self learning calculus and i found caculus by Swokowaki and olnick 6ed i bought it actualy for 1.5$ (used version) so i want to know is it a good book for self learning or there's better one? i actualy took couple courses in calculus ,algebra, geometry in collage...
  43. doktorwho

    How to Prove the Integral Property for Definite Integrals

    Homework Statement Today i had a test on definite integrals which i failed. The test paper was given to us so we can practise at home and prepare better for the next one. This is the first problem which i need your help in solving:: Homework Equations 3. The Attempt at a Solution [/B] As no...
  44. Greg Bernhardt

    Can The Complete Idiot's Guide to Calculus Really Simplify Learning Calculus?

    The Complete Idiot's Guide to Calculus INTRODUCTION I've never really been very good at math and when I found out I had to take a Calculus class I started to panic. Once I gathered myself I went to the local bookstore to see if I could get a book to read so i could get a heads start. We are...
  45. O

    Portions of Calculus to Review for Intro. to Prob & Stats

    Hey everyone, I'm a CS major, and haven't been using calculus 1 or 2 for over a year and a half now. I vaguely remember any of it other than basic concepts (what derivatives and integrals are, second derivative, inflection points, etc.). I am taking an Introduction to Probability and...
  46. W

    Connecting Vector Calculus to Maxwell's Equations

    I have recently finished an extensive review of vector calculus. I need to connect the exhaustive techniques of Surface Integrals and line integrals to quite a few problems involving Maxwell's Equations before I really feel certain that I am on board with both the math and the physics. I feel...
  47. D

    [Calc] First max and min values of an underdamped oscillation

    Homework Statement Determine the FIRST maximum and minimum values of the underdamped oscillation: y=e^(-x/2)(4sin(3x)+3cos(3x)) cm Homework Equations 3. The Attempt at a Solution [/B] I firstly differentiated the above equation and got: (-e^(-x/2)(22sin(3x)-21cos(3x)))/2 I checked this and...
  48. D

    Determine the truth of the following statements

    Homework Statement ##f(x) = \begin{cases} -\frac{1}{1+x^2}, & x \in (-\infty,1) \\ x, & x \in [1,5]\setminus {3} \\ 100, & x=3 \\ \log_{1/2} {(x-5)} , & x \in (5, +\infty) \end{cases}## For a given function determine the truth of the folowing statements and give a brief explanation: a) Function...
  49. doktorwho

    Integral that is reduced to a rational function integral

    Homework Statement Suggest an integral that is reduced to a rational function integral when this substitution is used: ##a)## ##t=\sin x## ##b)## ##t=\sqrt[6] {x+5}## ##c## ##\sqrt{1-9x^2}=-1+xt## Homework Equations 3. The Attempt at a Solution [/B] I found this to be a very interesting...
  50. jlmccart03

    Series: Determine if they are convergent or divergent

    Homework Statement I have a couple of series where I need to find out if they are convergent (absolute/conditional) or divergent. Σ(n3/3n Σk(2/3)k Σ√n/1+n2 Σ(-1)n+1*n/n^2+9 Homework Equations Comparison Test Ratio Test Alternating Series Test Divergence Test, etc The Attempt at a...
Back
Top