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. M

    MHB Calculus and Analytic Geometry

    My question is not a math question. I know about the calculus sequence (CAL 1, 2 and 3). I plan to go through all 3 in time. There is no rush for me. However, I know there is a course by the title of Calculus and Analytic Geometry. I want to know when this course is given. Is it given after...
  2. MermaidWonders

    MHB Integral Calculus - Spot the Error

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  3. MermaidWonders

    MHB True or False Integral Calculus Question #3

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  4. MermaidWonders

    MHB True or False Integral Calculus Question #2

    True or False: Let $F(x)$ be an antiderivative of a function $f(x)$. Then, $F(2x)$ is an antiderivative of the function $f(2x)$.
  5. MermaidWonders

    MHB True or False Integral Calculus Question #1

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  6. Pencilvester

    I Euler’s approach to variational calculus

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  7. KFSKSS

    B Need some help with understanding linear approximations

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  8. Dethrone

    MHB Layout Notation for matrix calculus

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  9. Phantoful

    How do I define a region in R3 with spherical/polar coords?

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  10. orangeraindrops

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  11. N

    I Solving Integral for All n≥2 | Evans PDE's (Page 48)

    In the book from Evans on PDE's (page 48) I came across this integral. Here r > 0 and \delta is an arbitrarily small number. Could you give me some hint on how to solve this integral for all integers n\geq2 , i.e why does it go to zero as t approaches zero from the right side.
  12. C

    Calculus 2 - Trig Integrals Question (Integrating cos^2x)

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  13. Roverse

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  14. ertagon2

    MHB Fundamental theorem of calculus and more....

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  15. M

    Calculating Eigenvectors for a 3x3 Matrix: Understanding the Process

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  16. A

    I Why does this concavity function not work for this polar fun

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  17. W

    Vector Calculus: Gradient of separation distance

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  18. W

    Vector Integration: Fundamental theorem use

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  19. S

    Maximizing Friction Coefficient for Block on a Wedge: A Calculus Approach

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  20. S

    I Relativistic Calculus Books & PDFs | Free Resources

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  21. V

    What went wrong with my simple differential equation?

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  22. D

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  23. GaussianSurface

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  24. GaussianSurface

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  25. M

    A Eigenvalue Problem and the Calculus of Variations

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  26. Mzzed

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  27. H

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  28. paulo84

    B How does calculus relate to dimensions?

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  29. Delta31415

    Calculus *BEST* Calculus 3 Textbook for self study

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  30. PeroK

    Demystifying the Chain Rule in Calculus - Comments

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  31. M

    I Directional Derivative demonstration

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  32. V

    Normalization constant for a 3-D wave function

    Homework Statement Show that the normalized wave function for a particle in a three-dimensional box with sides of length a, b, and c is: Ψ(x,y,z) = √(8/abc) * sin(nxπx/a)* sin(nyπy/b)* sin(nzπz/c). Homework Equations Condition for the normalization: ∫0adx ∫0bdy ∫0cdz Ψ*(x,y,z)Ψ(x,y,z) = 1...
  33. A

    MHB How Do You Calculate the Radius of Curvature for Complex Curves?

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  34. M

    B Self learn calculus for UK A-levels?

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  35. Saqib Ali

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

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  36. M

    Can somebody tell me what this topic is?

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  37. A

    I Calculation Of the energy Of beta decay in tritium

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  38. 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...
  39. Adgorn

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

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  40. lfdahl

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

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  41. 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...
  42. 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.
  43. 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 $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...
  44. 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...
  45. A

    Calculus Feynman's High School Calculus Book

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  46. K

    I Need a help in solving an equation (probably differentiation

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  47. 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...
  48. lichenguy

    Dimensional analysis problem

    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...
  49. 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.
  50. 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...
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