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

    Finding the Second Derivative of a Function with Two Variables

    Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2 starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2= [(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2 For the derivative in respect to y...
  2. L

    I Calculus problem differentiation.

    <Moderator's note: Member has been warned to show some effort before an answer can be given.> Hello all. This is my first post in this forum, I am asking for your understanding. I have a problem with the calculus task and I stuck in a dead endso I managed to find a solution on the internet. I...
  3. A

    What is the Correct Approach to Solve This Complex Integral?

    I split this to get \begin{equation} \int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx \end{equation} Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...
  4. S

    I Sum of Binomial Expansion | Spivak Chapter 2, Excercise 3 part d

    Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...
  5. JD_PM

    Looking for a bunch of solved Sympy problems (Calculus)

    Two weeks ago I had no idea on how to code using Python. Now I have completed an online course on functions, loops and strings. However, in that course I did not practice using the specific library called Sympy. Besides, I will use Python in the Physics-Math background, for solving problems like...
  6. JD_PM

    Understanding why we compute surface area as we do

    Homework Statement Homework Equations The Attempt at a Solution [/B] The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure. Question: 1) Why do we differentiate with...
  7. JD_PM

    Proving that a vector field is conservative

    Homework Statement Homework Equations $$F = \nabla \phi$$ The Attempt at a Solution Let's focus on determining why this vector field is conservative. The answer is the following: [/B] I get everything till it starts playing with the constant of integration once the straightforward...
  8. JD_PM

    Computing the Magnitude of F Using Gradient

    Homework Statement Homework Equations $$F = \nabla \phi$$ $$| F | = \sqrt{F \cdot F}$$The Attempt at a Solution I want to compute ##| F | = \sqrt{F \cdot F}## $$| F | = \sqrt{F \cdot F} = \sqrt{\frac{(km)^2 (r-r_0)^2}{|r-r_0|^6}} = \sqrt{\frac{(km)^2}{(r-r_0)}} =...
  9. P

    Variational Calculus: When Is dg(r=r+) ≠ dg(r=r++)?

    Homework Statement Question: If ##r_+ \neq r_{++}## and ## g(r=r_+) \neq g(r=r_{++}) ## When is it fulfilled that ## d g (r=r_+) \neq d g (r=r_+) ## ? Homework Equations ##r_+ \neq r_{++}## ## g(r=r_+) \neq g(r=r_{++}) ## The Attempt at a Solution I tried computing ## dg(r_+) =...
  10. K

    Why Does the Fourier Series of |sin(x)| Treat n=1 Differently?

    Homework Statement Hello, i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...
  11. JD_PM

    Python Python for Vector Calculus: Books & Online Resources

    I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus). If you also have online recommendations please feel free to write them.
  12. S

    How to Evaluate the 8th Derivative of a Taylor Series at x=4

    Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
  13. A

    I Faking a Formula For Movement Through Gravity

    I have a strange question. It's strange because I don't need a correct answer. I need an answer that seems correct and leads to predictable results. I'm making a multiplayer computer game where the players fire cannons in outer space. The cannon shells will move through the gravitational fields...
  14. matai

    Integral for the linear speed of the Earth

    I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...
  15. S

    I How to apply the disk/washer and shell methods

    In Calculus II, we're learning about solids of revolutions and computing their volumes. I'm unsure when to apply the appropriate methods and how to make the correct partitions. Please tell me if my reasoning is correct: The disk/washer method is applied when your partitions are perpendicular...
  16. matai

    Using Integrals to Calculate the Rotational Energy of Earth

    So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that...
  17. S

    Area of a bounded region using integration

    In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
  18. M

    Studying Learning calculus with a digital copy of a textbook

    Hi, I'm in 12 grade, and I always was a decent student but recently I became more fascinated by learning and I want to learn some math and physics beyond high school level. One or two weeks ago I started the calculus course on mit ocw (18.01sc) (at session 17 right now, right before lecture 7)...
  19. Q

    I Deriving the spherical volume element

    I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using $$dxdydz = \left (\frac{\partial x}{\partial r}dr +...
  20. WMDhamnekar

    MHB How Accurate Are the Calculations for Plane Speed and Cable Tension?

    1)An airplane heads due east at 300 mph through a tailwind whose velocity is given by $w=\langle20,20\rangle$ How fast is the tailwind blowing? In what direction? How fast is the plane flying? In what direction?Answer: The tailwind is blowing at 28.2843 mph approx.in $45^{\circ}$ direction...
  21. CharlieCW

    How do I deduce some basic thermodynamic identities using multivariate calculus?

    Homework Statement Let x, y and z satisfy the state function ##f(x, y, z) = 0## and let ##w## be a function of only two of these variables. Show the following identities: $$\left(\frac{\partial x}{\partial y}\right )_w \left(\frac{\partial y}{\partial z}\right )_w =\left(\frac{\partial...
  22. Q

    Maximum Horizontal Force of Relativistic Point Charge

    Homework Statement A charge q1 is at rest at the origin, and a charge q2 moves with speed βc in the x-direction, along the line z = b. For what angle θ shown in the figure will the horizontal component of the force on q1 be maximum? What is θ in the β ≈ 1 and β ≈ 0 limits? (see image) Homework...
  23. B

    Textbook for calculus of variations? Hamiltonian mechanics?

    I need to learn about Hamiltonian mechanics involving functional and functional derivative... Also, I need to learn about generalized real and imaginary Hamiltonian... I only learned the basics of Hamiltonian mechanics during undergrad and now those papers I read show very generalized version...
  24. M

    Is it possible to integrate acceleration?

    Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion: Mg*sin{α} - 1.5m*x(double dot)=0 I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...
  25. Savian

    Courses Is there a calculus requirement in Europe for Biophysics?

    I saw the roadmap of my university of and there is no calculus for biophysics, as soon as there is no calculus for medicine, as for example there is in the US. My question is if there is calculus for biophysics in Europe and in the US. Adendum: there is only calculus in Brazil for medicine in...
  26. S

    Calculus Problem: acceleration, speed, and displacement of a particle

    Homework Statement The acceleration of a particle given a=A√t where A=2.0 m/s5/2. At t=0, v=7.5 m/s and x=0. (a) What is the speed as a function of time? (b) What is the displacement as a function of time? (c) What are the acceleration, speed, and displacement at t=5.0s. Homework EquationsThe...
  27. N

    Calculus angular acceleration with respect to theta

    Homework Statement A disk with a 0.4 m radius starts from rest and is given an angular acceleration α = (10θ2/3)rad/s2 , where θ is in radians. Determine the magnitude of the normal (centripetal and tangential components of a point P on the rim of the disk when t = 4s. Homework Equations α =...
  28. dkeating

    Pressurized gas container gets opened

    Homework Statement I have an empty 2-Liter bottle. It contains 3 g of air inside with an initial air pressure of 1105 mb. When I open it (which is an adiabatic process), I release the pressure which is instantaneous. The pressure then becomes standard atmospheric pressure. What is the...
  29. CivilSigma

    Characteristic Function Integrand Evaluation

    Homework Statement [/B] I am trying to determien the characteristic function of the function: $$ f(x)= ae^{-ax}$$ $$\therefore E(e^{itx}) =\int_0^\infty e^{itx}ae^{-ax} dx = a \cdot \frac{e}{it-a} |_0 ^ \infty $$ But I am not sure how to evaluate the integral. Wolfram alpha suggests this...
  30. KF33

    Solving Calculus Homework: Stuck on #11 Riemann's Sum

    Homework Statement I am stuck on number 11 on my homework. Homework Equations Not Sure The Attempt at a Solution I know this has to have something to do with Riemann's Sum, but I am lost on where to start. I started by putting numbers in for p, but I think that is wrong.
  31. Adgorn

    Contradicting solutions to a basic movement problem.

    Homework Statement Hi, everyone. I came across a basic calculus problem concerning movement of 2 problems, I've attempted to solve it using vector analysis and got 1 answer, and then solved it with differentiation and got a different answer. I'll show a version of it I made which is a bit more...
  32. navneet9431

    Evaluating the Limit of Cosine Function Using L'Hospital's Rule - Explained

    <Moderator's note: Moved from a technical forum and thus no template.> $$\lim_{x \to 0} \cos(\pi/2\cos(x))/x^2$$ I tried to evaluate the limit this way, $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2$$ since $$\cos0=1$$ $$\lim_{x \to 0} \cos(\pi/2\cdot1)/x^2=\lim_{x \to 0} 0/x^2$$ Now apply...
  33. Adgorn

    B Questions regarding polynomial divisions and their roots

    Hello everyone, Going through calculus study, there is a vague point regarding polynomials I'd like to make clear. Say there's a polynomial ##f## with a root at ##a## with multiplicity ##2##, i.e. ##f(x)=(x-a)^2g(x)## where ##g## is some other polynomial. I define ##h(x)=\frac {f(x)} {x-a}##...
  34. H

    MHB Calculus: Understanding Infinity in Functions

    When have a function and I know by investigation of that it getting "bigger and bigger" or getting "smaller and smaller", how could I know that in infinity it continue by that way always?
  35. Philip Koeck

    I Applying a constraint in the calculus of variations

    I have an analytical function F of the discrete variables ni, which are natural numbers. I also know that the sum of all ni is constant and equal to N. N also appears explicitly in F, but F is not a function of N. F exists in a coordinate system given by the ni only. Should I carry out the...
  36. Q

    Cylindrical Coordinates: Line Integral Of Electrostatic Field

    Homework Statement An electrostatic field ## \mathbf{E}## in a particular region is expressed in cylindrical coordinates ## ( r, \theta, z)## as $$ \mathbf{E} = \frac{\sin{\theta}}{r^{2}} \mathbf{e}_{r} - \frac{\cos{\theta}}{r^{2}} \mathbf{e}_{\theta} $$ Where ##\mathbf{e}_{r}##...
  37. Adgorn

    Understanding the solution to a calculus problem (removable discontinuities)

    Homework Statement The problem (Spivak's Calculus, chapter 6, problem 17): "Let ##f## be a function with the property that every point of discontinuity is a removable discontinuity. This means that ##\underset {y \rightarrow x} {\lim} {f(y)}## exists for all ##x##, but ##f## may be...
  38. EEristavi

    Finding Length of a Curve: y2 = (x-1)3

    Homework Statement I have to find length of the curve: y2 = (x-1)3 from (1,0) to (2,1) Homework Equations s = ∫ √(1 + (f '(x) )2 ) dx where we have integral from a to b The Attempt at a Solution I'm bit confused: I'm thinking of writing function regarding x, f(x)...
  39. Adgorn

    Convergence of Roots at Infinity

    Homework Statement Hi everyone, I'm currently making my way through Spivak's calculus and got stuck in question 41 of chapter 5. It's important to note that at this point, the book has only reached the subject of limits (haven't reached continuous functions, derivatives, integrals, series...
  40. J

    B Geodesic dome parametric formula

    I've been researching for the calculus behind geodesic domes, and specifically calculus related to parametric surfaces. I've found http://teachers.yale.edu/curriculum/viewer/new_haven_06.04.05_u#f, but unfortunately, it comes short of providing me the most needed information, and so far I...
  41. CaptainAmerica17

    Need help on a proof from Spivak's Calculus

    Homework Statement I'm currently working through Spivak independently and have reached the problems at the end of ch. 1. The problem is: Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b Homework Equations Spivak's properties P1 - P12 The Attempt at a Solution I was...
  42. SebastianRM

    I What is the 'formal' definition for Total Derivative?

    A total derivative dU = (dU/dx)dx + (dU/dy)dy + (dU/dz)dz. I am unsure of how to use latex in the text boxes; so the terms in parenthesis should describe partial differentiations. My question is, where does this equation comes from?
  43. A

    Calculus Multivariable calculus without forms or manifolds

    Hi there all, I'm currently taking a course in Multivariable Calculus at my University and would appreciate any recommendations for a textbook to supplement the lectures with. Thus far the relevant material we've covered in a Single Variable course at around the level of Spivak and some Linear...
  44. J

    Courses Can I Succeed in Calculus and Maintain My GPA?

    I tried my best but the result doesn't match the effort I've put in. First when I know my result I freaked out and questioned my competent. Am I mathematically-inept? Am I cut out for a Physics Phd? End up with ~70% on this course and I'm taking Calc 2 this December. I need to...
  45. Adesh

    I How to convert the limit of a series into an integral?

    If I have a limit of a series then how can I convert it into integral. I know to convert a sum into an integral there must be Δx multiplied to each term and this must go zero. Can you please explain me the conversion of limit of series (normal series with no Δx) into an integral. Thank you.
  46. CaptainAmerica17

    Programs Don't like my calculus class, but want to be a math major....

    I am currently a junior in high school and recently, my guidance counselor has been asking me a lot of questions about what I want to major in. Within the last 6-8 months, I have been leaning heavily towards a math major. That was until I started my calc class this year... I'm in AP Calc BC...
  47. astroman707

    Studying What books are good to learn the math in intro physics?

    I'm struggling with the math used in my college's calc-based honors physics class, even though I've taken calculus 1. ---What are some good books/resources to learn the math used in introductory physics?--- Preferably, it'd be nice if the math was taught using examples in physics. Having that...
  48. astroman707

    Courses Is it okay to not understand the calculus in intro physics?

    I don't understand a good portion of the non-algebraic math behind much of the physics in my first semester college class. I understand everything with algebra, and can solve all problems, but I don't understand the relationships with vector cross/dot products, calculus derivations, DE, etc...
  49. D

    How Do You Apply the Chain Rule in This Multivariable Calculus Problem?

    I've been working on this one for a little bit, and I know I really just need to use the chain rule to solve it, but I can't seem to figure out how to set it up properly. Probably a dumb question, but I could really use some help on this!
  50. K

    Volume of revolution around the y-axis

    Homework Statement Hello, a bowl is created when rotating the function f(x) = \begin{cases} 0, & 0 \leq x \lt 6 \\ (12/\pi)arcsin(x-6), & 6\leq x \leq 7 \end{cases} around the y-axis. Find the height (h) and the volume (V) of the bowl.Homework EquationsThe Attempt at a Solution So, I graphed...
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