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. Frank Li

    I Can anyone explain the Gamma function to me?

    Γ(n) = ∫x→∞ tn-1 e-t dt?
  2. E

    I How can 'd' mean two different things?

    d is sometimes used to represent an infinitesimal change in a quantity and sometimes a small amount of a quantity. E.g dx vs dM. dV could mean a small volume element and also an infinitesimal change in volume. How can it be used for two different things? My suspicion is that while converting...
  3. Frank Li

    Calculus Any Calculus Starter Textbook suggestions?

    I would like to start learning my calculus course before in school. Are here any textbooks or science reading books that would help me with the situation?
  4. K

    B Is the theory of fractional-ordered calculus flawed?

    Let's talk about the function ##f(x)=x^n##. It's derivative of ##k^{th}## order can be expressed by the formula: $$\frac{d^k}{dx^k}=\frac{n!}{(n-k)!}x^{n-k}$$ Similarly, the ##k^{th}## integral (integral operator applied ##k## times) can be expressed as: $$\frac{n!}{(n+k)!}x^{n+k}$$ According...
  5. E

    I Why is this function constant in this interval?

    This question has a little bit of physics in it, but it's mostly maths. If I have force, or any function f(z), I was told that I can assume it to be constant only in the interval dz. However, in this case, I had to calculate the work done by the spring force as a function of y Over here, I...
  6. E

    I Why can I assume the force to be constant in this interval?

    If I have force, or any function f(z), I was told that I can assume it to be constant only in the interval dz. However, in this case, I had to calculate the work done by the spring force as a function of y Over here, I assumed the spring force, which is a function of its elongation x (F =...
  7. ItsTheSebbe

    I Difference Analysis and Calculus

    I'm a bit torn on what the difference between analysis and calculus is, I read somewhere that calculus is pretty much analysis without proofs? Either way, I see a lot of people mention problems being on calculus 1 or 2 level. I have finished Analysis 1 and 2 and covered stuff like (series, ODE...
  8. cg78ithaca

    A Inverse Laplace transform of a piecewise defined function

    I understand the conditions for the existence of the inverse Laplace transforms are $$\lim_{s\to\infty}F(s) = 0$$ and $$ \lim_{s\to\infty}(sF(s))<\infty. $$ I am interested in finding the inverse Laplace transform of a piecewise defined function defined, such as $$F(s) =\begin{cases} 1-s...
  9. cg78ithaca

    A Inverse Laplace transform of F(s)=exp(-as) as delta(t-a)

    This is mostly a procedural question regarding how to evaluate a Bromwich integral in a case that should be simple. I'm looking at determining the inverse Laplace transform of a simple exponential F(s)=exp(-as), a>0. It is known that in this case f(t) = delta(t-a). Using the Bromwich formula...
  10. cg78ithaca

    A Taylor/Maclaurin series for piecewise defined function

    Consider the function: $$F(s) =\begin{cases} A \exp(-as) &\text{ if }0\le s\le s_c \text{ and}\\ B \exp(-bs) &\text{ if } s>s_c \end{cases}$$ The parameter s_c is chosen such that the function is continuous on [0,Inf). I'm trying to come up with a (unique, not piecewise) Maclaurin series...
  11. S

    Finding length of an arc produced by projectile

    So I decided to try deriving a general formula for fun. Being a high school student, the calculus got scary very fast. At this point, I'm just curious as to what the best approach to this might be. The approach I used was finding y as a function of x and then inputting it into the arc length...
  12. J

    I Multi-dimensional Integral by Change of Variables

    Hi All, $$\int{\exp((x_2-x_1)^2+k_1x_1+k_2x_2)dx_1dx_2}$$ I can perform the integration of the integral above easily by changing the variable $$u=x_2+x_1\\ v=x_2-x_1$$ Of course first computing the Jacobian, and integrating over ##u## and ##v## I am wondering how you perform the change of...
  13. E

    The total tension acting on a rotating rod

    Homework Statement Find the total tension acting on a rod rotating about its end with an angular velocity of w as a function of its length x(length) Homework Equations F = ma[/B] The Attempt at a Solution Let the function be T(x) where x is the length of the rod. Considering an interval...
  14. Kaura

    Exploring Limits at (0,0) for xy/sin(x+y)

    Homework Statement limit (x -> 0 y -> 0) of xy/sin(x+y) Homework Equations None that come to mind but maybe Lopital's Rule The Attempt at a Solution I know that the limit does not exist but I am having trouble figuring out how to show that it does not using the line x=y gives x^2/sin(2x)...
  15. Math Henry

    How Long Until a Cursed Civilization's Population Reaches Zero?

    Homework Statement [/B] Summarizing: two civilizations hate each other, one of the civilizations throws a curse at the second. The second civilization succumbs to chaos and has a change in Population each week of ΔP= -√P. That is: Pn = Pn-1-√Pn-1 Homework Equations [/B] Considering that the...
  16. mr_persistance

    Apostol's Calculus I. 3.12 - Verify Solution.

    1. If x and y are arbitrary real numbers with x < y, prove that there is at least one real z satisfying x<z<y.2. I'll be using this theorem: T 1.32 Let h be a given positive number and let S be a set of real numbers. (a) If S has a supremum, then for some x in S we have x > sup S - h.The Attempt...
  17. F

    I Proof that lattice points can't form an equilateral triangle

    From Courant's Differential and Integral Calculus p.13, In an ordinary system of rectangular co-ordinates, the points for which both co-ordinates are integers are called lattice points. Prove that a triangle whose vertices are lattice points cannot be equilateral. Proof: Let ##A=(0,0)...
  18. Shakir

    Classical What Calculus Book Can Help Understand Goldstein's Classical Mechanics?

    Hello PF I have attached two screenshots from Goldstein's Classical Mechanics. Although I have done a course on multivariable calculus, I don't understand what is going on in this math part. Could you please provide some online resources or suggest a book so I can understand this sort of...
  19. P

    I Solve Challenging Integral with Proven Techniques | x>1 Integer Solution

    Hello. I am having a lot of trouble trying to solve/analyse this integral: $$\displaystyle \int_2^\infty \frac{x+y}{(y)(y^2-1)(\ln(x+y))} dy$$ I have tried everything with no result; it seems impossible for me to work with that natural logarithn. I have also tried to compute it, as it...
  20. Eri ep

    I Differentiating Biot-Savart Law

    Hello! I have the equation FM = qvBsinθ . As the end result, I am trying to figure out what B I need to change θ even a little bit. To do that, I was planning to find the minimum B by differentiating B=(μe/4π)(qv x R / R3) in terms of R and setting it equal to zero. . I am assuming that this...
  21. M

    Calculus of Variations: interesting substitution

    Homework Statement Find the externals of the functional $$\int\sqrt{x^2+y^2}\sqrt{1+y'^2}\,dx$$ Hint: use polar coordinates. Homework Equations ##x=r\cos\theta## ##y=r\sin\theta## The Attempt at a Solution Transforming the given functional where ##r=r(\theta)## yields...
  22. D

    Calculus Calculus of variation textbook 'not under a single integral'

    I have to find functions that maximise certain criterea. The problem can however not be put "under a single integral", for example I've to find ##f(t)##, ##g(t)## that maximise: ## \int_0^{t_e}f(t)^2dt\int_0^{t_e}g(t)^2dt - (\int_0^{t_e}f(t)g(t)dt)^2 ## With ## -1 \leq f(t)\leq1## and ## -1...
  23. L

    Partial derivative of inner product in Einstein Notation

    Homework Statement Can someone please check my working, as I am new to Einstein notation: Calculate $$\partial^\mu x^2.$$ Homework Equations 3. The Attempt at a Solution [/B] \begin{align*} \partial^\mu x^2 &= \partial^\mu(x_\nu x^\nu) \\ &= x^a\partial^\mu x_a + x_b\partial^\mu x^b \ \...
  24. doktorwho

    Examples of functions and sequences

    Homework Statement Give the example and show your understanding: [1][/B].Lets define some property of a point of the function: 1. Point is a stationary point 2. Point is a max/min of a function 3. Point is a turning point of a function If possible name a function whose point has properties of...
  25. K

    Should I retake pre-calc before calculus 1

    I have taken precalculus class a few years ago. I finally took calculus 1 and the same school just last quarter and got a 3.2. I had a pretty cool teacher. I'm transferring to a four year school and have to retake calculus 1, but I'm contemplating on retaking a pre calculus at the four year...
  26. doktorwho

    Evaluating Limits: x Approach 0 & Beyond

    Homework Statement ##\frac{e^x-1}{x}## Evaluate the limit of the expression as x approaches 0. Homework Equations 3. The Attempt at a Solution [/B] The question i have is more theoretical. I was able to solve this problem by expanding the expression into the talyor polynomial at ##x=0##. I...
  27. I

    Kinematics, deriving equations.

    Homework Statement "Derive the equations for position (in terms of acceleration, initial position, initial velocity, and time) and velocity (in terms of constant acceleration, a, initial velocity, v0, and time, t) from the definitions of position, velocity, and acceleration (derivative...
  28. doktorwho

    Help providing function examples

    Homework Statement This is one test question we had today and it asks as to provide examples of functions and intervals. Some may be untrue so we had to identify it. The test isn't graded yet so these are my question answers. Hopefully you'll correct me where necessary and provide a true...
  29. Z

    Programs Which engineering/science uses this the most?

    So I am currently a very indecisive mechanical engineering student, who can't figure out what to major in. I have found out that I am much more interested in solving problems that deal with a lot of equations, substitution, and differential equations than I am solving statics problems. I like...
  30. G

    Multivariable calculus: work in a line segment

    Homework Statement Compute the work of the vector field ##F(x,y)=(\frac{y}{x^2+y^2},\frac{-x}{x^2+y^2})## in the line segment that goes from (0,1) to (1,0). Homework Equations 3. The Attempt at a Solution [/B] My attempt (please let me know if there is an easier way to do this) I applied...
  31. SirHall

    B What form of calculus needs to be used?

    I have recently been attempting to solve a problem that has been bugging me for quite some time. I've gotten back into calculus and integrals to attempt to solve a little formula I'm trying to build for a simulation test. Over-all, if I have ##\int _0^bv^2x\ dx## I'd expect the outcome to be...
  32. S

    I How Does Summing Cubic Expansions Reveal the Formula for Sum of Squares?

    I found a deduction to determinate de sum of the first n squares. However there is a part on it that i didn't understood. We use the next definition: (k+1)^3 - k^3 = 3k^2 + 3k +1, then we define k= 1, ... , n and then we sum... (n+1)^3 -1 = 3\sum_{k=0}^{n}k^{2} +3\sum_{k=0}^{n}k+ n The...
  33. haushofer

    Teaching Calculus, exercises: just hints or worked out?

    Dear all, I'm currently teaching calculus courses with Stewart's book. I was wondering what other teachers their experiences are with giving fully worked out answers to the exercises versus giving just hints and the final answer. I have the feeling that the last approach activates students...
  34. R

    Calculus III (help sketching graph)

    Homework Statement Sketch the curve with the given vector equation. Indicate with an arrow the direction in with 't' increases r(t)=<t, 2-t, 2t>Homework Equations parametric equation (can't type the equation, too confusing to use the template) The Attempt at a Solution So far, I have <1, -1...
  35. B

    I Question on §3 of Einstein's 1905 paper

    Hi guys, This is my first time posting on PF! I have a question on §3 of Einstein's paper "On the electrodynamics of moving bodies."My problem is with the following mathematical statements: Hence, if x' is chosen to be infinitesimally small, or I have just finished high school, and...
  36. ManicPIxie

    Values of d where limit exists. lim x->d

    Homework Statement For which value of d does the following limit exist? lim x->d ln [ (x2-13x+30) / (x-d) ] Homework Equations None The Attempt at a Solution I understand how to find limits when the limit goes to a real number, and has a variable in the function to solve for, but not when...
  37. I

    As f(x) -> oo and g(x) -> c, if c > 0 then f(x)g(x) -> oo

    Homework Statement If ## \lim_{x \rightarrow a} f(x) = \infty## and ##\lim_{x \rightarrow a} g(x) = c ##, and if ##c>0## then prove that ##\lim_{x \rightarrow a} \left[ f(x)g(x)\right] = \infty~~ \text{if c > 0}## Homework Equations Epsilon Delta definition of the limit The Attempt at a...
  38. Vitani11

    Proving the second fundamental theorem of calculus?

    Homework Statement Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x). Homework Equations None The Attempt at a Solution I have ... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be...
  39. cathal84

    Find the equation of a plane perpendicular to a line and goes through a point

    Homework Statement find equation of plane P that is perpendicular to line L which passes through the point (-2,-2,3) Homework Equations ... The Attempt at a Solution [/B]line L passes through the points (1,2,1) and (0,0,-3) I have worked out the parametric equations of line L to be x=1-t...
  40. Alettix

    Antiderivative of 1/x: ln(x) or ln(|x|)?

    Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...
  41. ChrisisC

    Want to Learn Calculus Early? Check Out This Classic Book!

    I'm a 10th grade student in the United States and currently taking geometry which is a breeze, and if anyone else reading this is in the U.S. you know that 10th graders haven't reached calculus yet, not even physics. Since i know I'm going into quantum physics, i have thirst to learn calculus...
  42. P

    MHB What is the value of the triple integral for the given solid and region?

    Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.
  43. cathal84

    Show that f has a stationary point at (0, 0) for every k ∈ R

    Homework Statement Let f(x, y) = x^2 + kxy + y^2 , where k is some constant in R. i. Show that f has a stationary point at (0, 0) for every k ∈ R Homework Equations ... The Attempt at a Solution I may have the solution or i may have gone completely wrong I am not entirely sure. i first found...
  44. B

    Why Does Spivak Exclude Specific Rational Points in His Limit Proof?

    Homework Statement Consider the function $$ f(x) := \begin{cases} 0, & if~ x \in (0,1) - \mathbb{Q} \\ \frac{1}{q}, & if~ x = \frac{p}{q} \in (0,1) \cap \mathbb{Q} \mbox{ in lowest terms. } \\ \end{cases} $$ In his Calculus book, Spivak shows us that ##\lim_{x \rightarrow a} f(x) = 0## for...
  45. C

    I Visual interpretation of Fundamental Theorem of Calculus

    Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If...
  46. bwest121

    How do I calculate this integral?

    Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...
  47. S

    MHB Is the Russell's Paradox Resolved in Predicate Calculus?

    Prove (formall y) in predicate calculus : $\neg\exists y\,\forall x\,(x\in y\leftrightarrow \neg x\in x)$.
  48. cathal84

    Determining the domain and range of multi-variable function

    Homework Statement f(x,y) = 1/y^2-x find the domain of f. Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f. Homework Equations I know that the domain of the function is anywhere that the function is defined. The Attempt at a Solution in the case of...
  49. doktorwho

    What is this expression equal to?

    Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why: ##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...
  50. cathal84

    Finding length of vector with unknown variable

    Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...
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