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. T

    I Euler, Calculus of Variations and Mast on a ship

    From Wikipedia: "In 1727, [Euler] first entered the Paris Academy Prize Problem competition; the problem that year was to find the best way to place the masts on a ship." Does anyone know how he did this? Is there an on-line paper? (But what that is accessible with today's knowledge). And by...
  2. cwill53

    Integrating Mass of a Hollow Sphere: Multivariable Calculus Explained

    I know some multivariable calculus, I just want someone to walk me through the integration deriving the mass element dM and the integration of thin rings composing the hollow sphere. It would also be nice if you could show me doing it one way using the solid angle and one way without using the...
  3. barryj

    Please explain this calculus solution

    If f'(x) were a simpler function like f'(x) = cos(x) I would say f(x) = sin(x) + C and then evaluate C by knowing that 2 = sin(1) + C and then C would equal 2-sin(1) the f(x) = sin(x) + 2 - sin(1), f(0) = sin(0) + 2 - sin(1) = 0 + 2 -.841 = 1.58 However the more complicated problem has f'(x) -...
  4. WMDhamnekar

    MHB Does the Cosine Rule Apply to Vector Addition in 3-D?

    Hi, In $\mathbb{R^3} || v-w ||^2=||v||^2 + ||w||^2 - 2||v||\cdot ||w||\cos{\theta}$ But can we say $||v+w||^2=||v||^2 +||w||^2 + 2||v|| \cdot||w|| \cos{\theta}$ where v and w are any two vectors in $\mathbb{R}^3$
  5. mcastillo356

    Calculus Finding an Alternative to "Calculus" by Robert A. Adams

    Hi PF, Can you tell me about an alternative, substitute for "Calculus", written by Robert A. Adams, from University of British Columbia?. It's good, but I need more bibliography; I find this one too implicit: suggested but not communicated directly. I am now asking doubts to a lot of forums each...
  6. L

    Calculus and Kinematic equations--- seeing the logic

    Details of Question: ds/dt= v which becomes ds=v dt, where s=displacement, t =time, and v=velocity Then we can integrate both sides of this equation, and do a little algebra, and turn the above equation into: s − s0 = v0t + ½at2 My main question is about the integration of...
  7. A

    MHB Calculus 3 Help: Iterated, Double, Triple Integrals & More

    Kindly help me with: Iterated integrals Q1, 2,3 Double integrals in polar coordinates Q1, 2,3 Triple integrals Q1, 2,3 Triple integrals in cylindrical coordinates Q1, 2,3 Triple integrals in spherical coordinates Q1, 2,3 Change of variables Q7,8,9 Green's theorem Q1,2 Surface integrals...
  8. TheGreatDeadOne

    Calculating Gradient of 1/|r-r'|: Tips & Results

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  9. J

    What are the key topics in Advanced Calculus and Algebraic Geometry?

    Hello, I am a very experienced Mathematician with a BSc Honours degree in Mathematics and one year MSc studies in Operational Research in Sussex and London Universities respectively. I am interested in Advanced Calculus, Algebras, Positivity in Algebraic Geometry, The standard Model, and many...
  10. Hamiltonian

    B Basic doubts in vector and multi variable calculus

    If say we have a scalar function ##T(x,y,z)## (say the temperature in a room). then the rate at which T changes in a particular direction is given by the above equation) say You move in the ##Y##direction then ##T## does not change in the ##x## and ##z## directions hence ##dT = \frac{\partial...
  11. A

    MHB Master Calculus Questions with Expert Tips | Boost Your Grades

  12. Baums Mizushala

    Point on a graph nearest to the origin

    The Attempt at a Solution I know the answer is supposed to be ##(-1,0)##. However when I differentiate the above expression I get. $$ 2x+{\frac 5 2} $$ Then the shortest distance would be when the expression equates to 0. $$ 2x+{\frac 5 2}=0 $$ I should be getting ##x=-1## but solving for ##x##...
  13. ElieQuebec10

    I Calculus: What is the derivative of ##\phi## at a point?

  14. karush

    MHB 3.3.2 AP Calculus Exam interval from f'(x)

    screen shot to avoid typos OK the key said it was D I surfed for about half hour trying to find a solution to this but $f'(0)$ doesn't equal any of these numbers $e^0=\pm 1$ from the $e^{(x^2-1)^2}$ kinda ?
  15. karush

    MHB S8.7.2.1 AP calculus Exam (typo problem)

    ok I thot this was just observation to get b. but maybe not I saw some rather hefty substations to get different answers
  16. karush

    MHB What Is the Value of the Integral $\int_0^1 xe^x \, dx$?

    $\tiny{4.2.5}$ $\displaystyle\int^1_0{xe^x\ dx}$ is equal to $A.\ \ {1}\quad B. \ \ {-1}\quad C. \ \ {2-e}\quad D.\ \ {\dfrac{e^2}{2}}\quad E.\ \ {e-1}$ ok I think this is ok possible typos but curious if this could be solve not using IBP since the only variable is x
  17. Quarkman1

    Greetings from a 'late learner' and physics fan

    Hello! I am a 'mature' learner and am fascinated by all kinds of physics and math ideas. Learning is the key to enjoying science and keeping an open mind. I must admit, I am not very sharp on my physics skills and my calculus is pretty rusty now (I don't work in the science field, per se) so I...
  18. T

    Evaluate the Taylor series and find the error at a given point

    I have the following function $$f^{(0)}\left(x\right)=f\left(x\right)=e^{x}$$ And want to approximate it using Taylor at the point ##\frac{1}{\sqrt e} ## I also want to decide (without calculator)whether the error in the approximation is smaller than ##\frac{1}{25} ## The Taylor polynomial is...
  19. J

    A How Do You Solve This Alternating Series Involving Logarithms?

    Hi! Some time ago I came across a series and never solved it, I tried to give a new go because I was genuinely curious how to tackle it, which I thought would work, because it looks innocent, but there is something about the beast making it hard to approach for me. So need some help! Maybe this...
  20. JorgeM

    A How do I express an equation in Polar coordinates as a Cartesian one.

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  21. EchoRush

    Help with deriving the formula for kinetic energy (using calculus)

    Hello, I am learning how to use calculus to derive the formula for kinetic energy now, I understandthe majority of the steps in how to do this, however, there is one step where I get totally lost, I will post a picture of the steps and I will circle the part where I get lost. If you see the...
  22. jaychay

    MHB Calculus airplane related rates problem ( cosine rule)

    A student has test his airplane and he is far from the airplane for 5 meter.He start to test his airplane by letting his airplane to move 60 degree from the horizontal plane with constant velocity for 120 meter per minute.Find the rate of distance between the student and the plane when the plane...
  23. jaychay

    Related rates calculus problem about a water tank

    Summary:: Consider the rectangular water tank, at the base the length is the same for 200 cm. There are 100 holes for water to come out which each hole have the same flow rate. Find the amount of water that come out in each hole by using differential when we know that there is an error in the...
  24. jaychay

    MHB Related rates calculus problem about water tank

    Consider the rectangular water tank, at the base the length is the same for 200 cm. There are 100 holes for water to come out which each hole have the same flow rate. Find the amount of water that come out in each hole by using differential when we know that there is an error in the measurement...
  25. K

    Nabla operations, vector calculus problem

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  26. Bright Liu

    How do I derive this vector calculus identity?

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  27. R

    I Deriving Lorentz Transformations Using Calculus

    We take an arbitrary spacetime point ##(x,t)## in any observer's reference frame ##A##. Let ##(x(v),t(v))## be the co-ordinates of this same event as seen from a frame ##B## moving at a velocity ##v## wrt ##A##. As ##v## varies, the set of points ##(x(v),t(v))## constitute some curve ##C##. So...
  28. rxh140630

    Calculus Apostol's vol 1 calculus not as good as Stewart's calculus?

    Hello, all around the web and even on this website, I've been told countless times that Apostol/Spivak's calculus books are superior to Stewarts. Having personally read about a forth of Apostol's book, and having read half or more of Stewarts, I notice Stewart has better explanations, and better...
  29. karush

    MHB 1.6.1 AP Calculus Exam Limits with L'H

    $\displaystyle\lim_{x \to 0}\dfrac{1-\cos^2(2x)}{(2x)^2}=$ by quick observation it is seen that this will go to $\dfrac{0}{0)}$ so L'H rule becomes the tool to use but first steps were illusive the calculator returned 1 for the Limit
  30. S

    Finding the Determinant to find out if the matrix is invertible

    question: My first attempt: my second attempt: So I am getting 0 (the right answer) for the first method and 40 for the second method. According to the theorem, shouldn't the determinant of the matrix remain the same when the multiple of one row is added to another row? Can anyone explain...
  31. S

    MHB Single Variable Calculus Summary Rulesheet

    Students see my 20+ page calculus bundle on limits, derivatives and integrals and their applications. The summary notes are cleanly written, have background math grid paper, and summarize all major concepts, formulas, and procedures from calculus books. Please tell me what you think and if this...
  32. karush

    MHB Q1 Can you pass this 3 question AP calculus Quiz in 10 minutes

    1. $f(x)=(2x+1)^3$ and let g be the inverse function of f. Given that$f(0)=1$ what is the value of $g'(1)$? A $-\dfrac{2}{27}$ B $\dfrac{1}{54}$ C $\dfrac{1}{27}$ D $\dfrac{1}{6}$ E 6 2. given that $\left[f(x)=x-2,\quad g(x)=\dfrac{x}{x^2+1}\right]$ find $f(g(-2))$...
  33. S

    A Fractional Calculus - Variable order derivatives and integrals

    Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...
  34. Sabertooth

    "Astronomical Calculus" Spaceship Dilation problem

    Hi everyone. I have provided myself a problem that I insist on solving, however, I want to do it "the right way" where I can put every parameter into a calculator and get an answer quickly. I pondered doing it manually and figured that it could be done to a reasonable precision in an hour or...
  35. karush

    MHB 2.4.3 AP Calculus Exam Integration limits

    by observation I choose (c) since the limit values may not be =
  36. WMDhamnekar

    MHB Are $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ Parallel When m+n=1?

    Given $\vec{r}=t^m* \vec{A} +t^n*\vec{B}$ where $\vec{A}$ and $\vec{B}$ are constant vectors, How to show that if $\vec{r}$ and $\frac{d^2\vec{r}}{dt^2}$ are parallel vectors , then m+n=1, unless m=n? I don't have any idea to answer this question. If any member knows the answer to this...
  37. karush

    MHB What Does the Intermediate Value Theorem Guarantee for a Continuous Function?

    I thot I posted this before but couldn't find it ... if so apologize Let f be a function that is continuous on the closed interval [2,4] with $f(2)=10$ and $f(4)=20$ Which of the following is guaranteed by the $\textbf{Intermediate Value Theorem?}$ $a.\quad f(x)=13 \textit{ has a least one...
  38. samuelfarley

    Calculus problem: Questions about the function f (x) = - x / (2x^2 + 1)

    shown in attachment
  39. WMDhamnekar

    MHB How Can I Prove These Vector Calculus Relations?

    Hi, Let f(t) be a differentiable curve such that $f(t)\not= 0$ for all t. How to show that $\frac{d}{dt}\left(\frac{f(t)}{||f(t)||}\right)=\frac{f(t)\times(f'(t)\times f(t))}{||f(t)||^3}\tag{1}$ My attempt...
  40. densephysicist

    Calculus problem regarding Thermodynamics HW (entropy for C2H5OH at 348K)

    Summary:: Seems simple but has me stumped... [Thread moved from a technical forum, so no Homework Template is shown] Hello! I am struggling to use an equation given to me. To provide some context, I am trying to work out the entropy for C2H5OH at 348K. Using provided tabulated data, the...
  41. karush

    MHB 4.1.1 AP calculus Exam Int with U substitution

    Evaluate $\displaystyle\int{\dfrac{{(1-\ln{t})}^2}{t} dt=}$ $a\quad {-\dfrac{1}{3}{(1-\ln{t})}^3+C} \\$ $b\quad {\ln{t}-2\ln{t^2} +\ln{t^3} +C} \\$ $c\quad {-2(1-\ln{t})+C} \\$ $d\quad {\ln{t}-\ln{t^2}+\dfrac{(\ln{t^3})}{3}+C} \\$ $e\quad {-\dfrac{(1-\ln{t^3})}{3}+C}$ ok we can either expand...
  42. Leo Liu

    Calculus What book should I get for multivariable calculus after Stewart?

    Hi. I just finished the single variable part of Stewart's calculus book which helped me to master AP calculus. Now I am planning to move on to non-rigorous multivariable calculus. However, I have found reading his book a bit painful since the book mainly focuses on problem-solving techniques...
  43. O

    MHB Six assorted calculus problems

    I have 48 hours and i am bad, i am sorry but i want to understand how it is
  44. S

    Simplifying expressions using Euler's formula

    The following is the questions given. I solved the first one, which steps are shown below. But I am not sure if this is how the question wants me to solve the problem. Would you tell me if the way I solved the problem is the proper way of simplifying the expression using euler's formula...
  45. B

    Calculus 1 problems: functions, integrals, series

    Mentor note: Moved from technical section, so is missing the homework template. Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?
  46. A

    I The Ratio of Total Derivatives

    If we have two functions C(y(t), r(t)) and I(y(t), r(t)) can we write $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?
  47. T

    Multivariable Calculus, Line Integral

    The vector field F which is given by $$\mathbf{F} = \dfrac{(x, y)} {\sqrt {1-x^2-y^2}}$$ And the line integral $$ \int_{C} F \cdot dr $$C is the path of $$\dfrac{\ (\cos (t), \sin (t))}{ 1+ e^t}$$ , and $$0 ≤ t < \infty $$ How do I calculate this? Anyone got a tip/hint? many thanks
  48. Adesh

    I Checking the integrability of a function using upper and lowers sums

    Hello and Good Afternoon! Today I need the help of respectable member of this forum on the topic of integrability. According to Mr. Michael Spivak: A function ##f## which is bounded on ##[a,b]## is integrable on ##[a,b]## if and only if $$ sup \{L (f,P) : \text{P belongs to the set of...
  49. minimoocha

    MHB Exploring How Archimedes Discovered Quadrature of the Parabola w/o Calculus

    How did Archimedes discover the Quadrature of the parabola without the use of calculus? If someone could please explain, I would be eternally grateful.
  50. ttpp1124

    Calculus and Vectors - Limits and Derivatives

    if someone can concur that'd be great; also, is there any way for me to check myself in the future?
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