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. 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...
  2. samuelfarley

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

    shown in attachment
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. O

    MHB Is This Integral Divergent as t Approaches Infinity?

    I have 48 hours and i am bad, i am sorry but i want to understand how it is
  8. 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...
  9. 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?
  10. 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 $$\frac{\frac{dC}{dt}}{\frac{dI}{dt}}=\frac{dC}{dI}$$?
  11. 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
  12. 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...
  13. 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.
  14. 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?
  15. ttpp1124

    Calculus and Vectors - Limits and Derivatives

  16. F

    Thermodynamic equation of differentials (and how to work with them)

    Disclaimer: I am not a mathematician, I am a physicist. The thermodynamic identity is usually expressed in the following differential form $$ dU = TdS - PdV + \mu dN, $$ where U , T , S , P , V , \mu and N are the internal energy, temperature, entropy, pressure, volume, chemical...
  17. karush

    MHB 4.1.306 AP Calculus Exam Area under Curve

    $\textsf{What is the area of the region in the first quadrant bounded by the graph of}$ $$y=e^{x/2} \textit{ and the line } x=2$$ a. 2e-2 b. 2e c. $\dfrac{e}{2}-1$ d. $\dfrac{e-1}{2}$ e. e-1Integrate $\displaystyle \int e^{x/2}=2e^{x/2}$ take the limits...
  18. ttpp1124

    Calculus and Vectors - Determining Vector and Parametric Equations

  19. ttpp1124

    Calculus and Vectors - Determining intersection for lines and planes

  20. ttpp1124

    Calculus and Vectors - Lines and Planes

  21. ttpp1124

    Calculus and Vectors - Vector and Parametric Equations

    -
  22. P

    MHB Optimizing Truck Speed and Freight Load for Cost Efficiency on Interstate Travel

    A truck traveling interstate, driving at a constant speed of 110km/h, gets 7km/L efficiency and loses 0.1km/L in fuel efficiency for each km/h increase in speed. Costs include diesel ($1.49/L), truck drivers’ wage ($35/hour), and truck maintenance and repairs ($9.50/hour). This truck is mainly...
  23. jedishrfu

    B Bartlett's Calculus Paper: Confusing Calculus Notation & Differentials

    Bartlett’s Calculus Paper Reviewed in Mathematics Magazine https://mindmatters.ai/2020/03/bartletts-calculus-paper-reviewed-in-mathematics-magazine/
  24. O

    How to prove this statement about the derivative of a function

    My try: ##\begin{align} \dfrac{d {r^2}}{d r} \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \tag1\\ \dfrac{\partial r}{\partial p} = \dfrac{\partial {r^2}}{\partial p} \dfrac{1}{\dfrac{d r^2}{d r}}=\dfrac{p-a\cos\theta}{r} \tag2\\ \end{align}## By chain rule...
  25. K

    I Vector calculus identity format question

    I know there is an identity involving the Laplacian that is like ##\nabla^2 \vec A = \nabla^2 A## where ##\vec A## is a vector and ##A## is its magnitude, but can't remember the correct form. Does anyone knows it?
  26. karush

    MHB What is the AP calculus exam IVT question?

    Ok I thot I posted this before but after a major hunt no find Was ? With these options since if f(x) Is a curve going below the x-axis Is possible and () vs [] If this is a duplicate post What is the link.. I normally bookmark these Mahalo ahead
  27. ttpp1124

    How Do You Calculate the Tension in a Rope Holding a Chandelier?

    A 45 kg chandelier is suspended by two chains of lengths 5 m and 8 m attached to two points in the ceiling 11 m apart. Find the tension in the 5 m rope.
  28. ttpp1124

    Vectors Question using Calculus -- Swimmer crossing a River

    Hi! I have a physics question I need help with. Bob can swim at 4 m/s in still water. He wishes to swim across a river 200 m wide to a point directly opposite from where he is standing. The river flows westward at 2.5 m/s and he is standing on the South bank of the river. a. What is the speed...
  29. A

    Where is this function larger then zero?

    I do not know how to proceed.
  30. L

    What other information is needed to solve this equation involving b?

    the equation is not equal not to anything (example: 3x = 6) and i don't know how to resolve [e ^ - (b / 2m) t].
  31. karush

    MHB 6.1.1 AP Calculus Inverse of e^x

    If $f^{-1}(x)$ is the inverse of $f(x)=e^{2x}$, then $f^{-1}(x)=$$a. \ln\dfrac{2}{x}$ $b. \ln \dfrac{x}{2}$ $c. \dfrac{1}{2}\ln x$ $d. \sqrt{\ln x}$ $e. \ln(2-x)$ ok, it looks slam dunk but also kinda ? my initial step was $y=e^x$ inverse $\displaystyle x=e^y$ isolate $\ln{x} = y$ the...
  32. S

    Schwarzschild coordinate time integral

    I have tried integration by parts where, ##c dt = -\frac{1}{\sqrt{r*}} \frac{r^{3/2} dr}{r - r*} = \frac{1}{\sqrt{(r*)^3}} \frac{r^{3/2} dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##u = r^{3/2} \quad \quad dv = \frac{dr}{1 - \Big(\sqrt{\frac{r}{r*}} \Big)^2}## ##du = \frac{3}{2} r^{1/2} dr...
  33. karush

    MHB 4.5.1 AP Calculus Exam .... area of piece wise funcion

    ok well it isn't just adding the areas of 2 functions but is $xf(x)$ as an integrand Yahoo had an answer to this but its never in Latex so I couldn't understand how they got $\dfrac{7}{2}$
  34. G

    Calculus What is the best book for learning functional calculus for physicists?

    Hi, I'm looking for a book that explains more deeply (and a little bit more formal) the functional calculus than the typical introductions that I find in QFT books (like Peskin or Hatfield). Is there any good book for physicists to learn the mathematics behind functional calculus? Thanks
  35. S

    Uncertainty for n using calculus

    i'm thinking of differentiating the inside of both sin functions but I'm not sure what to do with the sin. if anything, I'm new to this sort of uncertainty calculation. I have calculated the uncertainty and values for both Dm and a in advance.
  36. karush

    MHB 2.7.2 AP Calculus Exam .... distance traveled estimate

    https://www.physicsforums.com/attachments/9542 ok all I did was count squares so about 21
  37. karush

    MHB 1.8.1 AP Calculus find k for critical point

    $\displaystyle g'=2xe^{kx}+e^{kx}kx^2$ we are given $ x=\dfrac{2}{3}$ then $\displaystyle g'=\dfrac{4}{3}e^\left(\dfrac{2k}{3}\right)+e^\left({\dfrac{2k}{3}}\right)\dfrac{4k}{9}$ ok something is ? aren't dx supposed to set this to 0 to find the critical point did a desmos look like k=-3 but ...
  38. cookiemnstr510510

    Turning a real world problem into a calculus problem

    So I am a bit confused on how to get started. So far my thought process is we have water flowing in and water evaporating from the pool. The part that I think we are interested in is the leakage. The leakage has the rate it is flowing out per unit time. I will call it change in volume, or dV...
  39. karush

    MHB 4.2.251 AP Calculus Exam......must be true on the interval 0<x<2

    https://www.physicsforums.com/attachments/9527 ok from online computer I got this $\displaystyle\int_0^x e^{-t^2}=\frac{\sqrt{\pi }}{2}\text{erf}\left(t\right)+C$ not sure what erf(t) means
  40. P

    Studying Should I read Introduction to Calculus and Analysis by Courant?

    Should i read Introduction to Calculus and Analysis by Courant? I have calculus background I want to study multivariable and single variable in rigorous way There is a lot of books in this subject like Spivak, apstoal,Courant,... . I am not sure what to choose
  41. U

    Other Is Richard Courant's Intro to Calculus Enough for the USAPhO Physics Exam?

    Hello, I am preparing for a physics exam which takes place next year. The scope of this test is mechanics, e&m, thermodynamics, relativity, waves, and modern physics. The exam doesn't require anything farther than Calculus 1, but it is still a rigorous exam. So I am looking for a calc 1 textbook...
  42. karush

    MHB 2.1.2 AP Calculus Exam particle at rest

    ok I chose e for the zeros
  43. karush

    MHB 1.4.293 AP Calculus Exam a(t) and v(t) t=8

    OK, this can only be done by observation so since we have v(t) I chose e but the eq should have a minus sign. here the WIP version of the AP Calculus Exam PDF as created in Overleaf https://documentcloud.adobe.com/link/track?uri=urn%3Aaaid%3Ascds%3AUS%3A053a75d8-ca5b-4447-bd65-4e580f0de793...
  44. Frigus

    B Calculus for Beginners: Basics & Examples

    My question is as it is as you have read in the heading and please tell me in taking mind that I am not a calculus master but just a beginner.
  45. karush

    MHB 4.1.310 AP calculus Exam Area under to functions

    ok I got stuck real soon... .a find where the functions meet $$\ln x = 5-x$$ e both sides $$x=e^{5-x}$$ok how do you isolate x? W|A returned $x \approx 3.69344135896065...$ but not sure how they got itb.? c.?
  46. karush

    MHB 2.6.284 AP calculus Exam Lamp and Shadow related rates

    yes I know this is a very common problem but likewise many ways to solve it ok I really have a hard time with these took me 2 hours to do this looked at some examples but some had 3 variables and 10 steps confusing to get the ratios set up... ok my take on it is here see if you can solve...
  47. karush

    MHB 1.6.365 AP Calculus Exam Limits

    ok I chose c being false, since a limit does not exist if f(x) is different coming from $\pm$
  48. Pispi Choudhury

    Calculus Book suggestions and good lecture notes on the calculus of variation

    I need suggestions on books and good lecture notes on calculus of variation. I've previously studied vector calculus and multivariable calculus.
  49. sergey_le

    A question not related to Calculus

    Homework Statement:: I don't understand what I need to write here Homework Equations:: I don't understand what I need to write here hello :) I recently posted a post and it was deleted because I did not comply with forum rules. Now I'm trying to figure out what to do right. So I want to ask...
  50. Saracen Rue

    I Intersections between this infinite power tower and a multifactorial

    For those unaware of multifactorial notation, it should be noted that there are some common mistakes made when first being introduced to the notation. For example, ##n! \neq (n!)!## and ##n! \neq (n!)! \neq (n!)! \neq ((n!)!)!##. Just to make sure we're all up to speed, here's a quick run down...
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