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. 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...
  2. ttpp1124

    Calculus and Vectors - Determining Vector and Parametric Equations

  3. ttpp1124

    Calculus and Vectors - Determining intersection for lines and planes

  4. ttpp1124

    Calculus and Vectors - Lines and Planes

  5. ttpp1124

    Calculus and Vectors - Vector and Parametric Equations

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  6. 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...
  7. 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/
  8. 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...
  9. 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?
  10. 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
  11. ttpp1124

    Vectors Question - Calculus

    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.
  12. 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...
  13. A

    Where is this function larger then zero?

    I do not know how to proceed.
  14. 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].
  15. 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...
  16. 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...
  17. 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 $\dfrac{7}{2}$
  18. 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
  19. 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.
  20. 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
  21. 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 ...
  22. 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...
  23. 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
  24. 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
  25. 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...
  26. karush

    MHB 2.1.2 AP Calculus Exam particle at rest

    ok I chose e for the zeros
  27. 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...
  28. 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.
  29. 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.?
  30. 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...
  31. 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 $\pm$
  32. 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.
  33. 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...
  34. 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...
  35. karush

    MHB 2.3.361 AP Calculus Exam of differentials of sin wave

    image due to graph, I tried to duplicate this sin wave on desmos but was not able to. so with sin and cos it just switches to back and forth for the derivatives so thot a this could be done just by observation but doesn't the graph move by the transformations well anyway?
  36. karush

    MHB 1.8.6 AP calculus Exam continuity

  37. karush

    MHB 4.2.251 AP calculus exam concave ?

    ok this one baffled me a little but isn't $g = e^{-t^2}$and the graph of that has an inflection point at x=1
  38. karush

    MHB 2.1.2 AP calculus Exan particle move along the x-axis

    A particle moves along the x-axis. The velocity of the particle at time t is $6t - t^2$. What is the total distance traveled by the particle from time $t = 0$ to $t = 3$ ok we are given $v(t)$ so we do not have to derive it from a(t) since the initial $t=0$ we just plug in the $t=3$ into $v(t)$...
  39. karush

    MHB 3.1.6 AP calculus Exam piece wise integral

    I tried to do this just by observation, but kinda hard with a piece wise function so would presume
  40. polytheneman

    What is the derivative of velocity with respect to position?

    I'm reading a book on Classical Mechanics (No Nonsense Classical Mechanics) and one particular section has me a bit puzzled. The author is using the Euler-Lagrange equation to calculate the equation of motion for a system which has the Lagrangian shown in figure 1. The process can be seen in...
  41. karush

    MHB 4.1.286 AP Calculus Exam .... table of f(t).

    ok these always baffle me because f(t) is not known. however if $f'(t)>0$ then that means the slope is aways positive which could be just a line. but could not picture this to work in the tables. Im sure the answer can be found quickly online but I don't learn by copy and paste. d was...
  42. karush

    MHB 1.8.4 AP Calculus Exam Integral of piece-wise function

    image due to macros in Overleaf ok I think (a) could just be done by observation by just adding up obvious areas but (b) and (c) are a litte ? sorry had to post this before the lab closes
  43. J

    G-force experienced by a jumper, using a DE

    We are given that ##v' = \frac{1}{10}v^2 - g##. I tried using implicit differentiation so that ##v'' = \frac{1}{5}vv' = \frac{1}{5}v(\frac{1}{10}v^2-g)## and set this equal to 0. Hence we have 3 critical points, at ##v= 0##, and ##v = \pm \sqrt{10g}##. Calculating ##v''(0)=-120##, we know the...
  44. karush

    MHB 3.3.300 AP Calculus f' interval

    image due to macros in overleaf well apparently all we can do is solve this by observation which would be the slope as x moves in the positive direction e appears to be the only interval where the slope is always increasing
  45. EchoRush

    Deriving the kinetic energy equation?

    Hello, I just have a quick question on deriving the kinetic energy formula using calculus. I understand most of it, I just have a question about one of the steps. here are the steps. Begin with the Work-Energy Theorem.The work that is done on an object is related to the change in its kinetic...
  46. karush

    MHB 2.2.1 AP Calculus Exam .... derivative with ln

    If $f(x)=7x-3+\ln(x),$ then $f'(1)=$ $a.4\quad b. 5\quad c. 6\quad d. 7\quad e. 8$ see if you can solve this before see the proposed solution
  47. karush

    MHB 303 AP calculus int with initial value

    ok image to avoid typo... try to solve before looking at suggested solutions ok I think you could do this by observation if you are careful with signs
  48. karush

    MHB AP calculus exam tikx graph of e (tan x ) -2

    The graph of $y=e^{\tan x} - 2$ crosses the x-axis at one point in the interval [0,1]. What is the slope of the graph at this point. A. 0.606 B 2 C 2.242 D 2.961 E 3.747ok i tried to do a simple graph of y= with tikx but after an hour trying failed doing this in demos it seens the answer is...
  49. fulton33

    Calculating the equations for the tangent/normal lines

    9. When I do this problem I know my slope is -3 because f'(2)=-3. I then went and substituted and got y+5=-3(x-2) which simplified to y=-3x+1 10. I get lost here because the tangent slope would be 0, which would give me the equation y=-2. The normal means perpendicular and the perpendicular...
  50. karush

    MHB 3.1.5 AP Calculus Exam average temperature

    309 average temperature $$\begin{array}{|c|c|c|c|c|c|c|} \hline t\,(minutes)&0&4&9&15&20\\ \hline W(t)\,(degrees Farrenheit)&55.0&57.1&61.8&67.9&71.0\\ \hline \end{array}$$ The temperature of water in a tub at time t is modeled by a strictly increasing, twice-differentiable function W. where...
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