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 2.7.3 AP calculus Exam Riemann sum

    ok basically t is 3 hours appart except between 7 and 12 of which I didn't know if we should intemperate. other wise it is just adding up the 4 $(t)\cdot(R(t))$s.
  2. M

    Find the volume of the solid formed by the rotation around the y=0

    Hi, I find this... Please tell me your opinion on this. Thanks.
  3. karush

    MHB 9.1.317 AP calculus exam multiple choice derivatives of sin wave

    ok just posted an image due to macros in the overleaf doc this of course looks like a sin or cos wave and flips back and forth by taking derivatives looks like a period of 12 and an amplitude of 3 so... but to start I was not able to duplicate this on desmos altho I think by observation alone...
  4. S

    MHB Dominant Terms in Calculus Limits

    Hello, I am having issues finding the dominant terms in the following expression: lim [(x^7)-9(e^x)] / [sqrt(10x-1)+8*ln(x)] x->infinity Prompt: Find the limit and the dominant term in the numerator and denominator.
  5. Gumbrain

    I Help With Simple Orbital Modeling

    I have yet to decide on values for the mass of the fixed object, M, the mass of the moving celestial body, m, the initial velocity, v, and the distance between the two objects, r. I will most definitely decide on a larger mass M because I would like the celestial body to spiral in towards the...
  6. EchoRush

    I Questions about the fundamental thoerem of calculus

    As you can see form my previous posts, I am in my first university level calculus class ever. It is going very well, and through the class I am asking good questions and trying to actually make connection with the stuff we arr doing - not just doing the math just for the sake of passing - I am...
  7. karush

    MHB 2.1.314 AP Calculus Exam a particle moves along the x-axis......

    ok again I used an image since there are macros and image I know this is a very common problem in calculus but think most still stumble over it inserted the graph of v(t) and v'(t) and think for v'(t) when the graph is below the x-axis that participle is moving to the left the integral has a...
  8. karush

    MHB 4.2.8 AP calculus Exam Integration limits

    ok I posted a image to avoid any typos but was wondering why the question has dx and options are in dt I picked C from observation but again that was assuming f was a horizontal line of which it could be something else that way the limits stay the same but the area is cut in halfopinions...
  9. karush

    MHB 5.1.313 AP Calculus Exam DE on bird weight

    I just posted a image due to overleaf newcommands and graph ok (a) if we use f(20) then the $B=0$ so their no weight gain. (b), (c), was a little baffled and not sure how this graph was derived...
  10. B

    Intro Physics Calculus based intro physics textbook recommendations?

    Hi, I'm a first year physics student who still hasn't found a textbook for his class. Our professor mentioned that any calculus based book is okay, but I can't seem to find anything! He suggested Halliday and Resnick's Fundamentals of Physics (extended edition), and although it covers the...
  11. karush

    MHB 2.1.312 AP Calculus Exam Int of half circle

    The function f is defined by $$f(x)=\sqrt{25-x^2},\quad -5\le x \le 5$$ (a) Find $f'(x)$ apply chain rule $$ \dfrac{d}{dx}(25-x^2)^{1/2} =\dfrac{1}{2}(25-x^2)^{-1/2}2x =-\frac{x}{\sqrt{25-x^2}}$$ (b) Write an equation for the tangent line to the graph of f at $x=-3$...
  12. karush

    MHB 9.2.2 AP Calculus Exam -- slope field for which DE

    ok probably if one did a lot of these this could be solved by observation others separate the variables and take to Integral to get the equation
  13. karush

    MHB 9.2.2 AP Calculus Exam Slope Fields

    I'm just going to post this image now since my tablet won't render the latex. This is a free response question.. But my experience is that the methods of solving are more focused here at mhb saving many error prone steps.. Mahalo ahead...
  14. karush

    MHB 4.2.5 AP Calculus Exam int of e

    calculator returned this but know sure why $\displaystyle2 \int _1^2e^udu$ note there might be a duplicat of this post ?
  15. karush

    MHB 3.3.04 AP Calculus Exam 2nd derivative

    Ok not sure if I fully understand the steps but presume the first step would be divide both sides deriving$$\dfrac{dy}{dx}=\dfrac{2x-y}{x+2y}$$offhand don't know the correct answer $\tiny{from College Board}$
  16. torito_verdejo

    Calculus Recommended book for advanced calculus

    I'm looking for recommendations about advanced calculus books. I'm interested in going further and deeper than nth-order linear differential equations, but overall as a Physics student I'm deeply interested in being very, very comfortable dealing with line, surface and volume integration...
  17. EchoRush

    I What can we learn from higher derivatives in calculus?

    As I have said before, I am in calculus class for the first time. I am doing really well in the class, however because of how my mind works, I’m always asking questions to know more, even when it’s too advanced for me. I just like to ponder and think about “what if” I know it’s probably not good...
  18. Adesh

    Calculus What are some books for learning the techniques of Calculus?

    We have so many great books available for Calculus, such as : Spivak's Calculus, Stewart Calculus, Thomas Calculus , Gilbert Strang's Calculus, Apostol's Calculus etc. These books are very nice but they teach you the concepts well and all the standard techniques that are available for solving...
  19. karush

    MHB 3.3.291 AP Calculus Exam Problem solve for k

    Let $g(x)$ be the function given by $g(x) = x^2e^{kx}$ , where k is a constant. For what value of k does g have a critical point at $x=\dfrac{2}{3}$? $$(A)\quad {-3} \quad (B)\quad -\dfrac{3}{2} \quad (C)\quad -\dfrac{3}{2} \quad (D)\quad {0} \quad (E)\text{ There is no such k}$$ ok I...
  20. EchoRush

    I Questions about implicit differentiation?

    I am new to calculus. I am doing well in my class. I just have a few questions about implicit differentiation. First, why do we call it "implicit" differentiation? Also, when we do it, why when we differentiate a term with a "y" in it, why do we have to multiply it by a dY/dX? What does that...
  21. W

    MHB Optimization calculus question (Difficult)

    A truck crossing the prairies at constant speed of 110km per hour gets 8km per litre of gas. Gas costs 0.68 dollars per litre. The truck loses 0.10 km per litre in fuel efficiency for each km per hour increase in speed. Drivers are paid 35 dollars per hour in wages benefits. Fixed costs for...
  22. dRic2

    Vector calculus identity and electric/magnetic polarization

    I spent a good amount of time thinking about it and in the end I gave up and asked to a friend of mine. He said it's a 1-line-proof: just "integrate by parts" and that's it. I'm not sure you can do that, so instead I tried using the identity: to express the first term on the right-hand side...
  23. S

    Calculus Homework Check (vectors/forces)

  24. Beelzedad

    I Swapping the order of surface and volume integrals

  25. EchoRush

    I Question about the quotient rule of derivatives

    Now, I understand how to use the quotient rule for derivatives and everything. I do not struggle with using it, my question is mostly about the formula itself...I very much enjoy WHY we do things in math, not just “here’s the formula, do it”...Here is the formula for the quotient rule of...
  26. WMDhamnekar

    MHB Using advanced calculus for finding values

    It is possible to find positive integers $A,B, C, D, E$ such that $\displaystyle\int_0^{\frac{2a}{a^2+1}} sin^{-1}\big(\frac{|1-ax|}{\sqrt{1-x^2}}\big)dx=\frac{A}{\sqrt{a^2+1}}sin^{-1}\big(\frac{1}{a^B}\big ) - C sin^{-1} \big(\frac{1}{a^D}\big) + \frac {Ea\pi}{a^2+1}$ for all real numbers $ a...
  27. egozenovius

    I Recovering some math notions: Variations

    I have a paper and on that paper I only can read: Let $$f:\mathbb{S^{1}} \to \mathbb{R^2}$$ be a function and $$f_{\epsilon}=f+\epsilon hn$$ and $$\mathbb{S^1}$$ is the unit circle. $$\dot{f_\epsilon}=\dot{f}+\epsilon\dot{h}n+\epsilon h\dot{n}$$ $$\delta\dot{f}=\dot{h}n+h\dot{n}$$ can you...
  28. M

    Ιntegral calculation : (sin(x))^4 * (cos(x))^6

    Summary: Ιntegral calculation : (sin(x))^4 * (cos(x))^6 Hi all, I tried to solve it, but I got stuck. An advice from my professor is to set: x=arctan(t) Τhanks.
  29. D

    B What are the general calculus concepts used in classical physics?

    Question.
  30. KF33

    B How do I differentiate vectors with derivatives and properties?

    Homework Statement: The homework problem is included below, but I am looking at the derivatives of vectors. Homework Equations: I have the properties of derivatives below, but not sure they help me here...
  31. Adgorn

    Limit of the remainder of Taylor polynomial of composite functions

    Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
  32. Celso

    I Curve Inside a Sphere: Differentiating Alpha

    Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.
  33. 0

    An identity to prove using calculus 1

    I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.
  34. K

    Mass Transfer in a Binary Star System

    Homework Statement: A binary star system consists of M1 and M2 separated by a distance D. M1 and M2 are revolving with an angular velocity w in circular orbits about their common center of mass. Mass is continuously being transferred from one star to the other. This transfer of mass causes...
  35. karush

    MHB 3.4.238 AP calculus exam Limits with ln

    238 Solve $$\displaystyle\lim_{h\to 0} \dfrac{\ln{(4+h)}-\ln{h}}{h}$$ $$(A)\,0\quad (B)\, \dfrac{1}{4}\quad (C)\, 1\quad (D)\, e\quad (E)\, DNE$$ The Limit diverges so the Limit Does Not Exist (E)ok the only way I saw that it diverges is by plotting not sure what the rule is that observation...
  36. karush

    MHB 4.2.236 AP calculus Exam integral with u substitution

    AP Calculas Exam Problem$\textsf{Using $\displaystyle u=\sqrt{x}, \quad \int_1^4\dfrac{e^{\sqrt{x}}}{\sqrt{x}}\, dx$ is equal to which of the following}$ $$ (A)2\int_1^{16} e^u \, du\quad (B)2\int_1^{4} e^u \, du\quad (C) 2\int_1^{2} e^u \, du\quad (D) \dfrac{1}{2}\int_1^{2} e^u \, du\quad...
  37. karush

    MHB 4.1.237 AP calculus exam find area

    $\tiny{237}$ $\textsf{The total area of the region bounded by the graph of $f(x)=x\sqrt{1-x^2}$ and the x-axis is}$ $$(A) \dfrac{1}{3}\quad (B) \dfrac{1}{3}\sqrt{2}\quad (C) \dfrac{1}{2}\quad (D) \dfrac{2}{3}\quad (E) 1 $$ find the limits of integration if $$f(x)=0 \textit{...
  38. endykami

    Differential calculus, solve for y: 4(y''y'')+(y'y')-1=0

    suppose y''=r^2=s y'=r 4(y''y'')-(y'y')-1=0=4(r^2)^2-(r^2)-1=4(s^2)-s-1 s=(-b±√(b^2-4ac))/2a s=(1±√17)/8 y=∫∫sdx=∫∫((1±√17)/8) dx=(1±√17)/8)(1/2)x^2+c1x+c2
  39. karush

    MHB 224 AP calculus Exam slope at a point (x,y)

    image to avoid typos
  40. C

    Schools Where should I take my calculus and linear algebra online?

    Has anyone taken these two courses online in a self-paced course for credit? If so, where and how was it in terms of quality? How about price? Opinions/thoughts are much appreciated. I'm working and the closest community college is a commute away, so that's out. I'm finding $1100-3000~ for...
  41. karush

    MHB 219 AP Calculus Exam Inverse function

    Let $f(x)=(2x+1)^3$ and let g be the inverse of $f$. Given that $f(0)=1$, what is the value of $g'(1)?$$(A)\, \dfrac{2}{27} \quad (B)\, \dfrac{1}{54} \quad (C)\, \dfrac{1}{27} \quad (D)\, \dfrac{1}{6} \quad (E)\, 6$ok not sure what the best steps on this would be but assume we first find...
  42. J

    Derivation of Lagrangian in the calculus of variations

    Hello. In a chapter of a book I just read it is given that ##\frac {d} {d\epsilon}\left. L(q+\epsilon \psi) \right|_{\epsilon = 0} = \frac {\partial L} {\partial q} \psi ## While trying to get to this conclusion myself I've stumbled over some problem. First I apply the chain rule...
  43. karush

    MHB 2.8.218 AP Calculus Exam f'(g(3))=

    image to avoid typos
  44. karush

    MHB 217 AP Calculus Exam continous function with k

    217 $f(x)=\begin{cases} \dfrac{(2x+1)(x-2)}{x-2}, &\text{for } x\ne 2 \\[3pt] k, &\text{for } x=2 \\ \end{cases}$$(A)\,0\quad(B)\,1\quad(C)\,2\quad(D)\,3\quad(E)\,5\quad$
  45. karush

    MHB 215 AP Calculus Exam Problem Power Rule

    If $y=(x^3-cos x)^5$, then $y'=$(A) $\quad 5(x^3-\cos x)^4$(B) $\quad 5(3x^2+\sin x)^4$(C) $\quad 5(3x^2+\sin x)^4$(D) $\quad 5(x^3+\sin x)^4(6x+\cos x)$(E) $\quad 5(x^3+\cos x)^4(3x+\sin x)$ ok I am sure this could be worded better. but I think many students take these tests and are not used...
  46. karush

    MHB 214 AP Calculus Exam velocity

    $\tiny{207 \quad DOY}$ 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 $t = 3$ ? $(A)\,3 \quad (B)\,6 \quad (C)\,9 \quad (D)\,18\quad (E) \, 27$ ok think this is correct...
  47. karush

    MHB 1.1.4 AP Calculus Exam Problem int sec x tan x dx

    $\tiny{213(DOY)}$ $\displaystyle\int \sec x \tan x \: dx =$ (A) $\sec x + C$ (B) $\tan x + C$ (C) $\dfrac{\sec^2 x}{2}+ C$ (D) $\dfrac {tan^2 x}{2}+C $ (E) $\dfrac{\sec^2 x \tan^2 x }{2}+ C$
  48. R

    I Does this ODE have any real solutions?

    The ODE is: \begin{equation} (y'(x)^2 - z'(x)^2) + 2m^2( y(x)^2 - z(x)^2) = 0 \end{equation} Where y(x) and z(x) are real unknown functions of x, m is a constant. I believe there are complex solutions, as well as the trivial case z(x) = y(x) = 0 , but I cannot find any real solutions. Are...
  49. karush

    MHB 2.2.212 AP Calculus Exam problem find increasing interval

    212 Let f be the function given by $f(x)=300x-x^3$ On which of the following intervals is the function f increasing (A) $\quad (-\infty,-10]\cup [10,\infty)$ (B) $\quad [-10,10]$ (C) $\quad [0,10]$ only (D) $\quad [0,10\sqrt{3}]$ only (E) $\quad [0,\infty]$ Steps ok this was a little...
  50. karush

    MHB 2.2.211 AP Calculus Exam practice problem

    211(DOY) If If $y=x \sin x,$ then $\dfrac{dy}{dx}=$ (A) $\sin x + \cos x$ (B) $\sin x + x \cos x$ (C) $\sin x + \cos x$ (D) $x(\sin x + \cos x)$ (E) $x(\sin x - \cos x)$ Solution ok this is a relatively simple problem but was wondering if $y'$ should be used in combination with...
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