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.

View More On Wikipedia.org
Recent contents Top users
  1. 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...
  2. 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...
  3. 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...
  4. 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{...
  5. 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
  6. karush

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

    image to avoid typos
  7. 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...
  8. 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...
  9. 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...
  10. karush

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

    image to avoid typos
  11. 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$
  12. 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...
  13. karush

    MHB What Is the Total Distance Traveled by the Particle from t = 0 to t = 3?

    $\tiny{207 \quad DOY}$ 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...
  14. 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) $\dfrac{\sec^2 x \tan^2 x }{2}+ C$
  15. 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...
  16. 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...
  17. 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...
  18. karush

    MHB 210 AP Calculus Exam problem tangent line to curve

    Find the slope of the tangent line to the graph of $$f(x)=-x^2+4\sqrt{x}$$ at $x=4$ (A) $8-$ (B) $-10$ (C) $-9$ (D) $-5$ (E) $-7$
  19. gary0000

    Rotating an ellipse to create a spheroid?

    I was able to find the equation of an ellipse where its major axis is shifted and rotated off of the x,y, or z axis. However, I could not find anywhere an equation for a spheroid that does not have its axis or revolution along the x,y, or z axis. How might I go about deriving such an...
  20. karush

    MHB 2.1.207 AP calculus practice exam problem Lim with tan(4X)/6x

    $\displaystyle\lim_{{x}\to{0}}\left(\frac{\tan 4x}{6x}\right)=$ (A) $\dfrac{1}{3}$ (B) $\dfrac{2}{3}$ (C) 0 (D) $-\dfrac{2}{3}$ (E) DNE solution direct substitution of 0 results in undeterminant so use LH'R so then after taking d/dx of numerator and denominator and factor out constant we...
  21. S

    Multivariate calculus problem: Calculating the gradient vector

    1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector $$\nabla{f} = \begin{bmatrix} f_x \\ f_y \end{bmatrix} = \begin{bmatrix} \frac{2\ln{(x)}}{x} \\ \frac{2\ln{(y)}}{y}...
  22. karush

    MHB 2.2.206 AP Calculus Practice question derivative of a composite sine

    206 (day of year number) If $f(x)=\sin{(\ln{(2x)})}$, then $f'(x)=$ (A) $\dfrac{\sin{(\ln{(2x)}}}{2x}$ (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ (C) $\dfrac{\cos{(\ln{(2x)}}}{2x}$ (D) $\cos{\left(\dfrac{1}{2x}\right)}$ Ok W|A returned (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$ but I didn't understand why...
  23. karush

    MHB 1.1.205 AP calculus exam practice question

    given that $\left[ f(x)=x-2, \quad g(x)=\dfrac{x}{x^2+1}\right]$ find $f(g(-2))$ (A) $\dfrac{-11}{5}$ (B) $\dfrac{-4}{17}$ (C) $-3$ (D) $\dfrac{14}{85}$ (E) $\dfrac{-12}{5}$ _____________________________________________________________________________ Solution find $g(-2)$...
  24. karush

    MHB 4.2.204 AP calculus practice question

    If $\displaystyle f(x)=\int_1^{x^3}\dfrac{1}{1+\ln t}\, dt$ for $x\ge 1$ then $f'(2)=$ (A) $\dfrac{1}{1+\ln 2}$ (B) $\dfrac{12}{1+\ln 2}$ (C) $\dfrac{1}{1+\ln 8}$ (D) $\dfrac{12}{1+\ln 8}$ ok I am little be baffled by this one due the $x^3$ in the limits since from homework you just take...
  25. karush

    MHB 1.4.203 AP calculus practice question on Limits

    I am posting some AP calculus practice questions on MeWe so thot I would pass them thru here first The solution is mine... any typos or suggestions... $\textbf{Find the Limit of}$ $\displaystyle\lim_{x\to \pi} \dfrac{\cos{x}+\sin{x}+1}{x^2-\pi^2}$ (A) $-\dfrac{1}{2\pi}$ (B) $\dfrac{1}{\pi}$...
  26. S

    I Calculus- Area between two curves (minimize it)

    Hi, This is my first question here, so please apologise me if something is amiss. I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...
  27. Beelzedad

    I Are Extra Conditions Affecting the Limit Definition?

    This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary. PRELIMINARY: Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...
  28. J

    I Quick question for Finding EOM with diff eq

    I have been going through my old books again, and found myself a little stuck. I am not entirely sure if this would be better in this one or diffy eq. The problem starts with having you find equation of motions when a= -bv, where b is constant and v = v(t) Using method of separable equations...
  29. silverfury

    Is There a Trick to Simplify Taylor Series Expansion?

    I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
  30. U

    Other I need advice on preparations for studying calculus

    Hello, I am new to physicsforums and I am still a high school student so I would like to have advice on what books should be relevant on preparation for calculus and more math beyond. I have basic algebra and geometry foundation and I would like to learn more high school math and up. So my plan...
  31. R

    B A proof of the fundamental theorem of calculus

    is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it? i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it. i know...
  32. sams

    A The δ Notation in Calculus of Variations

    On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows: $$\delta J = \frac{\partial J}{\partial...
  33. J

    Textbook question: Can I self-study with Velleman's Calculus?

    Summary: CALC BOOK QUESTION Hey I am going to be self studying calc AP BC because my school only offers AB. So I bought from a ton of reddit advice Vellemans Calculus: A rigorous first course, due to the fact where I want a challenge similar to AOPS however more into solving more problems no...
  34. J

    I What Are Some Examples of Symbolic Manipulation Not Included in Calculus?

    Not satisfied with the following definition of calculus. What is a better definition? More detailed? 1a : a method of computation or calculation in a special notation (as of logic or symbolic logic) b : the mathematical methods comprising differential and integral calculus —often used with the...
  35. B

    I Finding CDF given boundary conditions (simple stats and calc)

    I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...
  36. aligator11

    Multivariable Triple Integral - Calculus Physics/Math Problem

    Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...
  37. K

    How to find the deceleration of a mass colliding on a spring?

    Problem Statement: A known mass at a know velocity collides on a spring of known stiffness. What is the equation that governs the deceleration of the mass, so that the force on the spring could be found? Relevant Equations: 1/2 m*V^2 = 1/2*k*x^2 + 1/2*m*(Vo)^2 Kinetic energy of mass before...
  38. aligator11

    Particle Dynamics Problem (kinematics)

    Summary: Mechanics problem related with Calculus (differential equations) Hi everyone, I would like some help in that task, if anyone would be willing to help :) Namely I have a problem from particle dynamics. "D:" means given info... so, D: m,g,h,b, miu. We're looking for v0 and S as given...
  39. Eclair_de_XII

    Calculus What is a good book to review basic calculus?

    I want a book that isn't too wordy, explains things concisely, and covers single-variable and multi-variable calculus. I regret that I forgot so much of the stuff I learned in Calculus III and IV (especially the important theorems I learned but forgot in the latter), and now I want to review it...
  40. J

    I What is the proper format for solving this ODE using an Excel add-in calculator?

    I am attempting to solve an ODE using a Calculus add-in for Excel. I am an industry professional and I have not even thought about Differential Equations in 8 years. The equation that I am attempting to solve is in the form: (1) The ODE solver that I am using solves equations of the form...
  41. M

    How shall we show that this limit exists?

    Let: ##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'## where ##V'## is a finite volume in space ##\mathbf{r}=(x,y,z)## are coordinates of all space ##\mathbf{r'}=(x',y',z')## are coordinates of ##V'## ##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...
  42. B

    MHB How Do You Rearrange dS/dt = (n-x-rS)(L-1) to Solve for t?

    Hello, I have been given a general equation that is dS/dt = (n-x-rS)(L-1) and it needs to be rearranged in order to for the subject to be t. I have spent a while trying to find a way to attempt this but with no luck so I will leave this here. Thanking You, BSLAHi
  43. Arman777

    Proof that Variation of Integral is Equal to Integral of the Variation

    I actually don't know how to proceed. I tried something like this The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$ where ##f'(x)=F(x)## However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$ where ##f'(x)=F(x)##. For the...
  44. fazekasgergely

    Infinite series to calculate integrals

    For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
  45. E

    MHB Calculus Problem: Get Expert Help Here

    Dear all, I want your help in soving the attached problem. Thanks in advance for your valuable contribution.
  46. mishima

    Calculus of Variations, Isoperimetric, given surface area max volume

    My volume integral is... $$\pi\int y^2 dx$$ My surface area integral is... $$2\pi\int y \sqrt {1+x'^2} dy$$I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to...
  47. F

    B How Do Discrete Derivatives and Integrals Work in Calculus?

    Hello all. I've come across some math which consists of just applying the basic ideas of calculus (derivatives and integrals) onto discrete functions. (The link: http://homepages.math.uic.edu/~kauffman/DCalc.pdf ) The discrete derivative with respect to n is defined as ## \Delta_n f(n) = f(n+1)...
  48. M

    How to Solve a Derivative Presented in a Non-Standard Format?

    I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as ' a ' instead of ' a / b ' . ------ b
  49. K

    How to Find Fourier Coefficients for a Given Function

    Hello, I need help with question #2 c) from the following link (already LateX-formatted so I save some time...): https://wiki.math.ntnu.no/_media/tma4135/2017h/tma4135_exo1_us29ngb.pdf I do understand that the a0 for both expressions must be the same, but what about an and bn? I don't...
  50. majormuss

    Electrodynamics: Vector Calculus Question

    Why are the red circled Del operators not combining to become 'Del-squared' to cancel out the second term to give a net result of 0?
Back
Top