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 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) $-7$
  2. 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...
  3. 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...
  4. 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}...
  5. 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...
  6. 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 $g(-2)$...
  7. 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...
  8. 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) $\dfrac{1}{\pi}$...
  9. 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...
  10. Beelzedad

    I Does This Limit Exist?

    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...
  11. 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...
  12. 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
  13. 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...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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...
  19. 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...
  20. 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...
  21. 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...
  22. 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...
  23. 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...
  24. 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}##...
  25. B

    MHB Rearranging Algebraic Calculus

    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
  26. 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...
  27. 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.
  28. 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.
  29. 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...
  30. 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)...
  31. 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
  32. 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...
  33. 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?
  34. M

    Finding the Second Derivative of a Function with Two Variables

    Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2 starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2= [(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2 For the derivative in respect to y...
  35. L

    I Calculus problem differentiation.

    <Moderator's note: Member has been warned to show some effort before an answer can be given.> Hello all. This is my first post in this forum, I am asking for your understanding. I have a problem with the calculus task and I stuck in a dead endso I managed to find a solution on the internet. I...
  36. A

    What is the Correct Approach to Solve This Complex Integral?

    I split this to get \begin{equation} \int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx \end{equation} Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...
  37. S

    I Sum of Binomial Expansion | Spivak Chapter 2, Excercise 3 part d

    Hello, I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise. What I need to show is the following: $$ (a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j} $$ My attempt, starting from...
  38. JD_PM

    Looking for a bunch of solved Sympy problems (Calculus)

    Two weeks ago I had no idea on how to code using Python. Now I have completed an online course on functions, loops and strings. However, in that course I did not practice using the specific library called Sympy. Besides, I will use Python in the Physics-Math background, for solving problems like...
  39. JD_PM

    Understanding why we compute surface area as we do

    Homework Statement Homework Equations The Attempt at a Solution [/B] The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure. Question: 1) Why do we differentiate with...
  40. JD_PM

    Proving that a vector field is conservative

    Homework Statement Homework Equations $$F = \nabla \phi$$ The Attempt at a Solution Let's focus on determining why this vector field is conservative. The answer is the following: [/B] I get everything till it starts playing with the constant of integration once the straightforward...
  41. JD_PM

    Computing the Magnitude of F Using Gradient

    Homework Statement Homework Equations $$F = \nabla \phi$$ $$| F | = \sqrt{F \cdot F}$$The Attempt at a Solution I want to compute ##| F | = \sqrt{F \cdot F}## $$| F | = \sqrt{F \cdot F} = \sqrt{\frac{(km)^2 (r-r_0)^2}{|r-r_0|^6}} = \sqrt{\frac{(km)^2}{(r-r_0)}} =...
  42. P

    Variational Calculus: When Is dg(r=r+) ≠ dg(r=r++)?

    Homework Statement Question: If ##r_+ \neq r_{++}## and ## g(r=r_+) \neq g(r=r_{++}) ## When is it fulfilled that ## d g (r=r_+) \neq d g (r=r_+) ## ? Homework Equations ##r_+ \neq r_{++}## ## g(r=r_+) \neq g(r=r_{++}) ## The Attempt at a Solution I tried computing ## dg(r_+) =...
  43. K

    Why Does the Fourier Series of |sin(x)| Treat n=1 Differently?

    Homework Statement Hello, i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...
  44. JD_PM

    Python for Vector Calculus: Books & Online Resources

    I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus). If you also have online recommendations please feel free to write them.
  45. S

    How to Evaluate the 8th Derivative of a Taylor Series at x=4

    Homework Statement Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n## Evaluate: ##f^{(8)}(4)## Homework Equations The Taylor Series Equation The Attempt at a Solution Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
  46. A

    I Faking a Formula For Movement Through Gravity

    I have a strange question. It's strange because I don't need a correct answer. I need an answer that seems correct and leads to predictable results. I'm making a multiplayer computer game where the players fire cannons in outer space. The cannon shells will move through the gravitational fields...
  47. matai

    Integral for the linear speed of the Earth

    I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...
  48. S

    I How to apply the disk/washer and shell methods

    In Calculus II, we're learning about solids of revolutions and computing their volumes. I'm unsure when to apply the appropriate methods and how to make the correct partitions. Please tell me if my reasoning is correct: The disk/washer method is applied when your partitions are perpendicular...
  49. matai

    Using Integrals to Calculate the Rotational Energy of Earth

    So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that...
  50. S

    Area of a bounded region using integration

    In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given. I...
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