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. R

    Calculus III (help sketching graph)

    Homework Statement Sketch the curve with the given vector equation. Indicate with an arrow the direction in with 't' increases r(t)=<t, 2-t, 2t>Homework Equations parametric equation (can't type the equation, too confusing to use the template) The Attempt at a Solution So far, I have <1, -1...
  2. B

    I Question on §3 of Einstein's 1905 paper

    Hi guys, This is my first time posting on PF! I have a question on §3 of Einstein's paper "On the electrodynamics of moving bodies."My problem is with the following mathematical statements: Hence, if x' is chosen to be infinitesimally small, or I have just finished high school, and...
  3. ManicPIxie

    Values of d where limit exists. lim x->d

    Homework Statement For which value of d does the following limit exist? lim x->d ln [ (x2-13x+30) / (x-d) ] Homework Equations None The Attempt at a Solution I understand how to find limits when the limit goes to a real number, and has a variable in the function to solve for, but not when...
  4. I

    As f(x) -> oo and g(x) -> c, if c > 0 then f(x)g(x) -> oo

    Homework Statement If ## \lim_{x \rightarrow a} f(x) = \infty## and ##\lim_{x \rightarrow a} g(x) = c ##, and if ##c>0## then prove that ##\lim_{x \rightarrow a} \left[ f(x)g(x)\right] = \infty~~ \text{if c > 0}## Homework Equations Epsilon Delta definition of the limit The Attempt at a...
  5. Vitani11

    Proving the second fundamental theorem of calculus?

    Homework Statement Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x). Homework Equations None The Attempt at a Solution I have ... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be...
  6. cathal84

    Find the equation of a plane perpendicular to a line and goes through a point

    Homework Statement find equation of plane P that is perpendicular to line L which passes through the point (-2,-2,3) Homework Equations ... The Attempt at a Solution [/B]line L passes through the points (1,2,1) and (0,0,-3) I have worked out the parametric equations of line L to be x=1-t...
  7. Alettix

    Antiderivative of 1/x: ln(x) or ln(|x|)?

    Homework Statement Calculate the integral: ## \int_{a}^{b} \frac{1}{x} dx ## Homework Equations - The Attempt at a Solution In high school we learned that: ## \int_{a}^{b} \frac{1}{x} dx = ln(|x|) + C ## because the logarithm of a negative number is undefined. However, in my current maths...
  8. ChrisisC

    Want to Learn Calculus Early? Check Out This Classic Book!

    I'm a 10th grade student in the United States and currently taking geometry which is a breeze, and if anyone else reading this is in the U.S. you know that 10th graders haven't reached calculus yet, not even physics. Since i know I'm going into quantum physics, i have thirst to learn calculus...
  9. P

    MHB What is the value of the triple integral for the given solid and region?

    Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.
  10. cathal84

    Show that f has a stationary point at (0, 0) for every k ∈ R

    Homework Statement Let f(x, y) = x^2 + kxy + y^2 , where k is some constant in R. i. Show that f has a stationary point at (0, 0) for every k ∈ R Homework Equations ... The Attempt at a Solution I may have the solution or i may have gone completely wrong I am not entirely sure. i first found...
  11. B

    Why Does Spivak Exclude Specific Rational Points in His Limit Proof?

    Homework Statement Consider the function $$ f(x) := \begin{cases} 0, & if~ x \in (0,1) - \mathbb{Q} \\ \frac{1}{q}, & if~ x = \frac{p}{q} \in (0,1) \cap \mathbb{Q} \mbox{ in lowest terms. } \\ \end{cases} $$ In his Calculus book, Spivak shows us that ##\lim_{x \rightarrow a} f(x) = 0## for...
  12. C

    I Visual interpretation of Fundamental Theorem of Calculus

    Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If...
  13. bwest121

    How do I calculate this integral?

    Homework Statement We're given the gaussian distribution: $$\rho(x) = Ae^{-\lambda(x-a)^2}$$ where A, a, and ##\lambda## are positive real constants. We use the normalization condition $$\int_{-\infty}^{\infty} Ae^{-\lambda(x-a)^2} \,dx = 1$$ to find: $$A = \sqrt \frac \lambda \pi$$ What I want...
  14. S

    MHB Is the Russell's Paradox Resolved in Predicate Calculus?

    Prove (formall y) in predicate calculus : $\neg\exists y\,\forall x\,(x\in y\leftrightarrow \neg x\in x)$.
  15. cathal84

    Determining the domain and range of multi-variable function

    Homework Statement f(x,y) = 1/y^2-x find the domain of f. Given c ∈ R \ {0} find (x, y) ∈ R 2 such that f(x, y) = c. Finally determine the range of f. Homework Equations I know that the domain of the function is anywhere that the function is defined. The Attempt at a Solution in the case of...
  16. doktorwho

    What is this expression equal to?

    Basically i need to evaluate the limit of this expression ##\lim \sqrt[3]{n^6-n^4+5}-n^2=?## I want to know if this is correct and why: ##\lim \sqrt[3]{n^6-n^4+5}-n^2=\lim...
  17. cathal84

    Finding length of vector with unknown variable

    Finding length of vector with unknown variable. Purely for study purposes. Find the smallest possible length of the vector →v . Let vector V = (-2/3,b,16/7) Equation for finding length of vector : Sqrt(a^2+b^2+c^2)Question would be quite straight forward had there been no unknown variable but...
  18. T

    Finding Volume and Surface Area of a Banana Using Calculus

    Homework Statement We are given a Banana, and asked to find the volume and surface area of the function, using calculus. So far, we have learned elementary calculus (derivatives, limits, and integrals) as well as volumes of revolutions. We traced the banana on graph paper, plotted points on the...
  19. H

    Studying What Books Can Help Me Learn Physics on My Own?

    So, I am majoring in mechanical engineering with a minor in physics at San Jose State University. I want to learn as much as possible with physics by reading books and taking my future courses at SJSU, but I don't know what books to read. Any recommendations? I want to get pretty close to...
  20. H

    Studying What books can I learn from the most?

    Okay, so I'm currently a mechanical engineering major at San Jose State University, but I just want to become much more engaged with physics and mathematics. I do pretty good with calculus, math comes easy to me. I'm a first year student taking calculus 2. I was just wondering what books I can...
  21. C

    I Convergence of Taylor series in a point implies analyticity

    Suppose that the Taylor series of a function ##f: (a,b) \subset \mathbb{R} \to \mathbb{R}## (with ##f \in C^{\infty}##), centered in a point ##x_0 \in (a,b)## converges to ##f(x)## ##\forall x \in (x_0-r, x_0+r)## with ##r >0##. That is $$f(x)=\sum_{n \geq 0} \frac{f^{(n)}(x_0)}{n!} (x-x_0)^n...
  22. ParabolaDog

    Struggling immensely with tensors in multivariable calculus

    Homework Statement If f(x) is a scalar-valued function, show that ∂ƒ²/∂xi∂xj are the components of a Cartesian tensor of rank 2. Homework Equations N/A The Attempt at a Solution I don't even know where to begin. We began learning tensors in multivariable calculus (though I don't think this is...
  23. M

    Vector Calculus - Tensor Identity Problem

    Homework Statement Homework Equations The Attempt at a Solution I am really lost here because our professor gave us no example problems leading up to the final exam and now we are expected to understand everything about vector calculus. This is my attempt at the cross product and...
  24. dfklajsdfald

    Solve Antiderivative of xe^x | Step-by-Step Guide

    Homework Statement find the anti-derivative of xe ^x so its x to the power of e to the power of x Homework EquationsThe Attempt at a Solution i have 0 idea where to even start. this was a question on my quiz today
  25. W

    Derivative in spherical coordinates

    Homework Statement -here is the problem statement -here is a bit of their answer Homework Equations Chain rule, partial derivative in spherical coord. The Attempt at a Solution I tried dragging out the constant and partial derivate with respect to t but still I can't reach their df/dt and...
  26. B

    Absolutely Convergent, Conditionally Convergent, or Divergent?

    Homework Statement ∞ Σ (-1)n-1 n/n2 +4 n=1 Homework Equations lim |an+1/an| = L n→∞ bn+1≤bn lim bn = 0 n→∞ The Attempt at a Solution So I tried multiple things while attempting this solution and got inconsistent answers so I am thoroughly confused. My work is on the attached photo. I found that...
  27. PhotonSSBM

    B Finding Maxima/Minima of Polynomials without calculus?

    I'm tutoring a student who is in a typical precalculus/trig course where they're teaching her about graphing various arbitrary polynomials. Among the rules of multiplicity and intercepts they seem to be phrasing the questions such that they expect the students to also find the maxima and minima...
  28. S

    I Vector Calculus: What do these terms mean?

    In our section on path independence, we were asked to find the potential function given a vector field. Our teacher says to use only line integrals to find the potential function, and not any other method. Like if we have ##F=\left< M,N,P \right> ## The first step is to determine if the domain...
  29. N

    Schools Need Help: Calc, GPA, and Grad School.

    Hello everyone. I'm currently a sophomore working toward a BS in physics (and a minor in astronomy) at a top private engineering school. With the semester finishing up, I'm a little worried about where I am now and where I will be after graduation and I have a few questions. A little...
  30. G

    Fluid Pressure & Force Calculus 2

    Homework Statement A marine biology observation pod is being designed. It will be submerged, with vertical windows of various sizes and shapes. Calculate the total force being exerted on each of the windows described below. Density of water is 1000 kg/m^3...
  31. K

    I What is the boundary surface of a collimator?

    Hi everybody, I’m trying to calculate the shape of a boundary line f(x) between two mediums that collimates rays from a point light source. This requires the rays to hit the boundary line under a certain angle, so I calculated the slope m(φ) of the boundary line for a ray with polar angle φ (φ...
  32. G

    Finding maximum height of a string before it goes slack

    Homework Statement A mass m is suspended by a light elastic string. When the mass remains at rest it is at a point 0, which is a distance a + b below the point from which the string is suspended from the ceiling, where a is the natural length of the string. The mass is pulled down a distance h...
  33. S

    I Spivak's Definition of Integrals vs Stewart's

    Hello. I finished working through Spivak's Calculus 3rd edition chapters 13 "Integrals", and 14 "The Fundamental Theorem of Calculus". By that I mean that I read the chapters, actively tried to prove every lemma, theorem and corollary before looking at Spivak's proofs, took notes into my...
  34. O

    A Frechet v Gateaux Derivative and the calculus of variations

    Good Morning Could someone please distinguish between the Frechet and Gateaux Derivatives and why one is better to use in the Calculus of Variations? In your response -- if you are so inclined -- please try to avoid the theoretical foundations of this distinction (as I can investigate that by...
  35. M

    Electric field from a non-uniformly charged disk

    Homework Statement We are given a disk with negligible thickness, a radius of 1m, and a surface charge density of σ(x,y) = 1 + cos(π√x2+y2). The disk is centered at the origin of the xy plane. We are also given the location of a point charge in Cartesian coordinates, for example [0.5,0.5,2]. We...
  36. Cjosh

    Calculating the definite integral using FTC pt 2

    Homework Statement Sorry that I am not up on latex yet, but will describe the problem the best I can. On the interval of a=1 to b= 4 for X. ∫√5/√x. Homework EquationsThe Attempt at a Solution My text indicates the answer is 2√5. I have taken my anti derivative and plugged in b and subtracted...
  37. C

    I Find potential integrating on segments parallel to axes

    A simple method to find the potential of a conservative vector field defined on a domain ##D## is to calculate the integral $$U(x,y,z)=\int_{\gamma} F \cdot ds$$ On a curve ##\gamma## that is made of segments parallel to the coordinate axes, that start from a chosen point ##(x_0,y_0,z_0)##. I...
  38. T

    Work problem - Rope, pulley and brick (applied integration)

    If a brick is pulled across the floor by a rope thruogh a pulley, 1 meter above the ground - and work = W, where W = 10N , (in Newton).Show that the horizontal component of W, which is pulling the brick has the size \frac{10x}{\sqrt{1+x^2}} (*) Use this to calculate the amount of work needed...
  39. Kaura

    Taylor Series Error Integration

    Homework Statement Using Taylor series, Find a polynomial p(x) of minimal degree that will approximate F(x) throughout the given interval with an error of magnitude less than 10-4 F(x) = ∫0x sin(t^2)dt Homework Equations Rn = f(n+1)(z)|x-a|(n+1)/(n+1)![/B] The Attempt at a Solution I am...
  40. HRubss

    How Can I Deepen My Understanding of Calculus for Mechanical Engineering?

    So I am currently planning to major in Mechanical Engineering which is heavily involved with Math. I'm taking Calculus 1 this semester and so far I'm doing just above average on all my test. (All B's and 1 A for my limits exam). So far so good right? Eh, I only excel because I just know how to...
  41. Kaura

    How Can You Simplify the Taylor Series Calculation for cos(3x^2)?

    Homework Statement Determine the Taylor series for the function below at x = 0 by computing P5(x) f(x) = cos(3x2) Homework Equations Maclaurin Series for degree 5 f(0) + f1(0)x + f2(0)x2/2! + f3(0)x3/3! + f4(0)x4/4! + f5(0)x5/5! The Attempt at a Solution I know how to do this but attempting...
  42. Ketav

    Calculus: Verify Thick Walled Cylinder Equations

    Homework Statement I have a system of two ordinary differential equations as shown below. I have to prove that the Lame's exact solutions for a thick walled cylinder loaded by internal pressure satisfies the equations. The next step is to integrate the equations to obtain an equation for U...
  43. victorhugo

    Studying What is the best way to learn calculus?

    I didn't take calculus at school so I'm going to learn it during summer holidays before doing it at uni. The thing is, the only reason I got so good at General Mathematics is because I didn't follow the step by step to doing a problem and just remembering the rules, I took the time to ask lots...
  44. Y

    MHB What is the substitution for the definite integral ∫202x(4−x2)1/5 dx?

    Consider the definite integral ∫202x(4−x2)1/5 dx. What is the substitution to use? u= 4-x^2 Preview Change entry mode (There can be more than one valid substitution; give the one that is the most efficient.) For this correct choice, du/dx= -2x Preview Change entry mode If we make this...
  45. IntegralBeing

    Calculus I've just finished Stewart's calculus, now what?

    I hated that book so much; I had the opportunity to change to Spivak or Apostol in holidays but I didn't do it. I feel like I will have to read a good rigorous calculus text from the beggining since Stewart's textbook is sheer rubbish in many senses. Which book should I read to continue my...
  46. T

    I Derivative of A Def. Integral Equals Another Def. Integral?

    I'm going through the book "Elementry Differnetial Equations With Boundary Value Problems" 4th Eddition by William R. Derrick and Stanley I. Grossman. On Page 138 (below) ) The authors take the derivative of a definite integral and end up with a definite integral plus another term. How did...
  47. merricksdad

    Find the center of mass of a cone with variable density....

    Homework Statement Find the center of mass of an inverted cone of height 1.5 m, if the cone's density at the point (x, y) is ρ(y)=y2 kg/m. Homework Equations The formula given for this problem is rcm=1/M * ∫rdm, where M is total mass, r is position, and m is mass. The Attempt at a Solution...
  48. dykuma

    How Does Viscosity Affect Motion in a Fluid?

    Homework Statement A mass M falls under gravity (force mg) through a liquid with decreasing viscosity so that the retarding force is -2mv/(1+t). If it starts from rest, what is the speed, acceleration, and distance fallen at time t=1. Homework Equations F=ma The Attempt at a Solution F =...
  49. G

    Finding the time period using the potential

    Homework Statement A particle of unit mass moves in one dimension with potential V(x) = ½μ2x2 + εx4 (ε>0). Discuss the motion of the particle. If the particle released from rest at x=a (a>0) express the time period T for the particle to return to a in the form of an integral and show that when...
  50. C

    Maximum Torque / Evenly Spread across a lever.

    Homework Statement The maximum torque on a lever is 1.5 x 10^6 Newtons. How many people of weight 750N can stand evenly spaced on this lever, which has a length of 20 meters? Homework Equations T=FR Weight=mg W=Fd X = Number of people The Attempt at a Solution I have set 1.5x10^6 N =...
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