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. D

    Calculus II: Convergence of Series with Positive Terms

    Homework Statement https://imgur.com/DUdOYjE The problem (#58) and its solution are posted above. Homework Equations I understand that I can approach this two different ways. The first way being the way shown in the solution, and the second way, which my professor suggested, being a Direct...
  2. Phantoful

    Bead on a string, find y(x) if horizontal velocity is const.

    Homework Statement Homework Equations K = (1/2)mv2 U = mgh W=Fd Integration/Calculus The Attempt at a Solution I'm not sure what I should be doing for this question, if height changes how is it possible that velocity stays the same, according to the conservation of energy (frictionless wire)...
  3. V

    Using calculus to find gravity in space (dynamics)

    I am currently solving a problem where I need to find the gravity in the ISS (distance 400km from Earth with Radius 6371km). I am using the formula g=GM/R^2 . One way to solve it would be to find GM by multiplying g(which is 9.81) and R^2 (which is known) and then to use it in GM/(R+400)^2 and...
  4. G

    How many months should I give myself to learn Calculus I-III

    Hello, I'm interested in learning calculus in order to apply it to my current field of geology (by doing more advanced statistics and groundwater modeling). Considering that I'll study 10 hours per week, about how long should I realistically take to get through the material in Calculus I...
  5. N

    Maximum Velocity for different accelerations

    Homework Statement Force of jet= A(r(t))4/3 A: constant determined by the fighter model in the class being considered and the drag force on the plane r(t): the rate of fuel consumption as a function of time Consider 3 possible situations for r(t): 1. when the rate is constant for the...
  6. N

    Finding velocity with constant, incre., decrea, varying acce

    Homework Statement Force of jet= A(r(t))4/3 A: constant determined by the fighter model in the class being considered and the drag force on the plane r(t): the rate of fuel consumption as a function of time Consider 3 possible situations for r(t): 1. when the rate is constant for the...
  7. C

    Understanding the Gradient: Exploring Maximum Rate of Change in Scalar Fields

    Homework Statement Hi, I'm having some doubts about the gradient. In my lecture notes the gradient of a scalar field at a point is defined to point in the direction of maximum rate of change and have a magnitude corresponding to the magnitude of that maximum rate of change of the scalar field...
  8. J

    Schools Preparing for Physics C: Calculus or No?

    If I take physics C next year (I am a junior in high school), will I be at a disadvantage if I have not taken calculus yet?
  9. E

    Can you help me determine the convergence of these series?

    Homework Statement Determine whether the following series converge, converge conditionally, or converge absolutely. Homework Equations a) Σ(-1)^k×k^3×(5+k)^-2k (where k goes from 1 to infinity) b) ∑sin(2π + kπ)/√k × ln(k) (where k goes from 2 to infinity) c) ∑k×sin(1+k^3)/(k + ln(k))...
  10. F

    Substitution in a differential equation, independent variable

    Homework Statement $$y'=-\frac{1}{10}y+(cos t)y^2$$ when doing substitute for ##z=\frac{1}{y}## I understand this is ##z(t)=\frac{1}{y(t)}## I know t is independent variable and y is dependent variable but I want to know what is z role here, is it change the dependent variable? when...
  11. C

    Finding the Mass of a Hanging Rope (Wave Problem)

    Homework Statement "...you were presented with a geologist at the bottom of a mineshaft next to a box suspended from a vertical rope. The geologist sent signals to his colleague at the top by initiating a wave pulse at the bottom of the rope that would travel to the top of the rope. The mass of...
  12. Gene Naden

    Classical What are some recommended resources for learning general physics with calculus?

    Looking for a textbook or online pdf on general physics with calculus. I looked for Halliday and Resnick's Physics I and II from my past but did not find an affordable copy in good condition. If it includes advanced material, that is OK.
  13. Gene Naden

    Calculus Advanced undergrad text on Calculus and Differential eqns

    Hi, I have a masters in physics but it is decades old and I am a little rusty. Plus, I didn't study calculus and differential equations carefully or systematically, as I was young and arrogant (not to say that all young people are arrogant, but I was). Returning to physics now, I find that I can...
  14. K

    MHB Multivariable calculus line integral work

    calculate the work done by the force field $F(x,y)=(ye^{xy})i+(1+xe^{xy})j$ by moving a particle along the curve C described by gamma (γ):[0,1] in $R^2$, where gamma (γ)=(2t-1, t²-t)
  15. binbagsss

    Application of the Fundamental Theorem of Calculus (cosmological red-shift)

    Homework Statement [/B] I am stuck on the section of my lecture notes attached, where it says that equation 4.20 follows from 4.18 via an application of the fundamental theorem of calculus Homework Equations FoC: if ## f## is cts on ##[a,b]## then the function ...
  16. H

    How to solve this integral of an absolute function?

    Homework Statement Homework EquationsThe Attempt at a Solution I think the answer for number 1 , graph somewhat like this I get trouble for 2, 3, etc I (k) = ##\int_{-1}^{1} f(x) dx ## f(x) = ## \mid x^2 - k^2 \mid## 2) k < 1 for negative side ##\int_{-1}^{-k} (x^2 - k^2) dx +...
  17. M

    MHB When Should You Take Calculus and Analytic Geometry?

    My question is not a math question. I know about the calculus sequence (CAL 1, 2 and 3). I plan to go through all 3 in time. There is no rush for me. However, I know there is a course by the title of Calculus and Analytic Geometry. I want to know when this course is given. Is it given after...
  18. MermaidWonders

    MHB Integral Calculus - Spot the Error

    The big blue circle has been put there by my math prof to denote the location of the error in the following solution. Why is this an error? I'm lost. :(
  19. MermaidWonders

    MHB True or False Integral Calculus Question #3

    True or False: If $f(x)$ is a negative function that satisfies $f'(x) > 0$ for $0 \le x \le 1$, then the right hand sums always yield an underestimate of $\int_{0}^{1} (f(x))^2\,dx$. - - - Updated - - - Would it be true since right hand Riemann sums for a negative, increasing function will...
  20. MermaidWonders

    MHB True or False Integral Calculus Question #2

    True or False: Let $F(x)$ be an antiderivative of a function $f(x)$. Then, $F(2x)$ is an antiderivative of the function $f(2x)$.
  21. MermaidWonders

    MHB True or False Integral Calculus Question #1

    True or False: If $$h(t) > 0$$ for $$0 \le t\le 1$$, then the function $$H(x) = \int_{0}^{x} h(t)\,dt$$ is concave up for $$0 \le t\le 1$$.
  22. Pencilvester

    I Euler’s approach to variational calculus

    Hello PF, I’m going through a book called “A First Course in the Calculus of Variations.” I can’t remember who the author is at the moment, I’ll post it later. Anyway, I’m having trouble with one part: suppose we have a function ##y (x)## that gives a continuous polygonal curve from ##x = a## to...
  23. KFSKSS

    B Need some help with understanding linear approximations

    Hello. My problem is that I began with Linear Approximation and I'm terribly stuck. I have problems understanding its very concept and with calculations. (It may sound stupid but I'm autodidact and I'm studying mathematics in english (not my mother tongue) and sometimes it gets hard). It would...
  24. Dethrone

    MHB Layout Notation for matrix calculus

    Hi, I guess this could be a rather silly question, but I got a bit confused about the "numerator layout notation" and "denominator layout notation" when working with matrix differentiation...
  25. Phantoful

    How do I define a region in R3 with spherical/polar coords?

    Homework Statement Homework Equations x^2 + y^2 + z^2 = r^2 Conversion equations between the three coordinate systems The Attempt at a Solution I tried to solve this problem using spherical/cylindrical coordinates from the beginning, but that wouldn't work so I started with cartesian...
  26. orangeraindrops

    What does the area under a volume vs time graph represent?

    Homework Statement I have a function showing the volume of water in a bay at different times in the day, and I want to know what the area under this curve would represent (if it represents anything meaningful). I know how to integrate, so that isn't a problem. Homework Equations I am...
  27. N

    I Solving Integral for All n≥2 | Evans PDE's (Page 48)

    In the book from Evans on PDE's (page 48) I came across this integral. Here r > 0 and \delta is an arbitrarily small number. Could you give me some hint on how to solve this integral for all integers n\geq2 , i.e why does it go to zero as t approaches zero from the right side.
  28. C

    Calculus 2 - Trig Integrals Question (Integrating cos^2x)

    1. Here's the problem on trig integrating that I'm struggling with (Calculus 2 btw) 2. Wanted to see if I did everything right so far and what to do after all this. The part where I'm stuck is how to integrate (integral)cos^(2)udu and (integral)cos^(2)usin^(2)udu. I'm sure these are easy...
  29. Roverse

    An equilateral triangle's electric field at its center

    Homework Statement Three 18-cm long rods form an equilateral triangle. Two of the rods are charged to +10 nC, and the third to - 10 nC. What is the electric field strength at the center of the triangle? Homework Equations $$ \vec{E} = \frac{k*q}{r^2} $$ The Attempt at a Solution 1. Draw...
  30. ertagon2

    MHB Fundamental theorem of calculus and more....

    So as always I come here to make sure my maths homework is right and ask few questions to make sure I understand the topic. Here is my homework: Q.1 I'm fairly certain that this is correct, however, please check if I didn't do any stupid mistakes. Q.2 Same as above. Q.3 Now here is where the...
  31. M

    Calculating Eigenvectors for a 3x3 Matrix: Understanding the Process

    Hi, I am trying to find the eigenvectors for the following 3x3 matrix and are having trouble with it. The matrix is (I have a ; since I can't have a space between each column. Sorry): [20 ; -10 ; 0] [-10 ; 30 ; 0] [0 ; 0 ; 40] I’ve already...
  32. A

    I Why does this concavity function not work for this polar fun

    For the polar equation 1/[√(sinθcosθ)] I found the slope of the graph by using the chain rule and found that dy/dx=−tan(θ) and the concavity d2y/dx2=2(tanθ)^3/2 This is a pretty messy derivative so I checked it with wolfram alpha and both functions are correct (but feel free to check in case...
  33. W

    Vector Calculus: Gradient of separation distance

    Homework Statement Could someone explain how the property, $$\nabla (\frac{1}{R}) = -\frac{\hat{R}}{R^2}$$ where ##R## is the separation distance ##|\vec{r} - \vec{r'}|##, comes about? What does the expression ##\nabla (\frac{1}{R}) ## even mean? Homework EquationsThe Attempt at a Solution...
  34. W

    Vector Integration: Fundamental theorem use

    Homework Statement Could someone illustrate why $$\int_{V} \nabla \cdot (f\vec{A}) \ dv = \int_{V} f( \nabla \cdot \vec{A} ) \ dv + \int_{V} \vec{A} \cdot (\nabla f ) \ dv = \oint f\vec{A} \cdot \ d\vec{a}$$ ? Homework EquationsThe Attempt at a Solution I understand that the integrand can...
  35. S

    Maximizing Friction Coefficient for Block on a Wedge: A Calculus Approach

    Homework Statement A block of mass m is placed on a rough wedge inclined at an angle α to the horizontal, a distance d up the slope from the bottom of the wedge. The coefficient of kinetic friction between the block and wedge is given by µ_0x/d, where x is the distance down the slope from the...
  36. S

    I Relativistic Calculus Books & PDFs | Free Resources

    Just wanted any books / pdfs which introduce special relativistic calculus.
  37. V

    What went wrong with my simple differential equation?

    Homework Statement [/B] dy/dt = c - ky Homework Equations integral 1/y dy = ln(y) The Attempt at a Solution let y = c/k + z dy/dt = dz/dt = -kz dz/z = -kdt ln(z) = - kt z = e^(-kt) but z = y - c/k y = e^(-kt) + c/k + cons. answer should have been negative sign on the e term. I...
  38. D

    Solving 2nd order DE with initial condition

    Hello Guys, We haven't yet covered on how to solve 2nd order equation in class however we have this assignment given to us. Any tips would be appreciated for these 2 little problems. 1. Homework Statement We have this initial Equation: d2y/dt2−7dy/dt+ky=0, and we need to find the values of k...
  39. GaussianSurface

    How can I find this displacement?

    Homework Statement Your task is to estimate how far an object traveled during the time interval 0≤t≤8, but you only have the following data about the velocity of the object. *First image You decide to use a left endpoint Riemann sum to estimate the total displacement. So, you pick up a blue...
  40. GaussianSurface

    Calculating distance from speed

    Homework Statement The speed of a runner increased during the first three seconds of a race. Her speed at half-second intervals is given in the table. Find lower and upper estimates for the distance that she traveled during these three seconds. It follows the image's square. Homework Equations...
  41. M

    A Eigenvalue Problem and the Calculus of Variations

    Hi PF! Given ##B u = \lambda A u## where ##A,B## are linear operators (matrices) and ##u## a function (vector) to be operated on with eigenvalue ##\lambda##, I read that the solution to this eigenvalue problem is equivalent to finding stationary values of ##(Bu,u)## subject to ##(Au,u)=1##...
  42. Mzzed

    Using logarithms in vector Calculus

    Homework Statement My mentor has run me through the derivation of equation (3) bellow. I am unsure how he went from (1) to (3) by incorporating the log term from eq(2). In eq(3) it seems he just canceled the relevant n terms and then identified 1/n as the derivative of L however if this were...
  43. H

    Is Re-Learning Calculus Necessary for Future Mathematical Studies?

    Hi, I'm a high school senior and I'm wondering if I should re-learn calculus. This is already my third year learning calculus in my high school, and I'm currently taking some easy version of multi-variable calculus, but doubt my high school calculus class is solid enough as the foundations of...
  44. paulo84

    B How does calculus relate to dimensions?

    I am trying to understand what time^2 and velocity^2 mean in terms of how to visualize them? This wasn't explained in Physics or Mechanics (Further Mathematics) in high school, unfortunately. It seems likely it relates to matrices, maybe? Appreciate any replies! :)
  45. Delta31415

    Calculus *BEST* Calculus 3 Textbook for self study

    Hello, currently I am a high school senior who will be going to college in the fall and since my school ends in may and college starts in mid-August. I am planning on self-studying calculus 3, so I can test out of it and go straight into partial differential equations. The textbooks that the...
  46. PeroK

    Demystifying the Chain Rule in Calculus - Comments

    Greg Bernhardt submitted a new PF Insights post Demystifying the Chain Rule in Calculus Continue reading the Original PF Insights Post.
  47. M

    I Directional Derivative demonstration

    I find directional derivatives confusing. For example if there is a change in a direction and if this direction have both x and y components should not the change be calculated as square root of squares, i.e the pythogores theorem? Would you please provide a simple demonstration showing the...
  48. V

    Normalization constant for a 3-D wave function

    Homework Statement Show that the normalized wave function for a particle in a three-dimensional box with sides of length a, b, and c is: Ψ(x,y,z) = √(8/abc) * sin(nxπx/a)* sin(nyπy/b)* sin(nzπz/c). Homework Equations Condition for the normalization: ∫0adx ∫0bdy ∫0cdz Ψ*(x,y,z)Ψ(x,y,z) = 1...
  49. A

    MHB How Do You Calculate the Radius of Curvature for Complex Curves?

    1) Find the radius of curvature at any point of the cycloid x = a(\theta + sin\theta)y = a(1- cos\theta). 2) Find the radius of curvature at the point (3a/2 , 3a/2) for the curve x3 + y3 = 3axy
  50. M

    B Self learn calculus for UK A-levels?

    Hi, I am currently about to begin self studying for UK maths a-levels, however I am also wanting to gain a solid understanding of calculus. I know that calculus is covered in a-levels, but, the books for a-levels seem to be not as dense or as good as the US books I believe. My question is...
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