Calculus Definition and 1000 Threads

  1. 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...
  2. 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...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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...
  10. 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 =...
  11. 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...
  12. 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 =...
  13. A

    Unsure if this is a trig identity or calculus

    Homework Statement $$sinx - cosx = 1/3$$ solve for $$sin(2x)$$ Homework Equations $$sin^2x + cos^2x = 1$$ $$sin2x = 2cosxsinx$$ The Attempt at a Solution I think you can square both sides and get: $$sin^2x - cos^2x = 1/9$$ But how can I use this information to solve for sin2x? Is there a...
  14. S

    A Differential forms and vector calculus

    Let ##0##-form ##f =## function ##f## ##1##-form ##\alpha^{1} =## covariant expression for a vector ##\bf{A}## Then consider the following dictionary of symbolic identifications of expressions expressed in the language of differential forms on a manifold and expressions expressed in the...
  15. J

    I Extremal condition in calculus of variations, geometric

    Hi folks, I am a bit confused with the extreme condition used in the calculus of variations: δ = 0 I don't understand this rule to find extreme solutions (maximum or minimum) If in normal differential calculus we have a function y = y(x) and represent it graphically, you see that at the...
  16. A

    Calculus of Variations; Maximum enclosed area problem.

    The problem reads: "You are given a string of fixed length l with one end fastened at the origin O, and you are to place the string in the (x, y) plane with its other end on the x-axis in such a way as to maximise the area between the string and the x axis. Show that the required shape is a...
  17. INAM KHAN

    Calculus Where Can I Find Calculus by James Stewart 8th Edition?

    I need this book Calculus by James Stewart 8th edition.
  18. E

    Calculus Updating my Electricity and Magnetism --> Vector Calculus?

    Dear all, I'n an EE that finished his degree more than 10 years ago. I wanted to refresh my Electricity and Magnetism knowledge. I bough Purcells book some weeks ago (https://www.amazon.com/dp/1107014026/?tag=pfamazon01-20) and I'm kind of struggling through the maths (Vector calculus). I've...
  19. Z

    B Help with understanding Nature of Roots for Quadratic and Cu

    Hi I am writing my final Mathematics exams for Grade 12 in South Africa in 5 days. I am well prepared with an aim of getting 100%, but one concept in functions might prevent that - the concept of how the nature of roots are affected by vertical/horizontal shifts in a function, and how to...
  20. caters

    Best Tunnel Shape for X,Y,Z Coordinates

    Homework Statement X,Y,Z(coordinates) What function corresponds to the best tunnel shape? g = 9.8 m/s^2(earth gravity) Homework Equations F(x)=Y G(x)=X^2 in the xy plane G(z)= sin(X) in the xz plane H(x)= parabolic sinusoid(X^2 and sin(X) both in the xy plane) The Attempt at a Solution I have...
  21. MiLara

    I Why do some but not all derivatives have physical meaning?

    I know that taking the derivative of certain functions that explain physical phenomena can lead to another statement describing the physical system, the most famous being the derivatives of position. That is, position-->velocity-->acceleration-->jerk-->jounce...and taking any other further...
  22. P

    I Numerical Calculus of Variations

    I attempt to solve the brachistochrone problem numerically. I am using a direct method which considers the curve ##y(x)## as a Lagrange polynomial evaluated at fixed nodes ##x_i##, and the time functional as a multivariate function of the ##y_i##. The classical statement of the problem requires...
  23. K

    Why is the curl of the electric dipole moment equal to zero in the far field?

    Hello.Looking at Jackson's ch 9 on radiation, I am trying to calculate the fields E and B from the potentials in the far field but it is very confusing. Given now the approximation for he vector potential \textbf{A}_{\omega}(x) = -ik \frac{e^{ikr}}{r} \textbf{P}_{\omega} with...
  24. T

    Implicit Differentiation Question

    << Mentor Note -- thread moved from the technical math forums at OP request, so no Homework Help Template is shown >>[/color] x2y + xy2 = 6 I know we use the chain rule from here, so wouldn't that be: (d/dx)(x2y + xy2) = (d/dx)(6) so using the chain rule of g'(x)f'(g(x) and the d/dx...
  25. D

    [Multivariable Calculus] Implicit Function Theorem

    I am having trouble doing this problem from my textbook... and have no idea how to doit. 1. Homework Statement I am having trouble doing this problem from my textbook... Show that the equation x + y - z + cos(xyz) = 0 can be solved for z = g(x,y) near the origin. Find dg/dx and dg/dy (dg/dx...
  26. J

    Using integral calculus to find the equation of the quartic

    Homework Statement The question states Use integral calculus to find the euation of the quartic that has (1,23) and (3, 15) and a y-intercept of 24. Homework Equations The previous part of the question was A quartic has stationary points of inflection at x=1 and x=3. Explain why...
  27. M

    B Natural exponential function, calculus

    So I'm trying out various practice problems and for some reason I can't get the same answer when it comes to problems involving natural exponentials. Here's the problem A type of lightbulb is labeled as having an average lifetime of 1000 hours. It's reasonable to model the probability of...
  28. L

    Variational calculus or fluid dynamics for fluid rotating in a cup

    my first post having just joined! Problem statement - what curve describes the surface of a rotating liquid? Stirring my cup of coffee years ago sparked this thought. Question - is the way to solve this problem to use variational calculus, or fluid dynamics? I have always thought the former but...
  29. i_hate_math

    Area of Region Vector Calculus

    I have tried to apply greens theorem with P(x,y)=-y and Q(x,y)=x, and gotten ∫ F • ds = 2*Area(D), where F(x,y)=(P,Q) ===> Area(D) = 1/2 ∫ F • ds = 1/2 ∫ (-y,x) • n ds . This is pretty much the most common approach to an area of region problem. But here they ask you to prove this bizarre...
  30. I

    Weber-Fermat Problem, degenerate cases

    Homework Statement I have to prove some things on the Weber-Ferma problem. Here is the assignment : We want to find a point $$x$$ in the plane whose sum of weighted distances from a given set of fixed points $$y_1, ...,y_m$$ is minimized. 1-Show that there exist a global mimimum to the...
  31. ChloeYip

    Studying Are there any resources for studying calculus and physics?

    I am in year one, hope to get more exercises to work on practice... especially in calculus and physics (introductory level) My school is already provide webwork and masteringphysics as homework, but I don't think they are enough...(limited number of questions only...) Are there any more...
  32. weezy

    Proof of independence of position and velocity

    A particle's position is given by $$r_i=r_i(q_1,q_2,...,q_n,t)$$ So velocity: $$v_i=\frac{dr_i}{dt} = \sum_k \frac{\partial r_i}{\partial q_k}\dot q_k + \frac{\partial r_i}{\partial t} $$ In my book it's given $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
  33. karush

    MHB 10) AP Calculus linear functions

    $\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...
  34. A

    Calculus of Variations: Functional is product of 2 integrals

    Homework Statement Minimize the functional: ∫01 dx y'2⋅ ∫01 dx(y(x)+1) with y(0)=0, y(1)=aHomework Equations (1) δI=∫ dx [∂f/∂y δy +∂f/∂y' δy'] (2) δy'=d/dx(δy) (3) ∫ dx ∂f/∂y' δy' = δy ∂f/∂y' |01 - ∫ dx d/dx(∂f/∂y') δy where the first term goes to zero since there is no variation at the...
  35. Brian T

    Computational Looking for resources on Discrete Exterior Calculus and FEEC

    Does anyone have recommendations for reading/resources on Discrete Exterior Calculus and/or Finite Element Exterior Calculus? In particular, I want to learn the topics to use in a project for a course and so would like to learn how to implement these methods (specifically geared toward...
  36. weezy

    Verifying the Correctness of My Proof

    1. I have to show: 2. Given: 3. My attempt : I just want to verify if what I've done is correct or not. Thanks!
  37. S

    Prove r(t) moves in a line, if a and v are parallel

    Homework Statement A point moves on a curve \vec { r } with constant acceleration \vec { A } , initial velocity \vec { { V }_{ 0 } } , and initial position { \vec { { P }_{ 0 } } } b. if \vec { A } and \vec { { V }_{ 0 } } are parallel, prove \vec { r } moves in a line c. Assuming \vec {...
  38. M

    I A-level differentiation/derivative dilemma

    Hello, and thank you for your time. I just started my A-levels derivatives/differentiation , and I would be more than happy if you could help me clarify it. For example I know that y is a function in terms of x right? y=f(x) The derivative of it is f'(x)=dy/dx . This means it is the rate of...
  39. Tspirit

    A problem on calculus in Griffiths' book

    I can't understand the solution to Problem 1.4(a). The solution is the following: What puzzles me is that ρ(θ)dθ=ρ(x)dx ? Why are they equal?
  40. A

    Calculus Calculus 1 text book - Need review of precalculus

    Hi, so I'm taking calculus 1 this year however I haven't taken precalculus in several years. I don't remember any of it, and the textbook of the course doesn't review it at all(they just sample you questions) and I'm having issues solving the precalculus review questions(how necessary is it that...
  41. WeiShan Ng

    Find Antiderivative of y: y^2=x^2+1

    Homework Statement x=sec(t), y=tan(t), -π/2 ≤ t ≤ π/2 Try to find y in terms of x Homework EquationsThe Attempt at a Solution 1.[/B] ∂y/∂x = sec(t)/tan(t) y=∫sec(t)/tan(t)∂x =∫x/y∂x =(1/y)*∫x∂x =x2/2y + C 2y2=x2 + C When t=π/4, x=√2, y=1 2(1)2 = (√2)2 + C C=0 So y2 = x2/2 2. y/x = sin(t)...
  42. S

    I Properties of Direct Product of Half Open and Open Intervals

    The 2-D plane is usually constructed as "ℝxℝ" and ℝ is both open and closed. My question is, what is the direct product of a half open and an open interval? Is it also open or half open?
  43. TheDemx27

    How to Calculate Heat Current in a Spherical Shell?

    Homework Statement A spherical shell has inner and outer radii r_a and r_b, respectively, and the temperatures at the inner and outer surfaces are T_a and T_b. The thermal conductivity of he shell material is k. Derive an equation for the total heat current thought the shell in the steady...
  44. B

    I Understanding "Terrible" Math Notation: A Calculus Guide

    Is there some standardized math text with "proper universal notation" I could read for calculus? In one of my courses, $$\int\frac{dx}{x}$$ had a red mark through it, with a note that said "impossible" or something. I earned a zero on the question due to the above. In another instance...
  45. S

    MHB Proving Russell's Paradox in Predicate Calculus

    Can we prove in the predicate calculus,that there does not exist someone who can shave all those that do not shave themselfs?? (Russell's Paradox)
  46. S

    I What is the purpose of Arc-Length Parameterization?

    My teacher just briefly introduced arc length parameterization and went on to frenet serret frames, without any explanation or motivation. What is the purpose of arc length parameterization? What role does it play in TNB? What is the purpose of TNB frames anyways?
  47. casualluchador

    Courses Should I take calculus 2,3 and diff. eqs at CC while in HS?

    Hi I was wondering if taking calculus 2,3, and differential equations by the end of my senior year at the local community college would be a wise choice. Would taking these math classes before i use them in physics hinder my learning? (I want to be a physicist). Would I gain an advantage in...
  48. Elvis 123456789

    Integration by parts and approximation by power series

    Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...
  49. P

    Courses Is Skipping Calc I and II Worth It? Considerations for Freshman Physics Majors

    Hi! I'm a freshman and I plan to get my bachelor's in Physics. I couldn't get any Physics or Math classes this semester as most were either full or clashing with my mandatory History and English classes. I gave the Math Placement Test at my university and was able to skip Precalculus classes...
  50. S

    I Proving Theorem 1 in Spivak's Calculus: Tips & Tricks

    Hello I am struggling with proving theorem 1, pages 98-99, in Spivak's Calculus book: "A function f cannot approach two different limits near a." I understand the fact that this theorem is correct. I can easily convince myself by drawing a function in a coordinate system and trying to find two...
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