Calculus Definition and 1000 Threads

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...

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...

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...

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...

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##...

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...

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...

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! :)

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...

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

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...

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...

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

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...

I'm going to use Courant's volume 1 and 2. I don't want to try every problem in the book, so how do I go about choosing problems to know if I understand the material?

Homework Statement Could somebody link me to a youtube video explaining this topic, its from an exam paper at me college and I can't find notes on it.It think it has something to do with limits. Many thanks.

Hi! I hace been trying to calculate how many energy in form of beta radiation is emitted in one if those "Tritiglows" sold in Amazon. I did the following math (imatge) and got a really high energy. How is that posible? Where did I failed...

I am currently nearing the end of my Calculus 3 course and have been struggling all semester. First there is some background information. I passed Calc 1 and 2 with a B and C respectively. Over the summer I worked on my skills and felt prepared. Unfortunately my section was chosen for IBL...

So I've seen in several lectures and explanations the idea that when you have an equation containing a relation between certain expressions ##x## and ##y##, if the expression ##x## approaches 0 (and ##y## is scaled down accordingly) then any power of that expression bigger than 2 (##x^n## where...

Let $f$ be a positive and continuous function on the real line which satisfies $f(x + 1) = f(x)$ for all numbers $x$. Prove \[\int_{0}^{1}\frac{f(x)}{f(x+\frac{1}{2})}dx \geq 1.\]

Homework Statement Prove the convergence of this series using the Comparison Test/Limiting Comparison Test with the geometric series or p-series. The series is: The sum of [(n+1)(3^n) / (2^(2n))] from n=1 to positive ∞ The question is also attached as a .png file 2. Homework Equations The...

I need to solve: \dot{\mathbf{r}}=-kv\hat{r} - \dot{\mathbf{r}_s} However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.

1. The problem statement, all variables, and given/known data Find and categorize extremes of the following function: $$F(y)=\int_{y}^{y^{2}}\frac{1}{\ln^{2}x}dx$$ for ##y>1##. Homework Equations $$\frac{d}{dx}\int_{a}^{b}f(x,y)dy=\int_{a}^{b}\frac{\partial}{\partial x}\left(f(x,y)\right)dy$$...

Homework Statement I tried to answer the following questions is about the curve surface z= f (x, y) = x^2 + y^2 in the xyz space. And the three questions related to each otherA.) Find the tangent plane equation at the point (a, b, a^2+ b^2) in curved surface z . The equation of the...

Hi everybody. I enjoy looking at other people's handwritten notebooks, as well as what textbooks they learned math and physics from. This evening I came across this article about how Feynman learned calculus in high school by studying Calculus for the Practical Man by Thompson. He kept very...

I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...

When working a proof, I reached an expression similar to this: $$\int_{-\infty}^{\infty} \frac{\mathrm{e}^{-a^2 x^2}}{1 + x^2} \mathrm{d}x$$ I've tried the following: 1. I tried squaring and combining and converting to polar coordinates, like one would solve a standard Gaussian. However...

Homework Statement A block of mass ##m = 1.00 kg## is being dragged through some viscous fluid by an external force ##F = 10.0 N##. The resistive force can be written as ##R = -bv##, where ##v## is the speed and ##b = 4.00 kg/s## is a phenomenological constant. You may ignore gravity (we...

Reading through a bit the book seems nothing like what I learned in uni (Stewarts calculus) Will "Book of Proof" By Hammack --> "Basic Mathematics" by Lang be enough preparation for Calculus by Spivak? Thanks.

Homework Statement I have a system of coupled differential equations representing chemical reactions and given certain initial conditions for the equations I can observe oscillation behaviour when I solved the equations numerically using Euler's Method. However, then it asks to investigate the...

Homework Statement I'm given the integral show in the adjunct picture, in the same one there is my attempt at a solution. Homework Equations x = r.cos(Θ) y = r.sin(Θ) dA = r.dr.dΘ The Attempt at a Solution [/B] I tried to do it in polar coordinates, so I substituted x=r.cos(Θ) y=r.sin(Θ) in...

Homework Statement Homework Equations cos2x = (1+cos2x)/2 sin2x = (1-cos2x)/2 The Attempt at a Solution I believe you would use the double angle formula repeatedly but that is very tedious; is there a more concise way to solve the problem?

Homework Statement Homework EquationsThe Attempt at a Solution it seems like because theta 2 is independent, therefor, you can solve theta 3 by just using one equation from the system of equation? on a previous problem where its a 4 bar mechanism( which it didn't specify that theta 2 is...

Homework Statement Is the gradient of a plane, the normal to the plane? If so, why? Homework Equations No idea, just a question that popped up in my head eon of plane: n(x-x1)+n(y-y1)+n(z-z1) The Attempt at a Solution I found the partial derivative of each, and got the normal.[/B]

Hi, I was just wondering how one would arrive at the answers to these questions. I have the solution for parts a and b, but not for part c. Suppose that antibiotics are injected into a patient to treat a sinus infection. The antibiotics circulate in the blood, slowly diffusing into the sinus...

Homework Statement You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree. Homework Equations Line integral of Pdx + Qdy = Double integral of...

I recently found out the rule regarding the Taylor expansion of a translated function: ##f(x+h)=f(x)+f′(x)⋅h+\frac 1 2 h^ 2 \cdot f′′(x)+⋯+\frac 1 {n!}h^n \cdot f^n(x)+...## But why exactly is this the case? The normal Taylor expansion tells us that ##f(x)=f(a)+f'(a)(x-a)+\frac 1...

Homework Statement Why is that we can set two variables zero in an equation of a plane to find a point on that plane? What is the proof for this? Homework EquationsThe Attempt at a Solution

Homework Statement Homework Equations The Attempt at a Solution does Vpa has the same direction as Vp? i thought Vp is tangent to the circular path that point P creates( perpendicular to vector Rpa) but from the figure, Vp doesn't seem to be tangent to the path, but Vpa does...

Homework Statement the second solution is the correct, I know you can put C on both sides and it simplifed to C2 on one side, but why can't you put C2 on the right side? Homework EquationsThe Attempt at a Solution

Homework Statement Find the z -coordinate of the center of mass of the first octant of a sphere of radius R centered at the origin. Assume that the sphere has a uniform density. Homework Equations Mass = Integral of the density function Center of mass for z = Integral of density * z divided...

Homework Statement Consider a sphere of radius A from which a central cylinder of radius a (where 0 < a < A ) has been removed. Write down a double or a triple integral (your choice) for the volume of this band, evaluate the integral, and show that the volume depends only upon the height of the...

Homework Statement [/B] F =< 2x, e^y + z cos y,sin y > (a) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along a straight path. (b) Find the work done by the force in moving a particle from P(1, 0, 1) to Q(1, 2, −3) along the curved path given by C : r(t)...

Homework Statement (a) Consider the line integral I = The integral of Fdr along the curve C i) Suppose that the length of the path C is L. What is the value of I if the vector field F is normal to C at every point of C? ii) What is the value of I if the vector field F is is a unit vector...

i have attached the problem set. I have done the first three problems but number 4 is very difficult. Can someone help me out? Thanks [Editor's note: The PDF below contains the complete problem set from which #4 is as shown above.]

How can volumes and hypervolumes be related to Einstein's theory of special relativity and to quantum mechanics? Also, can volumes and hypervolumes of objects be used for modeling how different scenarios can change over time? Oh yeah, and hi my name is Sasha Jaffarove!

According to me this topic must be raised and discussed how fermat did it without calculus.What problems he faced since calculus was developed afterwards by Newton leibniz. http://aapt.scitation.org/doi/10.1119/1.1514235 Moderator's edit: File substituted by link due to potential copyright...

Hello! So I need some textbook recommendations for calculus based physics course. I have taken an algebra/ trigonometry physics course last semester, but now I'm taking a calculus based. I have a textbook which the professor told us will follow our course. The book is Fundamentals of Physics...

Homework Statement f(x+y) = f(x) + f(y) for all x,y∈ℝ if f is continuous at a point a∈ℝ then prove that f is continuous for all b∈ℝ. Homework Equations lim{x->a}f(x) = f(a) The Attempt at a Solution I do not understand how to prove the continuity, does f(x) = f(a) or does f(x+y) = f(a)

Homework Statement Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically. Homework Equations (x,y,z) = (x0,y0,z0) +s(a1.a2,b3) + t(b1,b2,b3) The Attempt at a Solution...