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.
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...
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. :(
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 $\int_{0}^{1} (f(x))^2\,dx$.
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Would it be true since right hand Riemann sums for a negative, increasing function will...
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...
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...
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...
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...
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...
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.
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...
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...
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...
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...
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...
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...
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...
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...
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...
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...