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.
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...
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...
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 $$\frac{\partial v_i}{\partial \dot q_k} = \frac{\partial r_i}{\partial q_k}$$...
$\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...
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...
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...
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 {...
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...
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...
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)...
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?
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...
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...
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?
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...
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...
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...
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...
This is my first year in college, and I am currently taking calculus 1, and physics w/ calculus.
My academic adviser told me that I would be okay taking the two together since I'd be learning the calculus as i went. This doesn't seem to be the case because for a lot of my problems we are already...
I am at the stage of my education where I am seeing calculus (mainly differential equations) popping up in my engineering courses. However, I have just started multi-variable calculus, so I have not taken differential equations yet. My professors basically show us how certain equations are...
Homework Statement
Let P be a point on the sphere with center O, the origin, diameter AB, and radius r. Prove the triangle APB is a right triangle
Homework Equations
|AB|^2 = |AP|^2 + |PB|^2
|AB}^2 = 4r^2
The Attempt at a Solution
Not sure if showing the above equations are true is the...
Hi!
First time poster, I'm about to enter first year calc and thought that I could get ahead of the curve by checking out some questions beforehand. This showed up on one of the university calculus exams but I couldn't figure out how to do it. I tried to finding a common denominator but then was...
I have a question regarding the usage of notation on problem 2-11.
Find ##f'(x, y)## where ## f(x,y) = \int ^{x + y} _{a} g = [h \circ (\pi _1 + \pi _2 )] (x, y)## where ##h = \int ^t _a g## and ##g : R \rightarrow R##
Since no differential is given, what exactly are we integrating with...
So I have a quick question that will hopefully yield some clarification. So the divergence of a dyadic ##\bf{AB}## can be written as,
$$\nabla \cdot (\textbf{AB}) = (\nabla \cdot \textbf{A}) \textbf{B} + \textbf{A} \cdot (\nabla \textbf{B})$$
where ##\textbf{A} = [a_1, a_2, a_3]## and...
What's the difference between f(x)=3 and f(x)=3x^0 ? and why Limit of the second function when x\rightarrow0 exists ? and is the second function continuous at x=0 ?
Hey everyone,
I've been stuck on this one piece of HW for days and was hoping someone could help me.
It reads:
The surface area, A, of a sphere with radius R is given by
A=4πR^2
Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
I saw the topic and have given it a lot of thought over the past few years, and my take is so different from the other thread I'm starting another.
Problems with the beginning math curriculum:
1. Too abstract and difficult
2. Totally unmotivated
These problems were unavoidable 50 years ago...
Hello,
do you know of any books similar in style to Callahan's Advanced Calculus book(a book that explains the geometrical intuition behind the math)?
This goes for any subject in mathematics(but especially for subjects like vector calculus, differential geometry, topology).
Thanks in advance!
Good afternoon,
I'm looking to complete a Mathematics Calculus Exploration for my course, before starting to write up my project I was hoping to find someone that could offer some advice or comments to make my project more successful.
First of all my project seems very simple at first glance...
This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it:
Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)].
Prove that at each point of C the velocity vectors associated with X and Y are parallel...
Which skill is more important for Physics in the modern era?
From what I know, Newton (along with some other fellows) developed Calculus in order to solve problems in Physics. Because of this, it's safe to say that Calculus is an essential branch of mathematics to understand when it comes to...
Hi everyone,
I will be junior next year in my high school and take Calculus BC. However, I also take Calculus 1&2 Honor in the community college which is known as Orange Coast College ( If anyone has studied here before, please tell me the difference between the content of this course and...
Homework Statement
Section 11-2, Page 651, Example 3
Homework Equations
y = sin^3t and dy = 3cos^2t sint dt. Shouldn't there be a Minus Sign there?
The Attempt at a Solution
Answer is correct. But isn't there a missing Minus Sign?
I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector.
Does that mean I will have to treat it as a Taylor expansion about two variables...
Hello everyone,
Im currently taking my first semester of College Physics with Calculus and i need a good book to help me study concepts and maybe have example problems that have step by step analysis.
Any Suggestions?
Hi, there is a book of dg of surfaces that is also about tensor calculus ?
Currently i study with Do Carmo, but i am looking for a text that there is also the tensor calculus!
Thank you in advance
A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant.
Two ratios are proportional if they change equally and are related by a constant of proportionality? Not sure about this definition, but please correct it if you can...
Homework Statement
You have been employed but(sic) the Mathematics Football League (MFL) to design a football. Using the volume of revolution technique, your football design must have a capacity of 5L ± 100mL. You must present a statement considering the brief below. Just a quick side note, I...
Do I substitute A_\mu + \partial_\mu \lambda everywhere A_\mu appears, then expand out? Do I substitute a contravariant form of the substitution for A^\mu as well? (If so, do I use a metric to convert it first?)
I’m new to Ricci calculus; an explanation as to the meaning of raised and lowered...
Hello! I am currently in Calculus 2 and we are dealing with hydrostatic force. I get the integration that happens but I always seem to have trouble with solving the word problems. With this problem, I realize that I use integrate by making an infinite number of rectangles on the 2 triangles. I...
Homework Statement
Let A be a dense set**. Prove that if f is continuous and f(x) = 0 for all x in A, then f(x) = 0 for all x.
**A dense set is defined, in the book, as a set which contains a point in every open interval, such as the set of all irrational or all rational numbers.Homework...
Homework Statement
X and Y 2 real numbers / |x| <1 and |y|<1
Prove that |x+y|<|xy+1|Homework EquationsThe Attempt at a Solution
|x+y|<2
I couldn't prove that |xy+1| >2
And couldn't find a way to solve the problem
Please help
Homework Statement
Decide whether the given function is bounded above or below on the given interval, and which take on their maximum or minimum value. (Notice that ƒ might have these properties even if ƒ is not continuous, and even if the interval** isn't closed)
**The interval is (-a-1...
Can someone help me with this?
(dA/dt)=1cm/s (cm^2 whatever...leave out trivial corrections).
A=pir^2
(dA/dt)=2pir(dR/dt)
Multiply through by (1/2pir)
(dA/dt)/(2pir)=dR/Dt
What is the rate of change of the radius for a circumfrance of 2
I just used the related rates formula that I derived for...
Homework Statement :
the question wants me to prove that the limit of f(x,y) as x approaches 1.3 and y approaches -1 is (3.3, 4.4, 0.3). f(x,y) is defined as (2y2+x, -2x+7, x+y).
[/B]
The attempt at a solution: This is the solution my lecturer has given. it's not very neat, sorry...