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.
Here is a picture of the problem:
I am firmly stuck on part b). I know that relative growth rate is defined at f'(x)/f(x), but I cannot seem to come up with the right answer. I've tried everything I could conceivably think of. I do not understand why my answer is incorrect, unless I'm...
The original problem for anyone that can read Chinese:
https://zerojudge.tw/ShowProblem?problemid=b221
The problem defines a convex polygon with multiple points located in the first quadrant and the required task is to find a linear function y = ax that can spilt the polygon into two parts each...
Okay, I never took calculus 3 so I don't know how to write/solve an integral for a power I made up.
I want to flesh out my game world with math because it's cool and I can. Like Chester wrote some of the physics for the Bizarro universe. Applying physics and math to a fictitious universe is...
Hi, I have decided I like to learn some physics as I know absolutely zero physics and also including chemistry or biology, and I’d like to learn a bit of all 3, to the high school level perhaps. It seems that from, what I’ve read if I want to learn chemistry, I’ll first have to know physics, and...
High school student here...
Recently, I've found an interest in topology and am trying to figure out the correct path for self-studying. I am familiar with set theory and some concepts of abstract algebra but have not really studied any form of analysis, which from what I've read is a...
Homework Statement
Use the integral test to show that the sum of the series
gif.latex
##\sum_{n=1}^\infty \dfrac{1}{1+n^2}##
is smaller than pi/2.
Homework EquationsThe Attempt at a Solution
I know that the series converges, and the integral converges to pi/4. As far as I´ve understood...
Hi guys!
Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?
In general, when we are solving power...
<Moderator's note: Moved from a technical forum and thus no template.>
I'm solving a problem using tuple relational calculus ( TRC) in DBMS.
Problem
Find the name of all the employees who work for XYZ Bank Corporation.
Solution ( as given in my book)
I don't understand why book is using...
Homework Statement
Show that the equation arctan(x) = x^2 have at least one solution. Then argue why the equation only has one, positive solution r.
Homework EquationsThe Attempt at a Solution
I know how to prove that the equation has at least one solution (IVT), but I do not know how to show...
Homework Statement
Let R be the area in the xy-plane in the 1st quadrant which is bounded by the curves y^2+x^2 = 5, y = 2x and x = 0. (y-axis). Let T be the volume of revolution that appears when R is rotated around the Y axis. Find the volume of T.
Homework EquationsThe Attempt at a Solution...
hi, I'm a high-school student that is just beginning to learn calculus.
in calculus we are learning how to apply integration and diffrentiaiton methods regarding kinematics.
there is this certain phrase i do not really understand in our textbook: e.g."in the second second"
how am i meant to...
Homework Statement
a. Find a point at where these lines intersect
b. Find the equation of a plane that contains the two lines.
Homework Equations
r[/B] = <1,3,0> + t<3,-3,2>
r = <4,0,2> + s<-3,3,0>
The Attempt at a Solution
I correctly found the point of intersection to be...
Homework Statement
a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1.
b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4.
The Attempt at a Solution
I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...
Homework Statement
A container with height 4.5 is created by rotating the curve y = 0.5x^2 0 \leq x \leq 3 around x = -3 and putting a plane bottom in the box. Find the volume V of the box.
Homework EquationsThe Attempt at a Solution
I want to solve this by using the shell method. I have...
Hi, my name is "Bob" I have a particular question about a specific book.
Preface...
It has been a while since I formally studied Physics. The type of Physics book I remember using last was I believe, Algebra-based. This was in high school. Based on my degree, I studied Algebra and Trigonometry...
the question:
My attempt:
The partial derivatives did not match so i simply tried to find f(x,y) I got the set of equations on the right but that's about it.
I have recently started studying multivariable calculus and I cannot quite visualise the concepts.Problem solving is not a problem but I want a true understanding of the concepts.Which book or online resources are great at developing visualisation in this course?
Homework Statement
Find a > 0 so the integral
int(exp(-ax)*cosx)dx from 0 to inf get as high value as possible.
The Attempt at a Solution
My way of solving this is to plot the integrand, i.e. exp(-ax)*cosx and check for different values of a. The larger a is, the smaller the area under the...
Hi there , well my questions is in the title but i have to say some things :
I have multiples mathematiques and physics courses calc-based and so i want to review the basic .
I already search around this forum and others and i find some books who looks popular here there are :
- Pre-Calculus...
Homework Statement
Find the smallest value of b so that the function f(x) = x^3 + 9x^2 + bx + 8 is invertible.
Homework EquationsThe Attempt at a Solution
I know that the function has to be only increasing/decreasing, and I think it is needed to find the derivative of the function. I do...
Homework Statement
By relative velocity, we mean velocity with respect to a specified coordinate system. (The term velocity, alone, is understood to be relative to the observer’s coordinate system.)
(a) A point is observed to have velocity Va relative to coordinate system A.What is its...
Hello!
Let ##I## be an interval of size ##L##, suppose we divide it into bits of ##dx## then ##L=dx+dx+... =\alpha.dx##
Since ##dx## is by definition infinitesimally small is it correct to imply that for each ##x## there's a corresponding ##dx## hence, here, ##\alpha## would be, theoretically...
Hello everyone.
This was originally a homework problem but I realized my misunderstanding stems from the explanation given before the problem so here we are. The thread deals with these 3 pages from Spivak's Calculus:
https://ibb.co/kAKyVU
https://ibb.co/jXVSPp
https://ibb.co/kwRdVU
I'm pretty...
Homework Statement
"Consider a cylinder with a generator perpendicular to the horizontal plane; the only requirement for a point ##(x,y,z)## to lie on this cylinder is that ##(x,y## lies on a circle: ##x^2+y^2=C^2##.
Show that the intersection of a plane with this cylinder can be described by...
I am confused when I should use the ∂ notation and the d notation. For example, on http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx, in Case 1, the author wrote dz/dt while in Case 2, the author wrote ∂z/∂t. Could anyone please explain to me when I should use the ∂ notation and the...
Homework Statement
Find k so that y - 36x = k is a normal to the curve y = 1 / abs(x-2).
Homework EquationsThe Attempt at a Solution
My problem is regarding the absolute value. I know that the tangent to the curve must be (-1/36). In the solutions manual, it is said that by knowing the sign...
What is the difference between: THOMAS' CALCULUS, media UPGRADE (11TH EDITION) [ISBN-13 9780321489876] and THOMAS' CALCULUS (11TH EDITION) [ISBN-13 9780321185587]? could you provide detailed answer please?
what does the title-addition "Media Upgrade" mean and what changes does it make on the...
Section 1 Indices and Logarithms
1) Solve Ln(e6x−4+7)=7 for x.
The answer (correct to 3 decimal places) is x =
Marks : 1
Section 2 Functions
2) Let f(x) = −4x+72x−2, find f-1(x).
Answer: (Correct to 3 decimal places) f-1(x) = (-2 x + ) / ( x + ).
Section 3 Roots of Equations
3) Solve...
Hey guys, I am working on a school project and have fired some water rockets at different angles. I am now trying to compare theoretical results vs experimental results. But I am struggling to produce an accurate mathematical model.
I have attached my attempt to this post. It is quite long and...
Homework Statement
Derive the volume of sphere using Calculus. i saw videos on this topic on youtube, but i want to do it by the method of integrating a circle at angle θ (Theta) . i am posting a photo where i explained every thing i did but i couldn't know what i am doing wrong.
Homework...
Hello,
I'm new to the language of calculus. I am learning about Newtons second law and I'm trying to understand it's forms in calculus. Excuse my notation, just trying to keep it as simple as possible.
F=m * dV/dt
I know V= dx/dt
My textbook says you can 'apply chain rule' to obtain the...
Homework Statement
https://www.physicsforums.com/attachments/229290
Homework Equations
The Attempt at a Solution
I am not sure what equation to use for the volume[/B]
hello,
Some background
I transfer to SUNY Albany in Fall 2017 and took the following courses
Organic chemistry 1: C+
Calculus 2: B-
A PHY 140 Physics I: Mechanics (3): E (like F)
An introduction to the fundamentals of physics: Classical Mechanics. Topics include the concepts of force, energy...
Hi PF!
Given a functional ##J[y]##, if the first variation is $$\delta J[y] = \int_D(ay+y'')y \, dV + \int_{\partial D} (y'+by)y\,dS$$
am I correct to think that when finding stationary points of ##J[y]##, I would solve ##ay+y''=0## on ##D## subject to boundary conditions, which would either...
Hi, I'd like to compute the de Rham cohomology of the 3 following objects :
-A connected sum of ##g \in \mathbb{N}## reals projectives plans ##P_{2}(\mathbb{R})##.
-A connected sum of ##g \in \mathbb{N}## torus without ##n## points ##\mathbb{T}^{2} - \{x_{1}, x_{2}, ..., x_{n}\}##.
- A...
Greg Bernhardt submitted a new PF Insights post
The 10 Commandments of Index Expressions and Tensor Calculus
Continue reading the Original PF Insights Post.
I am doing a problem from Schutz, Introduction to general relativity.The question asks you to find a coordinate transformation to a local inertial frame from a weak field Newtonian metric tensor ##(ds^2=-(1+2\phi)dt^2+(1-2\phi)(dx^2+dy^2+dz^2))##. I looked at the solution from a manual and it...
Since the two functions touch at $\displaystyle \begin{align*} A\left( 0, \frac{15}{2} \right) \end{align*}$ that means that this point lies on the cubic function. Thus
$\displaystyle \begin{align*} \frac{15}{2} &= a\left( 0 \right) ^3 + b\left( 0 \right) ^2 + c\left( 0 \right) + d \\...
Hey, I have been told to study calculus following Spivak's book.
I was in an Engineering program and I have moved to a Physics one, and I want to retake calculus to really get good at it.
The problem is, Spivak's seems to me like it's very proof based, and I'm having a hard time even with the...
Which calculus book self contains experiences in order, and is stable to the max, for all the problems known in the subject?
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I have got confused now. Math history seems to allow knowing all the data on how math formed, for any of a...
I’ve always been confused by the formula for the Total Derivative of a function. $$\frac{df(u,v)}{dx}= \frac{\partial f}{\partial x}+\frac{\partial f }{\partial u}\frac{\mathrm{d}u }{\mathrm{d} x}+\frac{\partial f}{\partial v}\frac{\mathrm{d}v }{\mathrm{d} x}$$
Any insight would be greatly...
I am currently self learning high school level AP Calculus BC over the summer. From a mathematical perspective, I have heard that high school level calculus is just pure and shallow computational work. For this reason, I seek a deeper understanding in calculus. So far, the BC curriculum has been...
Homework Statement
I have to construct 8 concrete right cylinders for hurricane protection on windows, but I need to know the height to fit the given criteria. Each has a 4 inch radius and each weighs 1200 pounds. Concrete weighs 150 pounds per cubic foot.
V= volume of right cylinder
r=radius...
Homework Statement
My problem is in integral calculus (I'm new to it).
I know what it is and how it works (basically. I'm not too advanced right now).
The problem is as following: (I will be posting comments/reasons along with what I've done and with what logic/understanding I've done it...
Okay. So I'm new to calculus. And this is the first time I'm solving a physics problem using integration.
I understood that ∫dt (or ∫1dt) will be equal to just 't+C.' (Just like f ' (t+C) = 1).
Though, that's not the problem. The problem is when I apply it this way:
Question: The expression for...
Homework Statement
The path integral from (0,0,0) to (1,1,1) of $$<x^2,2yz,y^2>$$.
I am a little confused about the setup.Homework Equations
$$\int_{a}^{b} v.dl$$The Attempt at a Solution
Here is how I set it up.
$$\int_{0}^{1}x^2 dx + \int_{0}^{1}2yz dy + \int_{0}^{1}y^2 dz$$
Since the...
i have one question concerning measure theory and this...could we map the natural numbers to the positive rationals...then observe the measure between the rationals being mapped ,as a value giving us a sense of how many irrationals are between them...then generate a function where we take the...