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.
I was able to find the equation of an ellipse where its major axis is shifted and rotated off of the x,y, or z axis. However, I could not find anywhere an equation for a spheroid that does not have its axis or revolution along the x,y, or z axis. How might I go about deriving such an...
$\displaystyle\lim_{{x}\to{0}}\left(\frac{\tan 4x}{6x}\right)=$
(A) $\dfrac{1}{3}$
(B) $\dfrac{2}{3}$
(C) 0
(D) $-\dfrac{2}{3}$
(E) DNE
solution
direct substitution of 0 results in undeterminant so use LH'R
so then after taking d/dx of numerator and denominator and factor out constant we...
1. We find the partial derivatives of ##f## with respect to ##x## and ##y## to get ##f_x = \frac{2\ln{(x)}}{x}## and ##f_y = \frac{2\ln{(y)}}{y}.## This makes the gradient vector
$$\nabla{f} = \begin{bmatrix}
f_x \\
f_y
\end{bmatrix} = \begin{bmatrix}
\frac{2\ln{(x)}}{x} \\
\frac{2\ln{(y)}}{y}...
206 (day of year number)
If $f(x)=\sin{(\ln{(2x)})}$, then $f'(x)=$
(A) $\dfrac{\sin{(\ln{(2x)}}}{2x}$
(B) $\dfrac{\cos{(\ln{(2x)}}}{x}$
(C) $\dfrac{\cos{(\ln{(2x)}}}{2x}$
(D) $\cos{\left(\dfrac{1}{2x}\right)}$
Ok W|A returned (B) $\dfrac{\cos{(\ln{(2x)}}}{x}$
but I didn't understand why...
If $\displaystyle f(x)=\int_1^{x^3}\dfrac{1}{1+\ln t}\, dt$ for $x\ge 1$ then $f'(2)=$
(A) $\dfrac{1}{1+\ln 2}$
(B) $\dfrac{12}{1+\ln 2}$
(C) $\dfrac{1}{1+\ln 8}$
(D) $\dfrac{12}{1+\ln 8}$
ok I am little be baffled by this one due the $x^3$ in the limits
since from homework you just take...
I am posting some AP calculus practice questions on MeWe so thot I would pass them thru here first
The solution is mine...
any typos or suggestions...
$\textbf{Find the Limit of}$
$\displaystyle\lim_{x\to \pi} \dfrac{\cos{x}+\sin{x}+1}{x^2-\pi^2}$
(A) $-\dfrac{1}{2\pi}$
(B) $\dfrac{1}{\pi}$...
Hi,
This is my first question here, so please apologise me if something is amiss.
I have two curves such that Wa = f(k,Ea,dxa) and Wb = f(k,Eb,dxb). I need to minimize the area between these two curves in terms of Eb in the bounded limit of k=0 and k=pi/dx. So to say, all the variables can...
This question consists of two parts: preliminary and the main question. Reading only the main question may be enough to get my point, but if you want details please have a look at the preliminary.
PRELIMINARY:
Let potential due to a small volume ##\delta## at a point ##(1,2,3)## inside it be...
I have been going through my old books again, and found myself a little stuck. I am not entirely sure if this would be better in this one or diffy eq.
The problem starts with having you find equation of motions when a= -bv, where b is constant and v = v(t)
Using method of separable equations...
I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
Hello, I am new to physicsforums and I am still a high school student so I would like to have advice on what books should be relevant on preparation for calculus and more math beyond. I have basic algebra and geometry foundation and I would like to learn more high school math and up. So my plan...
is there a rigorous version of this proof of fundamental theorem of calculus?if yes,what is it?and who came up with it?
i sort of knew this short proof of the fundamental theorem of calculus since a long while...but never actually saw it anywhere in books or any name associated with it.
i know...
On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows:
$$\delta J = \frac{\partial J}{\partial...
Summary: CALC BOOK QUESTION
Hey I am going to be self studying calc AP BC because my school only offers AB. So I bought from a ton of reddit advice Vellemans Calculus: A rigorous first course, due to the fact where I want a challenge similar to AOPS however more into solving more problems no...
Not satisfied with the following definition of calculus. What is a better definition? More detailed?
1a : a method of computation or calculation in a special notation (as of logic or symbolic logic)
b : the mathematical methods comprising differential and integral calculus —often used with the...
I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with.
For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...
Hello everybody.
If anyone could help me solve the calculus problem posted below, I would be greatful.
Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A
Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...
Problem Statement: A known mass at a know velocity collides on a spring of known stiffness. What is the equation that governs the deceleration of the mass, so that the force on the spring could be found?
Relevant Equations: 1/2 m*V^2 = 1/2*k*x^2 + 1/2*m*(Vo)^2
Kinetic energy of mass before...
Summary: Mechanics problem related with Calculus (differential equations)
Hi everyone, I would like some help in that task, if anyone would be willing to help :) Namely I have a problem from particle dynamics. "D:" means given info... so, D: m,g,h,b, miu. We're looking for v0 and S as given...
I want a book that isn't too wordy, explains things concisely, and covers single-variable and multi-variable calculus. I regret that I forgot so much of the stuff I learned in Calculus III and IV (especially the important theorems I learned but forgot in the latter), and now I want to review it...
I am attempting to solve an ODE using a Calculus add-in for Excel. I am an industry professional and I have not even thought about Differential Equations in 8 years. The equation that I am attempting to solve is in the form:
(1)
The ODE solver that I am using solves equations of the form...
Let:
##\displaystyle f=\int_{V'} \dfrac{x-x'}{|\mathbf{r}-\mathbf{r'}|^3}\ dV'##
where ##V'## is a finite volume in space
##\mathbf{r}=(x,y,z)## are coordinates of all space
##\mathbf{r'}=(x',y',z')## are coordinates of ##V'##
##|\mathbf{r}-\mathbf{r'}|=[(x-x')^2+(y-y')^2+(z-z')^2]^{1/2}##...
Hello,
I have been given a general equation that is dS/dt = (n-x-rS)(L-1)
and it needs to be rearranged in order to for the subject to be t.
I have spent a while trying to find a way to attempt this but with no luck
so I will leave this here.
Thanking You,
BSLAHi
I actually don't know how to proceed.
I tried something like this
The left side of the equation equals to $$\delta(\int_a^b F(x)dx)=\delta f(x) |_{a}^{b}$$
where ##f'(x)=F(x)##
However $$\delta f(x) |_{a}^{b}=f'(x)\delta x dx|_{a}^{b} = \delta (F(b)-F(a))$$
where ##f'(x)=F(x)##. For the...
For example integral of f(x)=sqrt(1-x^2) from 0 to 1 is a problem, since the derivative of the function is -x/sqrt(1-x^2) so putting in 1 in the place of x ruins the whole thing.
My volume integral is...
$$\pi\int y^2 dx$$
My surface area integral is...
$$2\pi\int y \sqrt {1+x'^2} dy$$I'm fairly sure the variable of integration on my volume and surface area integrals has to be the same, is that right? But when I change the variable in the surface area integral to...
Hello all. I've come across some math which consists of just applying the basic ideas of calculus (derivatives and integrals) onto discrete functions. (The link: http://homepages.math.uic.edu/~kauffman/DCalc.pdf )
The discrete derivative with respect to n is defined as ## \Delta_n f(n) = f(n+1)...
I have attached a word document demonstrating the working out cos i was too lazy to learn how Latex primer works and writing it like I did above would've been too hard too read. I tried to make it as understandable as possible, presenting fractions as
' a ' instead of ' a / b ' .
------
b
Hello,
I need help with question #2 c) from the following link (already LateX-formatted so I save some time...):
https://wiki.math.ntnu.no/_media/tma4135/2017h/tma4135_exo1_us29ngb.pdf
I do understand that the a0 for both expressions must be the same, but what about an and bn? I don't...
Quotient rule: z= f/g ------ z'= (f'g - g'f)/g^2
starting with finding the derivative in respect to x, i treated y^2 as constant 'a': z'= [(a*2x*cos a*x^2)(sin a*x^2) - (- a*2x*sin ax^2)]/cos(a*x^2)^2=
[(a*2x*cos a*x^2)(sin a*x^2)+(a*2x*sin ax^2)]/cos(a*x^2)^2
For the derivative in respect to y...
<Moderator's note: Member has been warned to show some effort before an answer can be given.>
Hello all.
This is my first post in this forum, I am asking for your understanding. I have a problem with the calculus task and I stuck in a dead endso I managed to find a solution on the internet. I...
I split this to get
\begin{equation}
\int ^{\infty} _{0} \dfrac{e^{ax}}{(1+e^{ax})(1+e^{bx})} \ dx - \int ^{\infty} _{0} \dfrac{e^{bx}}{(1+e^{ax})(1+e^{bx})} \ dx
\end{equation}
Then I tried to solve the first term (both term are similars). The problem is that I made a substitution (many ones...
Hello,
I am working through Spivak for self study and sharpening my math skills. I have become stuck on an exercise.
What I need to show is the following:
$$
(a + b) \sum_{j = 0}^{n} \binom nj a^{n-j}b^{j} = \sum_{j = 0}^{n + 1} \binom{n+1}{j} a^{n-j + 1}b^{j}
$$
My attempt, starting from...
Two weeks ago I had no idea on how to code using Python. Now I have completed an online course on functions, loops and strings. However, in that course I did not practice using the specific library called Sympy. Besides, I will use Python in the Physics-Math background, for solving problems like...
Homework Statement
Homework Equations
The Attempt at a Solution
[/B]
The solution to this problem is known. I want to use this exercise as a model to understand how to proceed when calculating the surface area of a geometric figure.
Question:
1) Why do we differentiate with...
Homework Statement
Homework Equations
$$F = \nabla \phi$$
The Attempt at a Solution
Let's focus on determining why this vector field is conservative. The answer is the following:
[/B]
I get everything till it starts playing with the constant of integration once the straightforward...
Homework Statement
Question:
If ##r_+ \neq r_{++}## and ## g(r=r_+) \neq g(r=r_{++}) ##
When is it fulfilled that ## d g (r=r_+) \neq d g (r=r_+) ## ?
Homework Equations
##r_+ \neq r_{++}##
## g(r=r_+) \neq g(r=r_{++}) ##
The Attempt at a Solution
I tried computing ## dg(r_+) =...
Homework Statement
Hello,
i am trying to do find the Fourier series of abs(sin(x)), but have some problems. As the function is even, bn = 0. I have calculated a0, and I am now working on calculating an. However, when looking at the solution manual, they have set up one calculation for n > 1...
I am looking for a book for learning Python so as to compute matrices, eigenvalues, eigenvectors, divergence, curl (i.e vector calculus).
If you also have online recommendations please feel free to write them.
Homework Statement
Given: ## f(x) = \sum_{n=0}^\infty (-1)^n \frac {\sqrt n} {n!} (x-4)^n##
Evaluate: ##f^{(8)}(4)##
Homework Equations
The Taylor Series Equation
The Attempt at a Solution
Since the question asks to evaluate at ##x=4##, I figured that all terms in the series except for the...
I have a strange question. It's strange because I don't need a correct answer. I need an answer that seems correct and leads to predictable results. I'm making a multiplayer computer game where the players fire cannons in outer space. The cannon shells will move through the gravitational fields...
I need to make an integral to fine the speed of the earth. Say the radius is 6378137 meters. How would I account for things closer to the axis that have a radius of 0.0001 meters? I don't think I can just take the speed at the radius. So I found that the Earth rotates at 6.963448857E-4 revs/min...
In Calculus II, we're learning about solids of revolutions and computing their volumes.
I'm unsure when to apply the appropriate methods and how to make the correct partitions.
Please tell me if my reasoning is correct:
The disk/washer method is applied when your partitions are perpendicular...
So I found the linear velocity by using the circumference of the Earth which I found to be 2pi(637800= 40014155.89meters. Then the time of one full rotation was 1436.97 minutes, which I then converted to 86164.2 seconds. giving me the linear velocity to be 465.0905584 meters/second. I know that...
In Calculus II, we're currently learning how to find the area of a bounded region using integration. My professor wants us to solve a problem where we're given a graph of two arbitrary functions, f(x) and g(x) and their intersection points, labeled (a,b) and (c,d) with nothing else given.
I...