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.
TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift?
Say you were given the equation :
How would you find : with a calculator that can only add, subtract, multiply, divide
Is there a general formula?
Hi, so I know how to find domain but how about range in this problem?
I don't understand the way he did it?
I always get answers wrong when it comes to range.
EXAMPLE 4 Find the area of the region ##R## lying above the line ##y=1## and below the curve ##y=5/(x^2+1)##.
Solution The region ##R## is shaded in Figure 5.24. To find the intersections of ##y=1## and ##y=5/(x^2+1)##, we must solve these equations simultaneously:
##1=\frac{5}{x^2+1}##
so...
I'm given the wavefunction
and I need to find the normalization constant A.
I believe that means to solve the integral
The question does give some standard results for the Gaussian function, also multiplied by x to some different powers in the integrand, but I can't seem to get it into...
Hi,
With respect to the techniques mentioned in point 2 and 3:
Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques.
Below excerpt shows problems 4 and 5 referenced in the above 2 points:
Using integration by parts:
$$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$
$$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$
Then how to continue?
Thanks
(a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##.
I also tried to calculate the limit directly (not using geometrical argument at...
I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...
Hello,
I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?
For this,
I first try to work out where function is increasing
My working is
##f'(x) = 12x^3 - 12x^2 - 24x##
For increasing,
##12x(x^2 - x - 2) > 0##
##12x > 0## and ##(x - 2)(x + 1) > 0##
##x > 0## and ##x > 2## and ##x > -1##
However, how do I combine those facts into a single domain...
Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
For this,
Does someone please know why we are allowed to take limits of both side [boxed in orange]?
Also for the thing boxed in pink, could we not divide by -h if ##h > 0##?
Many thanks!
For,
Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n##
Many thanks!
When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...
For this problem,
Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)?
Many thanks!
For part(a),
The solution is,
However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)##
Many thanks!
Is my solution correct? (I only have answers to odd-numbered exercises.)
Is it a good solution or have I overcomplicated things?
(a)
The forward force provided by the engine balances the air resistance force, so ##F_{engine}=F_{air} = \alpha v^2 + \beta /v{^2}##.
Let ##W_{engine}## be the...
We were taught that in cylindrical coodrinates, the position vector can be expressed as
And then we can write the line element by differentiating to get
.
We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
Here is an interesting book a student could do after after Algebra 1, or even integrate into an Algebra 1 course:
https://www.amazon.com/dp/B077VV95N3/?tag=pfamazon01-20
And a website:
https://www.calculussolution.com/
Several topics become easier, such as logarithms, when you know a...
While I was preparing for an integrals contest, I had a doubt about the following integral, I tried several substitutions but nothing worked.I would appreciate your support for this beautiful integral.
$$ \int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...) \ \mathrm{d}x$$
[mentor's note - moved from one of the homework help forums]
Homework Statement:: It's a question.
Relevant Equations:: Vector calculus.
Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector?
± can show possible directions in one dimension and...
I have found a new formula in Dirac calculus. The formula is elementary, so probably I'm not the first who found it. Yet, I have never seen it before. As many other formulas in Dirac calculus, it is not rigorous in the sense of functional analysis. Rather, it is a formal equality, which is only...
Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt:
(i) General Riemann Sums
Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
Multivariable calculus is a branch of mathematics that extends the concepts of single-variable calculus to functions of multiple variables. In this subject, vectors and partial derivatives are introduced to represent and manipulate multi-dimensional data. The gradient of a function represents...
This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example
## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ##
and not
## 2 \bf{v} \cdot \nabla \bf{v} ##
Is there an obvious way to see / think of this...
Hi PF
There goes the quote:
The Basic Area Problem
In this section we are going to consider how to find the area of the region ##R## lying under the graph ##y=f(x)## of a nonnegative-valued, continous function ##f##, above the ##x##-axis and between the vertical lines ##x=a## and ##x=b##, where...
Mentor note: For LaTeX here at this site, don't use single $ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane.
I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
I can get the domain, but getting the range seems impossible.
Domain
$$x-5=0$$
$$x =5$$
$$\therefore x \in (- \infty ,5) \cup (5, + \infty)$$
Range
I can simplify the function to the form below, but I don't know how to go from there.
$$ f(x)= x + 5 + \frac {1}{x-5}$$
I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...
If I were to read up to and including Chapter 3, would I be prepared enough to read Spivak's Calculus or at least some single variable calculus text based on proofs?
I'm asking because I plan on taking Real Analysis later and I'd like to gain a better understanding of Calculus.
I have read...
##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}##
I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)##
but now I have to show if it differentiable or not at ##(0,0)##.
According to answers it is not...
The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below:
In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
The following properties of big-O notation follow from the definition:
(i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##.
(ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...
What should I do when the f(x, y) function's second derivatives or Δ=AC-B² is zero? When the function is f(x) then we can differentiate it until it won't be a zero, but if z = some x and y then can I just continue this process to find what max and min (extremes) it has?
What I've done is...
My Progress:
I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change:
$$
\frac{\partial}{\partial\omega} f =...
IMPORTANT: NO CALCULATORS
I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did
f'(a) = f'(b):
1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1
2) 4a^3 - 4a = 4b^3 - 4b
3) 4(a^3 - a) = 4(b^3 - b)
4) a^3 - a = b^3 - b
5) a^3 - b^3 = a - b
6) (a...
Hi, I'm differentiating the "z" function to find extreme points but after solving the first partial derivatives with respect to "x" and "y" and also the "x" variable of the system, I can't find "y" (still in the system) using "ln" (natural logarithm).
The question is can I differentiate both...
Hi, I'm trying to calculate my own physics problem but didn't get it something.
When I'm trying to calculate the impulse of the object when it's hit by F=10N force in the smallest possible time, then should I write:
dP/dt = Fnet => dP = Fnet*dt ?
Another question: In general, if I calculate...
(expression given to be proven)
check for p(1)... 2=2
substitute (n+n) to
And here is the problem, I just can't find a way to continue solving this problem
Hi, last semester I "solved" a full differential equation and the answer was (see the picture). What does it mean? Can I make a graphic with it or what? I really don't get it.
*Arrows are just a continuation of the main formula*