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.
##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*
The point (1, 5) is on the curve: y=ax^2+bx+c. This point gives the linear equation: 5 = a + b + c. A second point on the curve, (2, 10) gives the linear equation 10=4a+2b+c. A student called Erika thinks that the point (2, 19) is also on the curve.
5 = a + b + c.
10=4a+2b+c
19=4a+2b+c
the...
This is a good book to read, free pdf version. Don’t judge the book by it’s cover. Read section 1, and 2, see if you learn something you weren’t expecting ~.^
This is the above link’s target:
https://www.gutenberg.org/files/33283/33283-pdf.pdf
Enjoy!
School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑
Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...
In Chapter 20 of Spivak's Calculus is the lemma shown below (used afterward to prove Taylor's Theorem). My question is about a step in the proof of this lemma.
Here is the proof as it appears in the book
My question is: how do we know that ##(R')^{n+1}## is defined in ##(2)##?
Let me try to...
I'm going to be starting my first Mathematics module (MST124 - Essential Mathematics I, at the Open University) and I have been looking or a Calculus textbook to use as a supplementary text.
I've found a couple of textbooks that I like the look of (Stewart's and Larson's), which both come in...
I was hoping to explore the Calculus of Variations.
How do we prove by Calculus of Variations that the minimum time for boat crossing a river (perpendicular to the current for starters) with current ##v_r##, and boat velocity in still water ##v_b## that the path will be a straight line?
I...
Summary: Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or calculus?
Hello! I'm an high school student and i want to study more math but I'm not sure where to start. Should i first study linear algebra or...
Dear members in this nice community,
Let me introduce myself first. My name is Derek Liang, and I am from Canada. I hold a bachelor degree in math education, a PhD in math from China and a PhD in applied math from Canada. I have more than twenty years of experience teaching and tutoring math at...
The Korean textbook standard defines the convexity of the function as an open section. Many textbooks and university calculus textbooks define the convexity of the curve as an open section. However, some textbooks define convexity as closed sections.
Do you think it is right to define the...
Hi,
I am going to be a 1st year college student in China this fall. I have a lot of interest for calculus and math in general, and I am wondering if the books I have in hand are good for self study.
High school math here stopped the at the derivatives, didn't went further.
The two books I...
I tried very hard studying calculus before my semester started. I self-taught myself for months and realized that I was actually good at it. I felt very confident, so I decided to take a online summer class. This was my first calculus class ever. Rather than a 15 week semester, the class is only...
Problem Statement : To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP.
Attempt 1 (without calculus): I mark some relevant lengths inside the circle, shown left. Clearly OS = 9 cm and SP = 12 cm using the Pythagorean...
Summary: Need a multivariable calculus textbook
For calculus I’ve been using James Stewart textbook as a guide, I find it really hard to follow so I just checkout the chapter titles and then use online courses that explain the chapters, for example professor Leonard and Michel Van biezen...
Summary: Trying to differentiate with respect to ## \theta ## is entangling me in cos and sec terms.
A simple problem I found, while looking for calculus practice. Roads between home and main road are 30 mph, main road is 60mph:
What is the optimum ## \theta ## to minimise journey time?
## t...
I'm having some problems using the chain rule and I'm not sure where the trouble lies. For example:
If I'm not mistaken, if we have the composite function f(g(n)) then \Delta f(g(n)) = \dfrac{ \Delta f(g) }{ \Delta g } \dfrac{ \Delta g(n) }{ \Delta n }
Let f(g(n)) = (n^2)^2. Then f(g) = g^2...
Hi, if the force is the derivative of potential energy, does it mean that the force is equal to mg and with a constant gravity, it will be the same at any height?
But in real life, F (or mg) would be different on the Earth's surface and 400 km above it (~8 m/s^2).
So, this formula is used to...
Hi.
I have the Marsden an Tromba vector calculus book 6th edition.
I was wondering which software was used to create the books graphs.
I attach two graphs as an example.
Thanks
Summary:: What calculus books do you recommend? Does Thomas Calculus include all the calculus topics?
Hi! I'm a 10th grader and preparing for physics olympiads. I'm planning to learn calculus this summer, i self learned prior required topics before calculus (trigonometry, logarithm etc.) ...
How to represent this system in state space form?
where:
$$ x' = Ax + Bu \text{ and
}y = Cx + Du$$
I am trying to create a state space model based on these equations on simulink, need to find A, B, C and D but like I mentioned, i cannot find the solution when the differentials are not of...
Hi, Hi,
Author said If we look at the graph of $ f (x, y)= (x^2 +y^2)*e^{-(x^2+y^2)},$ as shown in the following Figure it looks like we might have a local maximum for (x, y) on the unit circle $ x^2 + y^2 = 1.$
But when I read this graph, I couldn't guess that the stated function have a...
Question: Prove that the radius of the right circular cylinder of greatest curved surface area which can be inscribed in a given cone is half of that of the cone.
Answer:
Let r and h be the radius and height of the right circular cylinder inscribed in a given cone of radius R and height H. Let...
Hi PF, stucked with this proof:
Taylor's Formula
The following theorem provides a formula for the error in a Taylor approximation ##f(x)\approx{P_{n} (x)}## similar to that provided for linear approximation (...)
Taylor's Theorem
If the ##(n+1)##st-order derivative, ##f^{(n+1)} (t)##, exists...
Taking the variation w.r.t f(x) of the integral over some x domain of F[f(x), f'(x), df(x)/dt], why doesn't df(x)/dt need to be taken a variational derivative and is treated as if it were constant?
In Keisler Elementary Calculus page 39, example 4 it shows how to compute the standard parts of the following expression:
Example 4: If ##\epsilon## is infinitesimal but non zero, find the standard part of
##b=\frac {\epsilon} {5-\sqrt{25+ε}}##
Before calculating the standard parts the...
This is the question,
Now to my question, supposing the vectors were not given, can we let ##V=\vec {RQ}## and ##W=\vec {RP}##? i tried using this and i was not getting the required area. Thanks...
Hello all! I was thinking about strengthening my knowledge of Calculus after I finish the course I am taking in Multivariable Calculus. I am in a particularly unique situation, as I am only going to enter high school next year.
I took the AP Calculus BC Exam last year and got a 5. The course I...
Hey!
I have to show that the integral of the area of a sphere ##\int \frac{d\vec{a}}{d} = \frac{4}{3}\pi \vec{r}##, with ##d = |\vec{r} - \vec{R}|## and ##\vec{r} = r \hat{z}##
This is what I did.
##d\vec{a} = R^2sin\theta d\theta d\phi \hat{r}##, ##\hat{r} = sin \theta cos \phi \hat{x} + sin...
I am about to start a physics with theoretical physics major, I've taken calculus before but I've not been satisfied with the "memorization of formulas" type books.
I started to read Spivak and found it enjoyable, but since it's a major undertaking I am also concerned for the practical value...
A set is nothing more than a collection. To determine whether or not an object belongs to the set , we test it against one or more conditions. If it satisfies these conditions then it belongs to the set, otherwise it doesn't.
The geometric point of view of sets- a set can be viewed as being...
c) Why is the assertion ##\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)## obvious?
First of all I don't think it is obvious but here is an explanation of why the limits are the same.
##\lim\limits_{x\to0} f(x^3)=l_2## means we are looking at points with ##x## close to zero and...
I believe the x-axis is vertical here.
The graph is composed of
i) an infinite number of intervals that start on the ##y=x## line and finish at some ##x## with a decimal expansion ending in ##7\bar{9}##. E.g., from ##0.67## to ##0.67\bar{9}## which is considered ##0.68##.
Other examples of...