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.
\lim_{x \rightarrow 1} \frac{x^2 - 1}{x-1}
For this, we first divide the numerator and denominator by (x-1) and we get
\lim_{x \rightarrow 1} (x+1)
Apparently, we can divide by (x-1) because x \neq 1, but then we plug in x = 1 and get 2 as the limit. Is x = 1 or x \neq 1? What exactly is...
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...
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...
Hi everyone!
I just bought the courant John calculus volume-1 from uni bookstore. I need some tips like errors, topics which are not explained well and topics which are really important and need to focus more on etc..
My background:- have Spivak as a supplement. Knows how to prove and read...
If Tl;dr I am struggling in Math 171 and Physics 191 and throwing around the idea of declaring a geology major with an astronomy minor because the Physics major "juice is not worth the squeeze" at my age(29) anyone else out there who struggled with Calculus 1 when they first took it?
Hello...
I've been able to find the tangent line with most equations, but I don't have any idea how to do it with a range of values instead of being given a singular value.
Mentor note: Moved from technical section, so is missing the homework template.
Im doing some older exams that my professor has provided, but I haven't got the solutions for these. Can someone help confirm that the solutions I've arrived at are correct?
Homework Statement
Hello,
using Simpson´s method, one can calculate the number of intervals needed to achieve a given accuracy, through the error formula. Is there any other way to know that the required accuracy is achieved other than computing the integral and comparing it to the result from...
Homework Statement
Let Tn(x)=1+2x+3x^2+...+nx^(n-1)
Find the value of the limit lim n->infinity Tn(1/8).
The Attempt at a Solution
How do I solve this? I know how to write the polynomial as a series, but not sure how if this is the best way of finding the limit.
Homework Statement
Hi all, I'm currently studying the amazing Calculus by Spivak. Whenever I study textbooks I always attempt to do all the examples and proofs in the text before looking at the answers.
(Whether this is a good thing or a bad thing I don't know, the examples are similar to the...
Homework Statement
A trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 30 cm wide at the bottom, 80 cm wide at the top, and has a height of 50 cm. If the trough is being filled with water at the rate of 0.2m3/min, how fast is the water level rising when the...
Homework Statement
find maxima/minima of following equation.
Homework Equations
-(x+1)(x-1)^2
The Attempt at a Solution
(-x-1)(x-1)^2
Using product rule, we obtain,
-1(x-1)^2+(-x-1)*2(x-1)
I don't know where to go from here. The software's factoring I had never seen before.
Homework Statement
This is a chain rule problem that I can't seem to get right no matter what I do. It wants me to find the derivative of y=sqrt(x+sqrt(x+sqrt(x)))
Homework Equations
dy/dx=(dy/du)*(du/dx)
d/dx sqrtx=1/(2sqrtx)
d/dx x=1
(f(x)+g(x))'=f'(x)+g'(x)
The Attempt at a Solution
My...
Hello and thank you for trying to help.
In spite of the fact that this seems a very simple problem, I do not find myself able to get a solution. Here it goes:
Let $$f(x)=\displaystyle \sum_{k=3}^\infty a_k \frac{x^k}{k(k-1)(k-2)}$$ and $$g(x)=\displaystyle \sum_{k=0}^\infty a_k x^k$$. Express...
This is a simple exercise from Spivak and I would like to make sure that my proof is sufficient as the proof given by Spivak is much longer and more elaborate.
Homework Statement
Prove that \lim_{x\to a} f(x) = \lim_{h\to 0} f(a + h)
Homework Equations
The Attempt at a Solution
By the...
Homework Statement
Find the differential
Homework Equations
Chain rule : dy/du=dy/du*du/dx
Product rule: f(x)g'(x) + g(x)f'(x)
The Attempt at a Solution
I have tried to move the radical to the top of the equation by making it into an exponent (x^2+1)^-1/2. I then used the product rule and the...
Homework Statement
For ##x> 1##, prove that $$2\sqrt{x} > 3 - \frac 1 x$$
Homework Equations
Definition of increasing function
The Attempt at a Solution
Let ##f(x) = 2\sqrt{x} + \frac 1 x## defined on domain ##[1, \infty)##. Function is increasing if ## f'(x) > 0## on some interval ##I##...
Does a limit exist on a graph at (-1,0) if the point is solid, and has a right sided limit, but there is nothing left of the point?
I understand that if the left sided limit and the right sided limit are different then it doesn't exist, but on my graph it shows a line coming from the right...
I've always been the "Doesn't study but does well" type. Not only have I been informed of this being basically impossible in college, but I've also learned that my major, (Computer Engineering) is quite a bit of work.
This I do not mind. I enjoy the work. What I worry about, is being able to...
Homework Statement
If you throw a ball straight down from a building that is 443 meters tall with a velocity of 22 m/s how long would it take to reach the ground, and what would be its speed at impact?
Homework Equations
Since the ball is being thrown down with a velocity of 22 m/s I plug that...
Homework Statement
If f(x) = sin(x) +2x+1 and g is the inverse function of f, what is the value of g'(1)? Answer is 1/3. No calculator allowed. About 2 minutes allowed to solve.
Homework Equations
g'(a)=1/f'(g(a)) (from f(g(x))=x, taking derivative of both sides)
The Attempt at a Solution...
Hello,
Me and my friend were talking about differentiability of some piece-wise functions, but we thought of a problem that we could were not able to come to an agreement on. If the function is:
y=sin(x) for x≠0
and
y=x^2 for x=0,
Is this function differentiable? The graph looks like a normal...
Hey there, I will be taking calculus 2 this spring semester and I would like to know what topics from calc 1 I should be reviewing, I did receive a B+. I am also aware that trig is highly involved and my trig skills are not that good, so if there's anything from trig that is mandatory to review...
Just want to see if I actually understand what these all mean.
Partition: is like the x-coordinate values, also gives the number of times the graph was chopped up. We need them in order to find the distance or length of each rectangle. The distance is found by taking the further point minus...
Important:
Would you please tell me which pages or sections in the Humongous Book are Cal 1 ( which I need for my online pre-req course in Cal 1) and/or which pages or sections Relate to Cal 2 (which I can skip)?
Thank you :)
Not important:
I am taking an online course in Cal 1 but have to...
Hello everyone. I am about to start Calculus 1 and then Calculus 2. I want to get an idea of how the order of concepts of these to classes are generally laid out.
Thanks
Hello! I'm going to take calculus based physics and calculus 1 in the fall. At my school they're corequisites. But I'd like to do well in both of them.
It really worries me that I'm taking calculus based physics when I haven't ever taken any calculus before. Should I take cal based physics while...
Has anyone been through independent study calculus course ( DE through a university out of Utah)? I really miss the aspect of having an instructor but I do enjoy being able to focus on the aspects that I am not great at while passing over things that I've already learned. I would really like to...
So I started with limits and everything went well, and then into derivatives, and a problem started.
First I know how to find the derivative of basic things using like ( limit definition of derivative) or (rules for derivatives) or ( product rule/ quotient rule/ chain rule)...
My problem...
Homework Statement
The problem is:
f(x) = {ax^2 - b / (x - 2), x < 2
1/4(x^2) - 3c, x >= 2
If f is differentiable at x=2, what are the values of a, b, and c?
In case it is hard to see, I have provided an image
2. Homework Equations
I know that the limit definition of a...