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.
Homework Statement
Determine whether the following series converges absolutely, converges conditionally or diverges. Show your work in applying any tests used. sigma[k=1,inf] [(-1)^k*k/sqrt(k^4+2)]
Homework Equations
integral csc(x) dx = -ln|sec(x)+cot(x)| + C
csc(x) = 1/sin(x)
sec(x) =...
Homework Statement
The sereis sigma[k=1,inf] [(-1)^k/k^p] converges conditionally for
(a) p<1
(b) 0<p<=1
(c) p>1
(d) p=0
(e) None of the above
Homework Equations
The Attempt at a Solution
The answer key said that (b) was the correct answer and I'm having trouble...
Homework Statement
Which of the following functions are soltuions of the differntial equation y''+y=sin(x)?
a) y=cos(x) b) y=sin(x) c) y=1/2*xcos(x) d) y=1/2*x*sin(x)
Homework Equations
The Attempt at a Solution
I'm kind of lost on how to solve this problem. I don't think...
Homework Statement
Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x)
Homework Equations
The Attempt at a Solution
Find the power series for f(x) using the definition of taylor series expansion about a=9. f(x)=1/sqrt(x)
f(x) =...
Homework Statement
Find the radius of convergence and the interval of convergence of the series sigma[n=1,inf] ((3x-2)^n/(n^2*3^n))
Homework Equations
The Attempt at a Solution
sigma[n=1,inf] ((3x-2)^n/(n^2*3^n))
I applied the Root Test
p=lim n->inf |(3x-2)^n/(n^2*3^n)|^(1/n) = lim n->inf...
New Question (Changed Old one) - Taylor Polynomial - Upper Bound for Absolute Error
Homework Statement
(a) Find the 3-rd degree Taylor polynomial of sin(pix) centered at x=1.
(b) Use (a) to approximate sin(1.1*pi)
(c) Use the remainder term to find an upper bound for the absolute error in...
Homework Statement
Determine if the series
inf
Sigma n/(2n+1)
n=1
converges
Homework Equations
The Attempt at a Solution
When i did this I originally I thought I would just apply the divergence test
lim n/(2n+1) =/= 0
n->inf
there fore I thought by the...
Hi,
I was wondering if it was really necessary to evaluate improper integrals with limits? Could anyone really say I was wrong if I did something like
find the area bounded by the region y=1/x^2, x=2, and the x-axis
integral[1,inf] dx/x^2 = (-1/x)|[2,inf] = (-0)-(-1/2)=1/2
Like I don't...
Homework Statement
Hi,
I'm really struggling with trying to come up with the error bound when doing taylor series problems
Use the reaminder term to estimate the absolute error in approximating the following quantitites with the nth-order Taylor Polynomial cnetered at 0. Estimates are...
Hi,
If I'm given something like this for a problem,
Approximate the given quantities using Taylor polynomials with n=3
sqrt(101)
how do I know what I should set f(x) equal to? I could set it to many different things, sqrt(x), sqrt(x+100), sqrt(x+50). My answer would be very different...
Homework Statement
Hi,
I'm trying to solve the problem in the attachment. I was asked to evaluate the left hand side equation of the equal sign. I was unsure how to go about evaluating it so I consulted my solutions manual to look up the first step. The right hand side equation of the...
Homework Statement
Find the arc length of the following curves on the given interval by integrating with respect to x
y=x^4/4+ 1/(8x^2); [1,2]
Homework Equations
Let f have a continuous first derivative on the interval [a,b]. The length of the curve from (a,f(a)) to (b,f(b)) is...
Homework Statement
Hi,
I'm trying to solve this problem and guess I'm doing something wrong.
Evaluate Integral tan(x)^5*sec(x)^4 dx
Homework Equations
integral tan(x) dx = ln(|sec(x)|)
integral tan(x)^n dx = tan(x)^(n-1)/(n-1) - integral tan(x)^(n-1) dx
tan(x)^2+1=sec(x)^2
The Attempt at...
Homework Statement
Evaluate integral dx/(x^4-10x^2+9)
Homework Equations
The Attempt at a Solution
Ok so I'm trying to solve this problem for my calculus II course as a homework problem and am lost.
The first attachment you'll see how I attempted to evaluate the integral by...
August 22nd is my first day of school, I'll be taking Calc II. I'd like to prepare now.
It seems like Calculus I and II at any given school generally use the same textbook, my Calc I textbook was only covered about half-way in calc I. However, I'm transferring and it's no longer the same...
Homework Statement
Evaluate integral sqrt(x^2-9)/x dx
Homework Equations
The Attempt at a Solution
I get this for my answer sqrt(x^2-9)+3sin^(-1)(3/x)+c which is apparently wrong because when I take the derivative of it I don't get the integrand
my work is a attachment in this post
wolfram...
Homework Statement
Evaluate
integral dx/sqrt(1-2x^2)
Homework Equations
The Attempt at a Solution
My Answer: -cos^(-1)(sqrt(2)x)/sqrt(2) + c
See my work in attachment
Wolfram Alpha Answer: (sin^(-1)(sqrt(2) x))/sqrt(2)+constant...
Homework Statement
Evaluate
integral dx/sqrt(x^2-49)
Homework Equations
The Attempt at a Solution
SEE POST #4 BELOW WITH ATTACHMENT TO VIEW MY SOLUTION EASIER TO READ
i was trying to use a csc(theta) substitution and I don't see how my logic and math below is flawed but apparently I'm...
Evaluate
integral sqrt(9-x^2)/x dx
I get to here
-3 ln|csc(sin^-1(x/2)+cot(sin^-1(x/3))|+3cos(sin^-1(x/3)+c
I plugged in for theta because originally I had this
-3 ln|csc(theta)+cot(theta)|+3cos(theta)+c
and I set x equal to a expression
x=3sin(theta)
Hence i had to...
Homework Statement
Evaluate
integral csc(x)^4/cot(x)^2 dx
Homework Equations
The Attempt at a Solution
Apparently I'm doing something wrong, what I'm not sure, thanks for any help
My Answer: 2*tan(x) - (sec(x)^2*tan(x))/3 + c
integral csc(x)^4/cot(x)^2 dx
used fact that...
Homework Statement
Evaluate integral( sin^3(x) cos^5(x) ) dxHomework Equations
sin^2(x) + cos^2(x) = 1
integral x^n dx = x^(n+1)/(n+1) + c
d/dx cos(x) = -sin(x)
a^n*a^m=a^(n+m)
The Attempt at a Solution
I got -cos^6(x)/6+cos^8(x)/8+c
Apparently I did something wrong
SEE ATTACHMENT
Thank...
Homework Statement
Apparently I'm doing something wrong. I'm kind of lost as to what because I looked over my work several times.
Homework Equations
sin^2 x = ( 1 - cos 2x )/2
cos^2 x = ( 1 - sin 2x )/2
integral sin(x) dx = -cos(x)
integral cos(x) dx = sin(x)
The Attempt at a...
So I've apparently been given an assignment on Cauchy functions (it says here on the title), but I have no idea what that means. Nevertheless, here's my attempt to solve this problem:
Given (1):
f(x)=\frac{k}{\pi}\times\frac{1}{k^{2}+(x-\beta)^{2}}
and (2):
\hat{\beta}_{k}=arg\ max_{\beta}...
Homework Statement
I spoke to a few people and pretty much everyone but me seem to know what is going on with a few questions. This question was one of the ones I stared blankly for a few minutes and then wrote down an answer
\int_{0}^{2} e^{x^2} d\theta
The Attempt at a...
Homework Statement
This is the third problem of a project for a Calculus II course. My issue is with graphing. Below is the problem, and below that my tutors explanation via e-mail to try and help me. I understand everything thus far, except for how I go about graphing. I am not strong in...
I barely passed Calculus II, I got 50 in the whole course after finals and I had 56 before finals. The class avg was around a high D but I can't remember exactly now but that's besides the point. Calculus II at my college was integral calculus with some application of integrals(volume, surface...
To make a potentially novella-length post very short, I'll skip the useless back story and say only that I had perhaps the most haphazard and unsatisfactory introduction to Calculus I one can imagine. I completed most of the course in about four weeks, and three-quarters of my work remains...
1. Find the area of the surface generated by revolving y = (sqrt)(2x-x^2), x = [0,1]
a. About the x axis
b. About the y-axis
This problem is from the section about surface area revolving around an axis.
I actually did the problem but I am not sure on my answer.
For around the...
I worked out these two problems and got an answer but was not sure about the answer...
If you could just check it and if it is not write tell me what I did wrong or give me a step by step, it would be greatly appreciated.
1. A tank is formed by revolving y = 3x^2, x = [0,2] around the...
when going into university and completing first year calculus, how much harder is the proceeding levels of calculus, as in, are the concepts and methods just as hard to understand, or is it the same as calculus I all over again as if you have never seen calculus before?
Homework Statement
A smooth vector field F has divF(1,2,3) = 5. Estimate the flux of F out of a small sphere of radius 0.01 centered at the point (1,2,3).
Homework Equations
Cartesian Coordinate Definition of Divergence: If F= F1i + F2j +F3k, then divF=dF1/dx + dF2/dy + dF3/dz
The...
This is my situation, last week I took the AP Calculus BC exam, I feel like I did really poorly and even if they are going to curve it, I think I am going to end up with a 3 or 4.
Now in this summer, I going to take a Calculus III at a college who has granted me exemption from prerequisites...
So I just found out my grade for Calc I and I got a B-. I'm pretty disappointed in myself because I feel like I could've received at least a B+/A- if I wasn't so lazy. There were exams/quizzes that I seriously didn't study for at all besides a quick 10 minute glimpse in my notes.
So I think...
Homework Statement
Calculate the flux of the vector field through the surface: F = i + 2j through a square of side length 2, lying in the plane x + y + z = 1, oriented away from the origin.
Homework Equations
The Attempt at a Solution
So, I need to take the integral over the...
Homework Statement
F = 2i + 3j through a disk of radius 5 in the plane y = 2 oriented in the direction of increasing y.
Calculate the flux of the vector field through the surface.
Homework Equations
The Attempt at a Solution
I know that I need to calculate the area vector...
I'm double majoring in Physics and Mathematics at a small liberal arts school, and I am currently a freshman, with a 4.0 GPA so far. Right now I am taking Calculus II, and I am becoming quite frustrated. I definitely understand the material, and have little difficulty with homework or...
Okay so our calculus professor has given us an assignment to do a project involving calculus. He found out that I was a gymnast and liked physics so he said that my project should relate calculus, physics, and gymnastics.
I'm having problems coming up with an idea in which I can actually...
Homework Statement
\int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx
Homework Equations
The Attempt at a Solution
\int \frac{ \tan x \sec^2 x }{ \tan^2 x + 6 \tan x + 8 } dx
Okay I let...
u=tanx
du=sec^2x
Then I got
\int \frac{ u }{ (u+4)(u+2) } du
Then...
Hi, I'm taking Calculus II in the Spring and I'm wondering what you guys did to succeed in that class. I know it's different for each school, but in general, I've heard calc 2 was integration hell. Anyways, here are some topics that was listed in the Fall Calc 2 final review, what should I look...
I hear that calc II is the hardest course in the calculus sequence. I am not sure why this is, but it is something that I hear constantly. Why is it considered to be so difficult? Are there a lot of formulas or are there many different applications of formulas, ideas, and logic? Obviously people...
Thank you in advance for the help!
Homework Statement
Find the center of mass of a thin plate of constant density (delta) covering the given region.
The region bounded by the parabola y= x - x2 and the line y= -x
Homework Equations
See attachment question 15 p1
The Attempt...
Homework Statement
I have two relatively similar problems:
1.) Sigma n=1 to infinity ((ln n)^3 / n^2)
2.) Sigma n=1 to infinity (1 / sqrt(n) * (ln n)^4)
I'm to prove their convergence or divergence using either the direct comparison test or the limit comparison test.
I understand...
I was wondering what you should be familiar with before taking Calculus II. I just finished the Calc I summer class and was wondering if I should go on to Calc II this summer or just go over Calc I some more... I mean I can solve all of the problems that were presented but we hardly had any...
Hey... This really sucks. I am in Calculus 2 and I have had 3 in-class exams, all 3 were A's. This last exam is take-home and it is entirely Maclaurin and Taylor series.. The only thing in the class to go over my head.
Please help me out with these problems!
Homework Statement
Use...
I'm very unsure on how to do this equation...
Someone PLEASE HELP!
\intx^{3}\sqrt{}x^{2}+9
i do know that
if you...
let x = 3sec \vartheta
dx = 3 sec \vartheta tan \vartheta du
Do you think that taking Calculus II and Linear Algebra in the same semester is too much? Or should I be fine? How hard is Linear Algebra?
EDIT: I can't take them the same semester I just found out, do you think it is wise to take LA in a short semester over the summer, like 4 weeks?
Homework Statement
Evaluate the following definite integrals using the Fundamental Theorem of Calculus:
1. 1 \int 4 ( (5x2+7x+5)dx )
2. -5pi/6 \int 4pi/6 (−6sinx+7cosx) dx )
3. 2 \int 4 ( (e^-4x)/((e^(-4x))+7) dx )
If it's unclear, the number on the...
I'm taking calc 2 right now and I want to start planning what classes in going to take next semester.
I'm basically retaking the class for college credit, so I'm fairly certain I am going to get a B+/A-.
Some people are telling me to take vector calculus while the other half are saying...
Yes, that's right, I said it.
Most of the grad students, professionals, and upperclassmen are probably laughing at my title, either because they breezed through the course, or because they too, remember being a first semester freshman in engineering, and hearing such horrible things about...