What is Calculus 2: Definition and 117 Discussions
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.
Hi! I understand that this is an expanded Riemann sum but I'm having trouble determining its original form. I don't actually have any ideas as to how to find it, but I know that once I determine the original form of the Riemann sum, I will be able to figure out the values for a, b, and f.
If...
I self-studied Calculus 1 (AP Calc AB) in my sophomore year of high school and I got a 4 on the AP exam.
I took the class Calculus 2 (AP Calc BC) in my junior year of high school. I did not take the exam at the end of the year.
I didn't take any math in my senior year of high school.
Because...
I had already calculated the first partial derivative to equal the following:
$$\frac{\partial y}{\partial t} = \frac{\partial v}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial v}{\partial t}$$
Now the second partial derivative I can use the chain rule to do and get to...
I got the first part of it down, $$L=\int_1^5 \sqrt{1+(\frac{1}{x^2})}dx$$
I just want to know if it's right to make your ##f(x)=\sqrt{1+\frac{1}{x^2}}## then compute it's second derivative and find it's max value, for the trapezoidal error formula.
I have a calculus 2 midterm coming up and given the exam review questions, this seems like this question can potentially be on it.
I've tried to look it up, but I always find the famous painters example, which I don't find satisfying.
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...
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...
Homework Statement
##\sum_{k=0}^\infty \frac 4 k(\ln k)^2 ##
Homework Equations
The Attempt at a Solution
I tried to solve it using the integral test but since it's not continuous it doesn't work.
Hi guys!
Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay?
In general, when we are solving power...
Homework Statement
a. Write down a Riemann sum for the integral ∫x3dx from 0 to 1.
b. Given the following identity 13+23+33...+N3=(N(N+1)/2)2, show that the Riemann sums for ∫x3dx from 0 to 1 converge to 1/4.
The Attempt at a Solution
I believe I have gotten part a. I got ∑i^3/N^4 from i=0 to...
Homework Statement
Question attached in attachments
Homework Equations
Area enclosed by polar graph is ∫0.5r^2
where r is the radius as a function of angle theta
The Attempt at a Solution
I attempted to use the formula above and I subtracted the area of the inside from the outside but it...
Homework Statement
The problem is attached as pic
Homework Equations
∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)
The Attempt at a Solution
So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
Homework Statement
Please help me solve the calc problem pictured!
Homework Equations
y=3-x^2 and y=x+1
The Attempt at a Solution
My attempt is in one of the photos!
Homework Statement
determine the center of mass of a thin plate of density 12 and whose shape is the triangle of vertices (1,0), (0,0), (1,1). Then, using the appropriate pappus theorem, calculate the volume of the solid obtained by rotating this region around the line x = -2.
Homework...
1. Here's the problem on trig integrating that I'm struggling with (Calculus 2 btw)
2. Wanted to see if I did everything right so far and what to do after all this. The part where I'm stuck is how to integrate (integral)cos^(2)udu and (integral)cos^(2)usin^(2)udu. I'm sure these are easy...
Homework Statement
The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems.
For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
So, i am currently studying physics in a brazilian university. I am going to have a Calculus 2 course which, in Brazil, covers Ordinary Differential Equations and multi-variable differential calculus. So which challenging and rigourous books would you guys recommend for that? Thanks for the...
I am a high school student entering 11th grade soon. I will be taking calculus ab (calc 1) in school, but I am wondering if it is possible to self study calculus 2 and take it over the summer at a cc?(and do well!) If everything goes right I want to do calculus 2 the summer after junior year and...
I am currently studying calculus 3. However, once in a while, I like to review single variable calculus. I decided to tackle the following calculus 2 integration problem:
∫ tan^3 x dx
Solution:
Separate into two tangent functions.
(tan^2 x)(tan x)
Rewrite tan^2 x as sec^2 x - 1 using its...
Homework Statement
Calculate limit as x approaches infinity of (e^x - x^3)
Homework Equations
ln e^x = x
e^(ln x) = x
The Attempt at a Solution
I tried substituting x = ln e^x and got (e^x - (ln e^x)^3). I'm pretty much lost and this is my only attempt so far.
I'm thinking that this is an...
Homework Statement
Evaluate the indefinite integral as a power series. What is the radius of convergence (R)?
##\int x^2ln(1+x) \, dx##
Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1##
Homework Equations
Geometric series
##\frac {1} {1-x} =...
Homework Statement
Integrate x2(2+x3)4dx.
Show that Wolfram Alpha's answer is equivalent to your answer.
Homework Equations
No equations besides knowing that the integral of xpower is 1/power+1 * xpower + 1
The Attempt at a Solution
So I have the answer to the integral by hand as (2+x3)5)/15...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
I've encountered 2 problems in a row that involve lifting water out of tanks and finding the work needed. I am getting the incorrect answer.
w = ⌠ab pgA(y)D(y)dy
here is one of the problems:
A...
"It takes 100J of work to stretch a spring 0.5m from its equilibrium position. How much work is needed to stretch it an additional 0.75m."
Attempt: w = ⌠abF(x)dx
work = F x D
100J = F x 0.5m
F = 200J
0.75 + 0.5 = 1.25
w = ⌠0.51.25 200dx
w = 150 J
The correct answer: w = 525 J
what did I do...
Homework Statement
A marine biology observation pod is being designed. It will be submerged, with vertical windows of various sizes and shapes. Calculate the total force being exerted on each of the windows described below. Density of water is 1000 kg/m^3...
Homework Statement
I have a linear integral (e^xsiny-2)dx + (e^xcosy+x^2)dy
y≥0
2x=x^2+y^2
I used Green's theorem and got:
∬ (e^xcosy+2x) - (e^xcosy) dy dx
x bounds: from 0 to 2
y bounds: from 0 to sqrt(2x-x^2)
After solving all that stuff I get to:
∫ (2x) (sqrt(2x-x^2)) dx
x bounds: 0 to...
Homework Statement
Hi, I have a double integral with square root of 1-x^3 that I can't solve.
Here is the integral and attempt at the solution. I get stuck!
Homework Equations
I am in Calc II.
Mostly we used integration by parts, or substituting t=something for most "difficult" integrals...
The problem is to find the general term ##a_n## (not the partial sum) of the infinite series with a starting point n=1
$$a_n = \frac {8} {1^2 + 1} + \frac {1} {2^2 + 1} + \frac {8} {3^2 + 1} + \frac {1} {4^2 + 1} + \text {...}$$
The denominator is easy, just ##n^2 + 1## but I can't think of...
Hi everyone, in this days i was seeing a little of Fourier series and transform, and i wondered if it was necessary to better understand before the measure and Lebesgue integral before studying it. Or it's not necessary?
Hey everyone!
I started back at my university after a 3 & 1/2 year break and changed my major from Journalism to Physics last May. I am coming up on my finals in a couple of weeks for the Spring Semester and then I will be taking a class over the summer and doing research. My grades are great...
Let f(x) = (1+x)-4
Find the Taylor Series of f centered at x=1 and its interval of convergence.
\sum_{n=0}^\infty f^n(c)\frac{(x-c)^n}{n!} is general Taylor series form
My attempt
I found the first 4 derivatives of f(x) and their values at fn(1). Yet from here I do not know how to find the...
Homework Statement
Determine which of the sequences converge or diverge. Find the limit of the convergent sequences.
1) {asubn}= [((n^2) + (-1)^n)] / [(4n^2)]
Homework Equations
[/B]
a1=first term, a2=second term...an= nth term
The Attempt at a Solution
a) So I found the first couple of...
Homework Statement
I have to determine whether or not the following sequence is convergent, and if it is convergent, I have to find the limit.
an = (-2)n / (n!)
In solving this problem, I am not allowed to use any form or variation of the Ratio Test.
2. The attempt at a solution
I was...
Hey everyone,
First of all, let me start out by saying that I am not enjoying this class, not because I'm not doing well, but frankly because I find it boring. It's not that I'm not challenged, I am. The material can be meticulous. But, it just seems like a lot of algebraic manipulations and...
I'll be using a mass balance calc for a tank design at work. We are designing a tank that has an automatic float switch on it, attached to a valve which will start filling the tank when the water level drops below a certain point. This tank is intended to drain quickly (flow rate not known at...
Hello I am currently a sophomore in high school and am looking to graduate early next year. I need to take 4 additional classes this summer at my local community college in order to satisfy all the courses I need in order to graduate early. To give my self a boost I was looking into taking the...
1. My calculus 2 math teacher has us doing a project, involving calculus, on any topic we choose. I chose the topic of calculating work, as in Force x Distance. Specifically calculating work done by lifting a mass into orbit, or to escape velocity. A method is used in my textbook for calculating...
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...
I am in calc2 now and I just can't get excited about this series stuff. We went over many methods of testing for convergence/divergence and finally moved on to polar coordinates. Is series important in any type of CS field besides, I would imagine, creating software to solve series problems?
I've put the problem statement below and worked it out. I typically don't post questions like this as they're a lot to go through, but I am wondering if I have worked the problem correctly as my book does not have the solution and I feel like I am not understand the material correctly.
1...
Homework Statement
Homework Equations
trigonometric identities
The Attempt at a Solution
I did a trig substitution of u=tan(θ/2) and from that I could substitute cos(θ) = 1-u2/1+u2
dθ = 2/(1+u2)
du = 1/2 sec2(θ/2) dθ
I simplified a bit and changed the bounds to get 2du/(5u2 + 1)(1 + u2)2...
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
Hi,
I'm a college freshman studying computer science. I'm required to take up to Calc 3 and I'm currently in Calc 2. At the mid term I have a C( should be lower). I've always struggled with reading texts that are dense, I can't read for more than 5-6 minutes at a time without my mind running...
Homework Statement
Here are the problems
http://imgur.com/a/kbtPS
The problems I need help with are 1 and
4(a)
Homework Equations
The second fundamental theorem of calculus
The Attempt at a Solution
For problem 1, I calculated the areas under the curve (using remmien summs) and tried to find...
I am kind of bummed today because I got my finals result for calculus 2 and ended up getting an A- .Truth to be told it was the best Math course I have taken so far and the professor was outstanding. Its just that I am always hanging in the 90-92 range not just for math but for physics and...