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.

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  1. T

    How to be prepared for calculus 2?

    Just finished taking my calculus 1 final now I'm ready to take calculus 2 in the fall. The question is how should I prepare for calculus 2? What are the important trig and calculus 1 sections I should study beforehand? Thanks.
  2. M

    Calculus 2 and Real Analysis in one semester?

    I am in my first semester of university and currently taking Linear Algebra. I was planning on majoring in EECS but I lost interest in EE and engineering in general (except software) and gained a lot of interest in maths (especially statistics and financial mathematics) so I will double major in...
  3. C

    Calculus 2 second derivative of integral

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  4. I

    What's next in Calculus 2 after a successful semester of Calc 1?

    So...What's Calculus 2 like? So I just completed Calculus 1 with an A+ for the semester. What should I be expecting in Calculus 2 if the topics in Calc 1 were limits, differentiation, applications of differentiation, some integration, natural logs, logs, e, and hyperbolic functions?
  5. evinda

    MHB Exercises with Solutions of Calculus 2

    Hello everybody!Could you tell me where I can find exercises with solutions of Calculus 2 online?Thanks! :rolleyes:
  6. 1

    Calculus 2- Differential Equation Mixing Problem

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  7. D

    Calculus 2: Advice on Summer Learning/Teaching

    Hello users!,, What do you think about taking calculus 2 on summer? Im studying/teaching myself , and I want to know which topics should i focus to do very well in class.. Integration by parts?? Trig Sub? Thank you very much,, hope you ca help me.
  8. D

    Is Calculus 2 a Good Option for a Summer Course?

    Calculus 2 for summer?? Hello users!,, What do you think about taking calculus 2 on summer? Any of you have taken that course in summer? What can you say about it? Is it hard?? Also veryy important,, i got an A- in calc 1 I like math Im taking the course on july 11, it ends at...
  9. 1

    Calculus 2- Hooke's Law Spring problem

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  10. B

    MHB Convergence and Divergence - Calculus 2

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  11. R

    Calculus 2: Sequence Limits Question to the power n?

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  12. T

    Exploring the Challenges and Rewards of Calculus II: A Student's Perspective

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  13. A

    Need help with deciphering calculus 2 problem

    Homework Statement Use substitution to show that for any continuous function f, $$\int_0^{\pi/2} f(\sin x)\,dx = \int_0^{\pi/2} f(\cos x)\,dx.$$ Homework Equations $$\cos(\pi/2-x)=\sin x$$ The Attempt at a Solution My confusion is that f is inside the integral, and I have no idea...
  14. D

    MHB Center of Mass Calculus 2 Integration

    Uniform semi circle of radius R whose diameter is on the x axis. Since it is uniform and symmetric on the x axis, x = 0. For y, we have $$ y_{cm} = \frac{\int y\sigma dA}{\frac{\pi R^2}{2}} $$ In polar, $dA = rdrd\phi$. So the integral becomes $$ \int_0^{\pi}\int_0^R y^2\sin\phi drd\phi $$ How...
  15. I

    Calculus 2: Finding Work with Integrals

    Homework Statement Not sure if this goes here or physics, but this is for my calculus 2 class so I decided here would be best. #22 Homework Equations W=∫Fdx The Attempt at a Solution I think the limits of integration are from 0 to 1 since the water is right under the spout, but...
  16. A

    Which precal topics should I master before taking Calculus 2?

    I just finished taking Calculus 1 at a CC last semester. This coming semester, I will be taking Calculus 2, which will be my last semester of Calculus (woohoo!). I didn't take Precal seriously during junior year of high school (had an eating disorder that made me unable to concentrate and...
  17. S

    MHB Prove the Validity of Euclid's Elements

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  18. L

    Studying Gearing Up For Calculus 2: Need Advice!

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  19. L

    Calculus 2 book recommendation (exactly what I need inside)

    Next semester I will be taking Calculus 2. I have used Courant and a little of Spivak for Calculus 1, and loved it! Extremely complete and challenging. Great books. So, this is exactly what I will need to know: Ordinary differential equations Families of equations, initial value...
  20. I

    Should I Wait Until Completing Calculus 2 to Take Physics 1?

    My university's prerequisite is calculus 1 (which I've taken AP Calculus AB) to take Physics 1 (calculus based). My question is should I wait until I've completed calculus 2 to take physics 1? I've taken AP Physics B and did relatively well in it (although I don't remember everything perfectly..)
  21. E

    How Can I Solve These Calculus 2 Homework Problems?

    1) 1. y=x, y=sinx, x=-∏/4, x=∏/2 find the area between these limited curves. 2. ∫(upper function-(lower function) 3. ∫x-sinxdx(between x=-∏/4, x=0) + ∫sinx-xdx(between x=0,x=∏/2). Am I right? If it's okay,I don't have a problem in this integral. 2) 1. ∫sin(2x)/sin^2x(^=power)dx 2. I'm not...
  22. M

    Calculus 2, Series Convergence Questions?

    I have some problems here with Series and Convergence... Here are the problems and my guesses at it. http://img822.imageshack.us/img822/9523/23341530.png It won't tell me which one is wrong, but it just says one/all is wrong. Any help is appreciated. Attempts at solving, I tried...
  23. H

    Calculus 2 Find the volume problem

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  24. G

    United States Calculus 2 - Calculus in Polar Coordinates

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  25. G

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  26. G

    How to Calculate Remainder for Convergent Series in Calculus 2?

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  27. G

    Convergence of a Series with Calculus 2: (k^2-1)/(k^3+4)

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  28. G

    Does the series Ʃ[k=1,inf] tan(k)/(k^2+1) converge or diverge?

    Homework Statement Determine if the following series converges or diverges. Ʃ[k=1,inf] tan(k)/(k^2+1) Homework Equations The Attempt at a Solution I have no idea how to solve this problem. Now that I think of it, I have never solved a single question about series were I'm...
  29. G

    Does the series sum[k=1,inf] 3/(k(k+3)) converge or diverge?

    This is a similar question that I have to the other one that I posted. Determine if the following series converges or diverges. sum[k=1,inf] 3/(k(k+3)) I applied the limit comparison test with 1/k^2 also k(k+3))=k^2+3k lim k->inf [ 3/(k^2+3k) ]/[ 1/k^2 ] = lim k->inf (3k^2/(k^2+3k)...
  30. G

    Does the series sum[k=1,inf] 2/(k^2-1) converge or diverge?

    Determine weather or not the following series converges or diverges. sum[k=1,inf] 2/(k^2-1) I applied the limit comparison test with 1/k^2 2* lim k->inf [ 1/(k^2-1) ] /(1/k^2) = 2*lim k->inf k^2/(k^2-1) =H 2 then because sum[k=1,inf] 1/k^2 is a convergent p-series then...
  31. F

    Calculus 2 volume integration, y=axis

    Hello, so I've been having lots of trouble with a math problem. I can differentiate functions but I'm kinda confused on what to do when it is like y=4 So say you have the given: y= x^2 and then y=4, x=0, x=2. Rotate about the y-axis. I'm know you have to get it into respect to y so...
  32. G

    Does the series sum[k=1,inf] cos(k)/(k^2+1) converge or diverge?

    State if the following series converges or diverges sum[k=1,inf] cos(k)/(k^2+1) I applied the convergence test lim k->inf cos(k)/(k^2+1) =H lim k->inf -sin(k)/(2k) =H lim k->inf -cos(k)/2 = some undefined value bounded by -1 and 1 =/= 0 so the series diverges by the divergence test. I guess my...
  33. S

    Formal Proof for Predicate Calculus 2: Solving Complex Operations and Predicates

    Let: 1) P be one place operation 2) H be two place operation 3) G be two place predicate 4) k, m be two constants Let : The following assumptions : 1) \forall x [\neg G(x,k)\Longrightarrow G[H(P(x),x),m]] 2)\forall x\forall y\forall z[G(x,y)\Longrightarrow...
  34. G

    United States Calculus 2 - Infinite Series

    Homework Statement Determine how many terms of the following convergent series must be summed to be sure that the remainder is less than 10^-5. Although you do not need it, the exact value of the series is given. ln(128) = 7*sum[k=1,inf] of (-1)^(k+1)/k Homework Equations [b]3...
  35. G

    Evaluating the Limit of a SeriesWhat is the limit of the series .9+.09+.009+...?

    Homework Statement Find the sequence of partial sums {S_n} and evaluate the limit of {S_n} for the following series .9+.09+.009+... What is .9+.09+.009+... equal to? Homework Equations The Attempt at a Solution For the first part of the question (find the sequence of...
  36. G

    Calculus 2 - Infinite Series Question - Estimating Series with Positive Terms

    Homework Statement Consider the following convergent series. Then complete parts a throw d below. sum[k=1,inf] 5/k^7 a. Find an upper bound for the remainder in terms of n Homework Equations Estimating Series with Positive Terms Let f be a continuous, positive, decreasing...
  37. G

    Calculus 2 - Introduction to Differential Equations

    Homework Statement One model for the spread of rumor is that the rate of spread is proportional to the product of the fraction y of the population who have heard the rumor and the fraction who have not heard the rumor. i) Write a differential equation that is satisfied by y ii) Solve the...
  38. A

    Calculus 2 question (∫cos^3(x)dx)

    Homework Statement ∫cos^3(x)dx Homework Equations http://college.cengage.com/mathematics/larson/calculus_analytic/8e/students/derivatives/derivative_integrals.pdf http://college.cengage.com/mathematics/larson/calculus_analytic/8e/students/trig_review/trigonometry.pdf This is the...
  39. R

    Calculus 2 Book: Advice for Portuguese Students

    Hello I am having calculus 2 at university in my country ( portugal ). The topics are basically double and triple integrals all the way through stokes. Do you guys have any advice on a book that could help me study this course? Thank you very much and I am sorry about my english
  40. K

    Calculating Derivatives of Composite Functions

    Homework Statement Given F(2)=5, F'(2)=6, F(4)=3, F'(4)=6 and G(3)=6, G'(3)=3, G(4)=2, G'(4)=1 A. H(4) if H(x)=F(G(x)) = ? B. H'(4) if H(x)=F(G(x)) = ? C. H(4) if H(x)=G(F(x)) = ? D. H'(4) if H(x)=G(F(x)) =? E. H'(4) if H(x)=F(x)/G(x) =? Homework Equations My teacher did not teach...
  41. K

    Derivative of Composite Trigonometric Functions

    Homework Statement What is the derivative of f(x) = cos^5(sin(8x)) Homework Equations trig, product rule, chain rule The Attempt at a Solution f(x) = cos^5(sin(8x)) Answer: 5-sin^4(cos(8x))(8)
  42. K

    Derivative of f(t) = te^(2-7t)

    Homework Statement What is the derivative of f(t) = te^(2-7t) Homework Equations We would use product rule and chain rule, I believe The Attempt at a Solution (t)(e^(2-7t) (1)(e^(2-7t)(-7) answer: -7(e^(2-7t))
  43. J

    Is Self-Studying Calculus 2 Over Summer a Good Idea?

    Hello everyone! Alright, my current high school math class is IB Higher Level Mathematics. It's a two year long course. During the first year we have focused primarily on Advanced Algebra, Trigonometry, and we are about to start Vectors. Year 2 consists of Calculus I, Statistics, and then...
  44. J

    Starting Calculus 2 this Spring

    Hi all! I'm starting Calculus 2 this Spring at my college and I was wondering what to look out for. Any specific series or problems that might be a headache? What do you suggest I prepare "better" for? Thanks for your time!
  45. J

    Calculus 2 Series Question: Prove the inequality

    This was already posted by someone else but an answer wasn't received so I thought I'd repost. Any help is appreciated. Homework Statement Use the Maclaurin series for cosx and the Alternating Series Estimation Theorem to show that \frac{1}{2} - \frac{x^2}{24} < \frac{1-cosx}{x^2} <...
  46. S

    What are the best resources for practicing integrals in calculus 2?

    I bombed my first test this semester and had to withdraw from my calc 2 class because I did not want to get below a B for the class. Before next semester starts I would like to get a lot of practice in and I was curious if there were any websites or books that people recommend that just have...
  47. K

    Online Calculus 2 / Differential Equations for Credit?

    Hello all! As it turns out, I have a condition that requires me to not be at a physical university this semester, but I would like to try and catch up a bit and also need to make my time off more productive. I would like to take a couple of math courses, particularly Calculus 2 and...
  48. N

    I just wanted to know what is in Calculus 2?

    Hey I just finish Essentials Of Calculus. I'm going to take Honors Calc 1 in the fall. I just wanted to know what is in Calculus 2? I heard there are a ton of formulas to remember. lol
  49. N

    Courses Calculus 2 summer online courses?

    As a sophomore in college, I looking to take a calculus 2 course over the summer to fulfill requirements for course credit. I'm looking for online courses. I just haven't been able to successfully find a place that offers Calculus 2 online. Searching online has just been so confusing. Also...
  50. Totalderiv

    Self-Study Books for Calculus 2 & 3 | Examples & Questions

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