What is calculus i: Definition and 52 Discussions

Calculus 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 the latter 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. Later work, including codifying the idea of limits, put these developments on a more solid conceptual footing. Today, calculus has widespread uses in science, engineering, and social science.

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  1. S

    What type of mathematics did you find the most difficult?

    Mathematics tends to be more progressive than any other field. But I've heard some people say that some math classes that they took were more difficult than the most advanced math classes that they took. For instance, I've heard people say that Calculus II is more difficult than Calculus III...
  2. J

    Studying Reading "How to Prove It" right before taking Calculus I -- Question

    If I were to read up to and including Chapter 3, would I be prepared enough to read Spivak's Calculus or at least some single variable calculus text based on proofs? I'm asking because I plan on taking Real Analysis later and I'd like to gain a better understanding of Calculus. I have read...
  3. J

    Courses I only got a B in Calculus I :(

    I tried my best but the result doesn't match the effort I've put in. First when I know my result I freaked out and questioned my competent. Am I mathematically-inept? Am I cut out for a Physics Phd? End up with ~70% on this course and I'm taking Calc 2 this December. I need to...
  4. O

    MHB Calculus I - Relative Growth Rate help

    Here is a picture of the problem: I am firmly stuck on part b). I know that relative growth rate is defined at f'(x)/f(x), but I cannot seem to come up with the right answer. I've tried everything I could conceivably think of. I do not understand why my answer is incorrect, unless I'm...
  5. havonasun

    Courses Deciding Between Calculus I & Calculus w/ Analytic GeometryI

    I work full-time, so evening or online classes are all I can do. I need to take calculus from the community college I've been attending, but it is not offered in the evening. In fact, none of the local colleges or universities offer it in the evening, so I must do it online. So I go to register...
  6. S

    Integral Equation (or I think so) Calculus I problem

    Homework Statement Find a continuous funciton ##f## such that $$ f(x) = 1+ \dfrac{1}{x} \int_{1}^{x} f(t)dt $$ I think I solved it but I would like to see if it's right. Well, first of all, by the fundamental theorem of calculus I know that $$ \left( \int_{1}^{x} f(t)dt \right) ' = f(x) $$...
  7. mr_persistance

    Apostol's Calculus I. 3.12 - Verify Solution.

    1. If x and y are arbitrary real numbers with x < y, prove that there is at least one real z satisfying x<z<y.2. I'll be using this theorem: T 1.32 Let h be a given positive number and let S be a set of real numbers. (a) If S has a supremum, then for some x in S we have x > sup S - h.The Attempt...
  8. D

    Should I Retake Calculus I over the Summer to Prepare for Calculus II?

    Hello, I am a Computer Science major still knocking out some of my gen eds. Unfortunately, today I found out that a made a 79 on one of my final Calculus I tests. This has brought my letter grade down to a mid-high B. I was planning on taking Calculus II over the summer, however, now I am...
  9. A

    MHB Calculus I Simplifying and Substitution

    Hey guys, I have a couple more questions about this problem set I've been working on. I'm doubting some of my answers and I'd appreciate some help. Question: For the first one, I evaluated by replacing x for 2+h for f(x). Then I substituted into the givern expression and simplified to get...
  10. QuantumCurt

    Suggestions for Calculus I honors projects?

    Hey everyone, I was hoping I could get some input on this. I'm taking an honors section of Calculus I this semester, and part of the course involves a cumulative honors project. It has to be a minimum 8 page paper, at least a third of which has to be actual calculations, about a topic that...
  11. A

    Solving Indeterminate Forms: Evaluating Limits with Exponential Functions

    lim of (3^y-5^y)/(2^y-7^y) as y->0 Evaluate the above limit. Okay, so, I was flabbergasted at how challenging this problem is. I realized it was an indeterminate form [0/0]. I was trying to find a way to cancel y out of the numerator and denominator, in vain. I tried finding a pattern to...
  12. F

    Exploring the Honeycomb: Calculating Routes from Cell A to Cell B

    Hi there, I've got a bonus question from my calc class that I've been working on. First off I AM NOT LOOKING FOR AN ANSWER. What I really need is for someone to break my algorithm and point me in the right direction so I can keep working on it and hopefully--if it's not too far over my head--I...
  13. B

    Physics I and Calculus I Together

    I'll be taking these two classes together this fall (along with an intermediate college writing class and a one-hour-per-week mathematics computing lab). Am I biting off more than I can chew? I am getting an A- in precalculus, which I admit I only study the bare minimum and am about one or...
  14. K

    What should I study to prepare for Calculus I and II

    I recently took a calculus readiness given by a special program for high school students in my area called the math and physics institute (MPI) to see which math and physics classes I would be taking my senior year. I scored well enough to be placed in the most difficult section of the program...
  15. K

    Calculus I review for Calculus II

    I'm scheduled to take Calculus II this Fall semester, but it will have been over a year since I'd taken Calculus I. I am aware that I need to review, but what are some are the major concepts I should review to adequately prepare for the infamous Calc II?
  16. K

    Should I take Calculus I, or Calculus I and II?

    Hi, my name is Matthew, and I am a junior in high school. I will soon be faced with the decision of which classes to take as a senior. At my school, once you have taken Trigonometry, you can choose to go to this other school called MPI (math and physics institute) for the first two hours of...
  17. B

    Determine if EVT applies (Calculus I)

    Homework Statement Determine if the EVT (Extreme Value Theorem) applies to the following. on the interval [2, 8] The Attempt at a Solution For the Extreme Value Theorem to apply, the function must be on an interval, and the function must be continious. I believe the domain for this...
  18. P

    Derivatives Problem (Calculus I)

    Homework Statement find d/dt for a rectangle.Homework Equations A=bh product rule for derivatives (the first times the derivative of the second plus the second times the derivative of the first) Chain rule for derivatives The Attempt at a Solution If b is a constant, then I know that dA/dh = b...
  19. D

    Schools How to study for Calculus I (College)

    I recently got back my first Calculus I exam in college, and I received a B. To study, I watched Patrick JMT, read over my notes, and did a few practice problems. When I got to the exam, I found it hard to recall the steps that I practiced for working out problems, but I remembered most later...
  20. T

    Find the General Anti-derivative (Calculus I)

    Homework Statement given f(x) = [x^3+5sqrt(x)]/x^2, find the anti-derivative Homework Equations The Attempt at a Solution Hi I have attempted to solve this by re-writing the equation as a sum of two fractions: x^3/x^2 + 5sqrt(x)/x^2, simplying gives = x + 5/x^3/2 I then...
  21. K

    Calculus I - Line and extrema problem

    Hi again, my calculus exam is on monday so trying hard to understand the things I'm struggling with. I think I'm almost there with part a - well I can solve it through drawing a graph or using a graphing program - but I'd rather solve it without graphs if possible. Part b however is causing me a...
  22. T

    Studying Self-Studying Calculus: Tips for Success from a High School Book

    Could anyone give me some tips on self studying? I have a Calculus book from my high school ( I am alloud to use it for the remainder of the year...but I am NOT in calculus) So could anyone give me some tips on selfstudying skills, it would be extreemly helpfull, I just started on chapter 1 and...
  23. N

    Is physics I harder than calculus I?

    Is it me or is physics I much harder than calculus I? The world of mathematics makes me feel much more at comfort than with physics. In calculus I everything is set down in hard stone and you can, in theory at least, perform the same scheme to many problems. In contrast, physics you need much...
  24. A

    What do students usually struggle with in Calculus I and II

    I am applying for a tutoring position at my schools math learning center, and part of the application process is a little examination based on questions which tend to be asked by students visiting the center. I am just curious what the biggest issues for students tend to be so that I can do a...
  25. G

    Calculus I - Volume by Slicing

    Hi, I'm trying to solve this problem. "Use the general slicing method to find the volume of the following solids. #13. The tetrahedron (pyramid with four triangular faces), all of whose edges have length 4" Ok So I don't see how I am suppose to use "the general slicing method" to find the...
  26. G

    Calculus I - Area Between Curves - Mistake on Answer Key

    This is the answer key to one of my quizzes If you notice in the question, see attachment, were it says to integrate with respect to y the integral is integrating from 0 to 25 but this produces a negative area so this is technically wrong, yes? You don't just simply integrate from the lower...
  27. Dembadon

    Calculating Limits for a Quadratic Function

    Homework Statement Consider the following function: f(x) = 4x^2 - 8x. Find the limit. \lim_{{\Delta}x\rightarrow 0} \frac{f(x+{\Delta}x)-f(x)}{{\Delta}x} Given: The limit exists. The Attempt at a Solution Since the limit exists, I know that I need to do some algebraic...
  28. mccoy1

    Calculus I vs Probability & Statistics

    Which one do think is better for physical science?...one which help the most!
  29. C

    Calculus I: Monotinic Functions and First Derivative Test

    Homework Statement Using the derivative f'(x) = (x-1)^{2}(x+2)^{2} answer the following: a) What are the critical points of f? b) On what intervals is f increasing or decreasing? c) At what points, if any, does f assume local maximum and minimum values? Homework Equations n/a...
  30. R

    Preparing for Calculus I: Essential Precalculus Skills for Success

    Hey guys, I was wondering what Precalculus skills I should brush up on before the spring semester when I'll be taking Calculus I?
  31. W

    What is the correct solution to \int_{0}^{-2} (9 f(x)- 2)dx?

    Homework Statement Let \int_{-4}^{2} f(x) dx =9, \int_{-4}^{-2} f(x) dx=2, \int_{0}^{2} f(x)dx =3 Find (I have solved this one) \int_{-2}^{0} f(x)dx= 4 Find \int_{0}^{-2} (9 f(x)- 2)dx= ? Homework Equations \int_{a}^{b} f(x) dx = F(b) - F(a) The Attempt at a Solution I...
  32. E

    Calculus I: Is it possible to teach myself over winter break?

    I have never been particularly fond of math. It's not that I'm bad at it, I just don't enjoy it, and have been procrastinating taking any math classes as a result. Unfortunately, my procrastination has finally caught up to me, and I am faced with a calculus-based microeconomics class that I...
  33. R

    Calculus I - Double Derivitive

    Homework Statement Find dy/dx and d^2y/dx^2 y = x / (x^(2)+1)Homework Equations d/dx (f/g) = (g d/dx f - f d/dx g) / g^2 The Attempt at a Solution Finding d/dx: d/dx y = (x^(2)+1) d/dx (x) - (x) d/dx (x^(2) + 1) / (x^(2)+1)^2 = (x^2 + 1) - (2x^2) / (x^(2)+1)^2 So that's my first...
  34. R

    What is the derivative of sin2x / (sin2x + secx) with respect to x?

    Homework Statement sin2x / (sin2x + secx) d/dx = ? Homework Equations d/dx (f/g) = (g d/dx f - f d/dx g) / g^2 d/dx sec u = sec u tan u d/dx u d/dx cos u = -sin u d/dx u d/dx sin u = cos u d/dx u The Attempt at a Solution d/dx y= [(sin2x + secx) d/dx (sin2x) - sin2x d/dx...
  35. R

    How do I differentiate (x^(3)+2x) / (x^(2)-5)?

    Homework Statement (x^(3)+2x) / (x^(2)-5) d/dx = ? Homework Equations d/dx (f/g) = (g d/dx (f) - f d/dx (g)) / g^2 The Attempt at a Solution d/dx y = { (x^(2)-5) d/dx (x^(3)+2x) - (x^(3)+2x) d/dx (x^(2)-5) } / (x^(2)-5)^2 This is where it gets complicated for me. {...
  36. P

    I learned calculus I and II using stewart's. I also taught myself

    I learned calculus I and II using stewart's. I also taught myself vector calculus using Paul C. Matthew's book. However, I'm not very satisfied and would like to relearn it proper. Is it a good idea to start on Apostol Volume 2? Thanks
  37. A

    Calculus I - Velocity/Distance/Time

    Homework Statement In 1939, Joe Sprinz of the San Francisco Seals Baseball Club attempted to catch a ball dropped from a blimp at a height of 800 ft (for the purpose of setting a record). (a) How long does it take for a ball to drop 800 ft? (b) What is the velocity of a ball in miles per hour...
  38. A

    How do I find tangent lines passing through a given point on a graph?

    Homework Statement A) Find all points on the graph of f(x)=x^2 with tangent lines passing through the point (5,0). B) Show that no line tangent to g(x)=1/x passes through the origin. These are problems from a lab problem solving assignment. Homework Equations y=mx+b f`(x) =...
  39. S

    What's an acceptable grade for Calculus I in university?

    I was just wondering what grade people here passed their first university calculus class with..I guess I posted this in the right section.. I have had some things going on at home and haven't been able to make it to my morning Calc I class for about 2.5 weeks. I've been taking my own notes...
  40. B

    Calculus I problem I can't solve

    Here's the problem: d/dx(f(3x^5)) = 8x^2 Find f'(x) After applying the chain rule: f'(3x^5)(15x^4) = 8x^2 f'(3x^5) = 8/(15x^2) It's not apparent to me how I proceed from here to find f'(x). I tried dividing the expression on the right by three and taking the fifth root but that does not...
  41. D

    Are there any good video lectures for learning Calculus I?

    Don't know if it's already been posted somewhere else, my apologies if it has been, but I found http://press.princeton.edu/video/banner/8351.html" video lectures by Adrian Banner quite educational. It's not very advanced stuff, but everything is explained very clearly. Good stuff.
  42. S

    Calculus I - without a calculator

    I'm taking Calculus I in the spring and the instructor sent out out an email outlining the course. He stated that he did not allow calculators on the final. Is this common? I did well in my precalc maths, but I like to be able to check my answers with a calculator - especially during a test...
  43. S

    Is Michael Spivak's Calculus I book worth the investment?

    well, i would like to know whether it is worth paying $75 for the book of Michael Spivak Calculus I?? what i want to say, is that i have heard that it is a great book, so since i have never seen it personally, then would someone who has seen it or even worked with it explain to me what are some...
  44. D

    Can I Take Linear Algebra Without Calc. II and Multi-Variable?

    I am finishing up a Calc. I course @ columbia university and I'm planning on taking Calc II back @ school (High school - 10th grade). I was wondering if it is possible to take Linear Algebra without having taken courses in Calc. II and Multi-Variable. In Calc I, I've done fairly well...
  45. S

    Calculus I - max/min, concavity

    This problem is really frustrating me. Any help is appreciated! Answer the following questions for the function f(x) = \sin^{2}{\frac{x}{6}} defined on the interval [-6π + 0.8, 6π/4 - 0.3]. A. f(x) is concave down on the region ____ to ____. B. A global minimum for this function occurs...
  46. E

    Calculus I Problem Involving Tangents

    Ok, I cannot figure out how to this problem. It is the last on my practice test and have had no real problems with any of these other ones. I have been using this formula: \lim_{x\rightarrow a} \frac{f(x)-f(a)}{x-a} Here is the problem, find the equation of the line tangent to the...
  47. D

    Calculus I Practice Problems with Limits and Derivatives

    1.) lim x --> 1(left) (x^2 + |x| -2)/(|1-x|) Since the limit is from the left, i made all absolute values pnegative, therefore numerator = x^2 - x - 2, and similarly denom. = (x-1). Then, by inspection, the limit would equal to + infinite. 2.) Find numbers a and c such that lim x --> 0...
  48. tony873004

    Finding Online Calculus I Resources to Refresh Your Memory

    Are there any good online courses for Calculus I. Or maybe some VHS or DVD's I can buy that will basically give me what a first semester of what Calculus I would teach me? I want to take Calculus II. I already took and passed Calc I a few years ago. The problem is that I don't remember...
  49. Orion1

    Is Fractional Calculus Possible According to 'Fractional Calculus I'?

    I have located a paper claiming that it is possible to fractionally differentiate, called 'Fractional Calculus I' Orion1 derivative integer factorial theorem: \frac{d^n}{dx^n} (x^n) = n! Is this paper correct? is 'Fractional Calculus' really possible...
  50. T

    Advanced Calculus I (Analysis)

    Hi. I was wondering... this course seems pretty rigorous and tough. Was this course proven extremely difficult when you guys took it? I'm planning to take it during Summer quarter, but need some feedback on how hard this course will be. Thanks in advance.