What is Calculus 1: Definition and 96 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. Rhdjfgjgj

    Proof that velocity of image by a plane mirror is negative of object

    We were studying reflection due to plane mirrors and our sir derived the relation between velocity of image and velocity of object in case of a plane mirror. He took the following case as bare for deriving the formula. Following is the detailed working Now I was wondering if I could derive it...
  2. Another

    Confused between the units of a constant and the units of the integral boundary conditions

    I want to integrate this function ## \int_{0.8um}^{1.8um} A e^{B/E(x)} \, dx ## But A has a unit as ## 1/cm ##. Should I change ##1/cm## to ##1/um## by multiplying ##1/10^{4}## For this function, I decided to integrate using the online numerical integral, This side . I am just curious that...
  3. jambaugh

    Teaching Differential Calculus as the Limit of Discrete Calculus

    I'm teaching Calc I. this semester and we're now covering the derivatives of power function and exponential functions as well as the basic rules, e.g. linearity and product rule. Some years back I ran across an exposition of umbral calculus in the appendix of a reference. I cannot help but...
  4. KungPeng Zhou

    A question about the derivation of upper bound integrals

    From the question,we know that the variable is x First,we can get f(x)=x^{2}\int_{0}^{x}\sin(t^{2})dt,then\frac{df(x)}{dx}=2x\int_{0}^{x}\sin(t^{2})dt+x^{2}sint^{2},but I can't deal with \int_{0}^{x}\sin(t^{2})dt,If I do the second differentiation, I can indeed deal with integrals, but there is...
  5. murshid_islam

    I Question about Limit: \lim_{x\rightarrow1} (x+1)

    \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...
  6. T

    What are the two different senses of since and how are they used?

    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...
  7. stivodivo

    First time taking Calculus 1 and I only got a B

    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...
  8. Mr.Husky

    Calculus Just got Courant's calculus text -- Need some tips to work through it

    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...
  9. aspiringastronomer

    Struggling in my freshman year of Physics at university

    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...
  10. U

    Finding the horizontal tan() lines of this equation

    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.
  11. B

    Calculus 1 problems: functions, integrals, series

    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?
  12. 0

    An identity to prove using calculus 1

    I have a feeling that I forgot to copy something from the black board, maybe some f' because as it is I'm not seeing a solution.
  13. K

    Simpson's method, error estimate

    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...
  14. K

    Limit of Taylor Polynomial for Tn(x) as n Approaches Infinity

    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.
  15. N

    An alternative proof (Hopefully not an alternative fact)

    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...
  16. rishi kesh

    Please explain this limit problem

    Please explain the above limit problem.i am able to understand last steps but can't get initial 4 steps.
  17. starstruck_

    Related rates (prism with a trapezoid base)

    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...
  18. O

    Find maxima/minima of polynomials

    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.
  19. S

    Finding the Derivative of y=sqrt(x+sqrt(x+sqrt(x)))

    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...
  20. A

    Schools Taking Calculus 1 over the summer -- question?

    I was talking to my friend and she told me how she was taking Calculus 1 over the summer in 5 weeks. Everyone else averaging a 30 percent as their overall grade. The teacher was planning to raise the grade. So if you get a 30%, you'll get a C. However, my question is that is it okay to pass...
  21. P

    Expressing series in terms of a Power Series

    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...
  22. harpazo

    MHB How can self-study lead to long-term learning compared to classroom instruction?

    As I study calculus 3, I often revisit calculus 1 and 2. The following application is from single variable calculus, partucularly calculus 1, called RELATED RATES. I have not seen a related rates problem since the 2015. I am a bit rusty with the set up. Can someone help me set it up? I can take...
  23. M

    Ε-δ proof: lim x->a f(x) = lim h->0 f(a + h)

    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 EquationsThe Attempt at a Solution By the...
  24. waffletree

    Finding the derivative of a function with a radical

    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...
  25. I

    Prove 2 sqrt(x) > 3 - (1/x) for x > 1

    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##...
  26. K

    Should I retake pre-calc before calculus 1

    I have taken precalculus class a few years ago. I finally took calculus 1 and the same school just last quarter and got a 3.2. I had a pretty cool teacher. I'm transferring to a four year school and have to retake calculus 1, but I'm contemplating on retaking a pre calculus at the four year...
  27. A

    Calculus Calculus 1 text book - Need review of precalculus

    Hi, so I'm taking calculus 1 this year however I haven't taken precalculus in several years. I don't remember any of it, and the textbook of the course doesn't review it at all(they just sample you questions) and I'm having issues solving the precalculus review questions(how necessary is it that...
  28. K

    Does a limit exist on a graph at (-1,0) if the point

    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...
  29. bakerino

    Schools Scared about college -- First time I've been scared of academics

    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...
  30. N

    High School Calculus Velocity question

    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...
  31. B

    Pre-Cal to Calc 1: Which Option is Best?

    Hi everyone, I am new here. I know this has been asked before so forgive me, but ima ask again. :) So I'm n pre cal right now and it's going fairly well. I think I'll pass with a B. Next I need to take calculus 1..then later calc 2. So I have 2 options. 1. I can take calculus 1 this summer. I...
  32. nomadreid

    F(x) = sin(x)+2x+1, show (f^(-1))'(1)=1/3

    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...
  33. E

    Calculus 1 by Coursera: Is it a Good Course?

    Hello everyone. I was wondering if the Calculus one course by Coursera (OSU) is a good course to self teach Calculus. Thank you for all the responses in advancee
  34. 1

    Differentiability of piece-wise functions

    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...
  35. T

    What type of calculus is in physics 1?

    I was just wondering what type of calculus is in cal based physics 1? I'm a little rusty at calculus, so I was just curious...
  36. R

    Calc II Tips: Revise Calc 1 & Trig Before Semester Starts

    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...
  37. N

    What are the definitions and properties of Riemann sums?

    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...
  38. G

    How is calculus 1 over the summer?

    How is calc 1 over the summer? How should I prepare myself? I don't care what grade I get, as long I get a C or B. Any suggestions? Are there helpful websites that cane help me?
  39. Bactoschool

    Which pages are Cal 1 only in the Humongous book of Calculus

    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...
  40. Ghostcrown

    Trouble with basic algebra in Calculus 1.

    It's rather strange, isn't it? I did just fine in Algebra I, and even better in Algebra II, but now that I'm in Calculus I I'm finding that my algebra skills aren't as sharp as I thought they were. Often times i'll get the calculus portions of problems correct, but when it comes down to...
  41. Y

    What is the limit of the given fraction in Calculus 1 homework?

    Homework Statement The limit ##\lim_{x\to\pi}\frac{xcos\frac{x}{2}}{\pi^{2}-x^{2}}## Can be expressed as a fraction. Solve 2. Relevant equation 3. The Attempt at a Solution EDIT See new post for solution
  42. Y

    {Calculus 1} Optimization Problem

    Homework Statement A wire is divided into two parts. One part is shaped into a square, and the other part is shaped into a circle. Let r be the ratio of the circumference of the circle to the perimeter of the square when the sum of the areas of the square and circle is minimized. Find r...
  43. N

    Order of concepts taught in calculus 1 & 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
  44. A

    Multivariable Calculus 1 Problem

    Homework Statement Let P be the tangent to the graph of g(x,y) = 8-2x^2-3y^2 at the point (1, 2, -6). Let f(x,y) = 4-x^2-y^2. Find the point on the graph of f which has tangent plane parallel to P.Homework Equations g(x,y) = 8-2x^2-3y^2 at (1, 2, -6) f(x,y) = 4-x^2-y^2 The Attempt at a...
  45. T

    Calc based physics with no calc experience/how to learn?

    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...
  46. JesseJC

    Taking linear algebra after only Calculus 1

    hey all, I've begun my engineering degree and have been enjoying it thoroughly, my first semester naturally wasn't too hot, I've got around a 2.7 so far, but my school states calculus 1 as a prerequisite for linear algebra, I hardly went to class and flipped classes but consistently scored above...
  47. K

    Calculus Re-Learning Calculus 1 - Tips for Starting University

    Hello! I will be starting university in August and I plan on starting with Calc 2 and skipping Calc 1 to save myself some time. I received a 5 on my Calc AB exam but it has been awhile and I have forgotten a lot. The courses I will be taking my first semester require Calculus 1 as prerequisite...
  48. R

    Maximize Success in Independent Study Calculus: Tips from Experienced Students

    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...
  49. Drakkith

    Maximizing Success in Calculus 1: Tips and Tricks for Math 220

    I'm starting calculus 1 (math 220) next week and I was wondering if there were any specific things to keep in mind or watch out for. Things that I should pay a little more attention to than normal or things that are confusing unless you look at them a certain way. That kind of stuff. Thanks.