What is Calculus: Definition and 1000 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. PeaceMartian

    How to find integrals of parent functions without any horizontal/vertical shift?

    TL;DR Summary: How to find integrals of parent functions without any horizontal/vertical shift? Say you were given the equation : How would you find : with a calculator that can only add, subtract, multiply, divide Is there a general formula?
  2. I

    Calculus Problem: Blowing Up a Spherical Balloon

    I'm struggling with section a. This is my calculation: The expression remains depend on the variable t, while in the answer is a concrete number:
  3. Mohmmad Maaitah

    How to find range inside square root

    Hi, so I know how to find domain but how about range in this problem? I don't understand the way he did it? I always get answers wrong when it comes to range.
  4. mcastillo356

    B Why is this definite integral a single number?

    EXAMPLE 4 Find the area of the region ##R## lying above the line ##y=1## and below the curve ##y=5/(x^2+1)##. Solution The region ##R## is shaded in Figure 5.24. To find the intersections of ##y=1## and ##y=5/(x^2+1)##, we must solve these equations simultaneously: ##1=\frac{5}{x^2+1}## so...
  5. casparov

    Help Solve for the normalization constant of this QM integral

    I'm given the wavefunction and I need to find the normalization constant A. I believe that means to solve the integral The question does give some standard results for the Gaussian function, also multiplied by x to some different powers in the integrand, but I can't seem to get it into...
  6. 1

    Integration Substitution Techniques for quadratic expressions under square roots

    Hi, With respect to the techniques mentioned in point 2 and 3: Can someone explain or even better, post a link for an explanation or a videos showing the use of these two techniques. Below excerpt shows problems 4 and 5 referenced in the above 2 points:
  7. S

    Solving this definite integral using integration by parts

    Using integration by parts: $$I_n=\left. x(1+x^2)^{-n} \right|_0^1+\int_0^{1} 2nx^2(1+x^2)^{-(n+1)}dx$$ $$I_n=2^{-n} + 2n \int_0^{1} x^2(1+x^2)^{-(n+1)}dx$$ Then how to continue? Thanks
  8. S

    Radii of stacked circles inside the graph of y = |x|^1.5

    (a) The hint from question is to used geometrical argument. From the graph, I can see ##r_1+r_2=c_2-c_1## but I doubt it will be usefule since the limit is ##\frac{r_2}{r_1} \rightarrow 1##, not in term of ##c##. I also tried to calculate the limit directly (not using geometrical argument at...
  9. carlsondesign

    I What is the official name for a Field Series in mathematics/physics?

    I've been working on developing infinitesimal recursion (what I call continuous hierarchy), but I ended up arriving at "field series" instead. My searches didn't seem to come up with anything reasonable (battlefield the video game series), so I'm wondering what the official name for a field...
  10. Mohmmad Maaitah

    L'Hopital's Rule case: How does x^(-4/3) equal 0 when x approches infinity?

    I'm talking about the x^(-4/3) how does it equal 0 when x approch infinite?? so I can use L'Hopital's Rule
  11. P

    I Tensor Calculus (Einstein notation)

    Hello, I realize this might sound dumb, but I'm having such a hard time understanding Einstein notation. For something like ∂uFv - ∂vFu, why is this not necessarily 0 for tensor Fu? Since all these indices are running through the same values 0,1,2,3?
  12. C

    Finding where this function is increasing or decreasing

    For this, I first try to work out where function is increasing My working is ##f'(x) = 12x^3 - 12x^2 - 24x## For increasing, ##12x(x^2 - x - 2) > 0## ##12x > 0## and ##(x - 2)(x + 1) > 0## ##x > 0## and ##x > 2## and ##x > -1## However, how do I combine those facts into a single domain...
  13. A

    Calculus Does Apostol Calculus Volume 2 cover sufficient multivariate calculus?

    Hello. I am currently doing a high school univariate calculus book, but I would like to go through Apostol's two volumes to get a strong foundation in calculus. His first volume seems great, and I've heard great things about his series, but I am not sure if his second volume contains sufficient...
  14. C

    Why Can We Take Limits of Both Sides? [Answered]

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  15. C

    Proof of ##M^n## (matrix multiplication problem)

    For, Does anybody please know why they did not change the order in the second line of the proof? For example, why did they not rearrange the order to be ##M^n = (DP^{-1}P)(DP^{-1}P)(DP^{-1}P)(DP^{-1}P)---(DP^{-1}P)## for to get ##M^n = (DI)(DI)(DI)(DI)---(DI) = D^n## Many thanks!
  16. bhobba

    B How can hyperreal numbers make infinitesimals logically sound in calculus?

    When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...
  17. C

    Why is continuity necessary before applying the Extreme Value Theorem?

    For this problem, Why cannot we say that ##f(2.999999999) ≥ f(x)## and therefore absolute max at f(2.99999999999999) (without reasoning from the extreme value theorem)? Many thanks!
  18. C

    H'(x) of h(x) = 3f(x) + 8g(x)

    For part(a), The solution is, However, why do they not take the derivative of the inner function (if it exists) of f(x) or g(x) using the chain rule? For example if ##f(x) = \sin(x^2)## Many thanks!
  19. Sailor Al

    I Can Air Resist a Piston? A Serious Discussion

  20. M

    Intro Math Wanting to study Functions from scratch for my Calculus preparation

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  21. Argonaut

    Maximizing Range/Time in Air of an Airplane: Solving with Calculus

    Is my solution correct? (I only have answers to odd-numbered exercises.) Is it a good solution or have I overcomplicated things? (a) The forward force provided by the engine balances the air resistance force, so ##F_{engine}=F_{air} = \alpha v^2 + \beta /v{^2}##. Let ##W_{engine}## be the...
  22. chiyu

    I Vector calculus: line element dr in cylindrical coordinates

    We were taught that in cylindrical coodrinates, the position vector can be expressed as And then we can write the line element by differentiating to get . We can then use this to do a line integral with a vector field along any path. And this seems to be what is done on all questions I've...
  23. bhobba

    Unlock the Power of Calculus: Algebra 1 to Boaz for Students

    Here is an interesting book a student could do after after Algebra 1, or even integrate into an Algebra 1 course: https://www.amazon.com/dp/B077VV95N3/?tag=pfamazon01-20 And a website: https://www.calculussolution.com/ Several topics become easier, such as logarithms, when you know a...
  24. YAYA12345

    I Integral Bee Preparation -- Trouble with this beautiful integral

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  25. MatinSAR

    Vector Calculus in 1D: ± to Show Magnitude?

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  26. WMDhamnekar

    Stochastic Calculus: Conditional Expectation

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  27. WMDhamnekar

    Stochastic Calculus: Conditional Expectation

  28. Demystifier

    A Found a new formula of Dirac calculus

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  29. A

    Calculus Calculus book between Stewart & Spivak levels

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  30. mcastillo356

    I Express the limit as a definite integral

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  31. T

    Calculus Multivariable calculus PDF books

    Multivariable calculus is a branch of mathematics that extends the concepts of single-variable calculus to functions of multiple variables. In this subject, vectors and partial derivatives are introduced to represent and manipulate multi-dimensional data. The gradient of a function represents...
  32. binbagsss

    I Using the Chain Rule for Vector Calculus: A Tutorial

    This is probably a stupid question, but I have never realised that there's an order things should be done in the chain rule , so for example ## \nabla(\bf{v}.\bf{v})=2\bf{v} (\nabla\cdot \bf{v}) ## and not ## 2 \bf{v} \cdot \nabla \bf{v} ## Is there an obvious way to see / think of this...
  33. mcastillo356

    I The Basic Area Problem (introduction to the topic of integrals)

    Hi PF There goes the quote: The Basic Area Problem In this section we are going to consider how to find the area of the region ##R## lying under the graph ##y=f(x)## of a nonnegative-valued, continous function ##f##, above the ##x##-axis and between the vertical lines ##x=a## and ##x=b##, where...
  34. R

    I Multivariable function optimization inconsistency

    Mentor note: For LaTeX here at this site, don't use single $ characters -- they don't work at all. See our LaTeX tutorial from the link at the lower left corner of the input text pane. I have a function dependent on 4 variables ##f(r_1,r_2,q_1,q)##. I'm looking to minimize this function in the...
  35. V

    Is it possible to find the range of this function?

    I can get the domain, but getting the range seems impossible. Domain $$x-5=0$$ $$x =5$$ $$\therefore x \in (- \infty ,5) \cup (5, + \infty)$$ Range  I can simplify the function to the form below, but I don't know how to go from there. $$ f(x)= x + 5 + \frac {1}{x-5}$$  
  36. V

    Limit question to be done without using derivatives

    I am confused by this question. If I try applying the theorem under Relevant Equations then it seems to me that the theorem cannot be applied since the limit of the denominator is zero. This question needs to be done without using derivatives since it appears in the Limits chapter, which...
  37. 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...
  38. P

    A Vector calculus - Prove a function is not differentiable at (0,0)

    ##f\left(x\right)=\begin{cases}\sqrt{\left|xy\right|}sin\left(\frac{1}{xy}\right)&xy\ne 0\\ 0&xy=0\end{cases}## I showed it partial derivatives exist at ##(0,0)##, also it is continuous as ##(0,0)## but now I have to show if it differentiable or not at ##(0,0)##. According to answers it is not...
  39. N

    I Is It Possible? Solving Calculus Questions with Ease

    Hi, just wondering if that's possible in calculus. (See the attachment)
  40. Saracen Rue

    I How to evaluate the enclosed area of this implicit curve?

    The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below: In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
  41. mcastillo356

    I Understanding Theorem 13 from Calculus 7th ed, R. Adams, C. Essex, 4.10

    The following properties of big-O notation follow from the definition: (i) if ##f(x)=O(u(x))## as ##x\rightarrow{a}##, then ##Cf(x)=O(u(x))## as ##x\rightarrow{a}## for any value of the constant ##C##. (ii) If ##f(x)=O(u(x))## as ##x\rightarrow{a}## and ##g(x)=O(u(x))## as ##x\rightarrow{a}##...
  42. N

    I Finding Max and Min Extremes of a Function with Second Derivatives Equal to Zero

    What should I do when the f(x, y) function's second derivatives or Δ=AC-B² is zero? When the function is f(x) then we can differentiate it until it won't be a zero, but if z = some x and y then can I just continue this process to find what max and min (extremes) it has? What I've done is...
  43. N

    I Why can't we just integrate a simple function?

    Can anyone explain to me why the second one is the right? (See the attachment)
  44. luqman

    Coordinate Transformation (multivariable calculus)

    My Progress: I tried to perform the coordinate transformation by considering a general function ##f(\mathbf{k},\omega,\mathbf{R},T)## and see how its derivatives with respect all variable ##(\mathbf{k},\omega,\mathbf{R},T)## change: $$ \frac{\partial}{\partial\omega} f =...
  45. G

    Find the two points on the curve that share a tangent line

    IMPORTANT: NO CALCULATORS I assumed two points, (a, f(a)) and (b, f(b)) where b is greater than a. Since the tangent line is shared, I did f'(a) = f'(b): 1) 4a^3 - 4a - 1 = 4b^3 - 4b - 1 2) 4a^3 - 4a = 4b^3 - 4b 3) 4(a^3 - a) = 4(b^3 - b) 4) a^3 - a = b^3 - b 5) a^3 - b^3 = a - b 6) (a...
  46. N

    I Can I Solve y in y^2=ln(1-y^2)?

    Hi, I'm differentiating the "z" function to find extreme points but after solving the first partial derivatives with respect to "x" and "y" and also the "x" variable of the system, I can't find "y" (still in the system) using "ln" (natural logarithm). The question is can I differentiate both...
  47. N

    How do I calculate physics formulas containing derivatives and real numbers?

    Hi, I'm trying to calculate my own physics problem but didn't get it something. When I'm trying to calculate the impulse of the object when it's hit by F=10N force in the smallest possible time, then should I write: dP/dt = Fnet => dP = Fnet*dt ? Another question: In general, if I calculate...
  48. fresh_42

    Insights Counting to p-adic Calculus: All Number Systems That We Have

    Continue reading...
  49. A

    Prove by the principle of induction

    (expression given to be proven) check for p(1)... 2=2 substitute (n+n) to And here is the problem, I just can't find a way to continue solving this problem
  50. N

    What does the differential equation answer mean?

    Hi, last semester I "solved" a full differential equation and the answer was (see the picture). What does it mean? Can I make a graphic with it or what? I really don't get it. *Arrows are just a continuation of the main formula*
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