Calculus Definition and 1000 Threads

  1. M

    Why Is the Chain Rule Not Used in Differentiating 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!
  2. Sailor Al

    I Can Air Resist a Piston? A Serious Discussion

  3. M

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

    Does Precalculus by James Stewart teaches function from scratch ?
  4. 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...
  5. 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...
  6. 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...
  7. YAYA12345

    I Integral Bee Preparation -- Trouble with this beautiful integral

    While I was preparing for an integrals contest, I had a doubt about the following integral, I tried several substitutions but nothing worked.I would appreciate your support for this beautiful integral. $$ \int\limits_{0}^{1/2} \cos(1-\cos(1-\cos(...(1-\cos(x))...) \ \mathrm{d}x$$
  8. M

    Vector Calculus in 1D: ± to Show Magnitude?

    [mentor's note - moved from one of the homework help forums] Homework Statement:: It's a question. Relevant Equations:: Vector calculus. Is it true to say that in one dimension I can show vector quantities using ±number instead of a vector? ± can show possible directions in one dimension and...
  9. WMDhamnekar

    Stochastic Calculus: Conditional Expectation

    Are my following answers correct?
  10. WMDhamnekar

    Stochastic Calculus: Conditional Expectation

  11. Demystifier

    A Found a new formula of Dirac calculus

    I have found a new formula in Dirac calculus. The formula is elementary, so probably I'm not the first who found it. Yet, I have never seen it before. As many other formulas in Dirac calculus, it is not rigorous in the sense of functional analysis. Rather, it is a formal equality, which is only...
  12. A

    Calculus Calculus book between Stewart & Spivak levels

    Hi, Are there calculus books that lie between Stewart (or Thomas) level and Spivak (Courant/Apostol) level? Thanks.
  13. mcastillo356

    I Express the limit as a definite integral

    Hi, PF, there goes the definition of General Riemann Sum, and later the exercise. Finally one doubt and my attempt: (i) General Riemann Sums Let ##P=\{x_0,x_1,x_2,\cdots,x_n\}##, where ##a=x_0<x_1<x_2<\cdots<x_n=b##, be a partition of ##[a,b]##, having norm ##||P||=\mbox{max}_{1<i<n}\Delta...
  14. 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...
  15. 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...
  16. 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...
  17. 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...
  18. 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. $$ f(x)= x + 5 + \frac {1}{x-5}$$  
  19. 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...
  20. 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...
  21. 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...
  22. N

    I Is It Possible? Solving Calculus Questions with Ease

    Hi, just wondering if that's possible in calculus. (See the attachment)
  23. 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...
  24. 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}##...
  25. 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...
  26. 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)
  27. 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 =...
  28. 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...
  29. N

    I Can I Differentiate Both Sides to 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...
  30. 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...
  31. F

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

    Continue reading...
  32. 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
  33. 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*
  34. E

    B Attempt to solve this system of three linear equations

    The point (1, 5) is on the curve: y=ax^2+bx+c. This point gives the linear equation: 5 = a + b + c. A second point on the curve, (2, 10) gives the linear equation 10=4a+2b+c. A student called Erika thinks that the point (2, 19) is also on the curve. 5 = a + b + c. 10=4a+2b+c 19=4a+2b+c the...
  35. benorin

    Calculus An interesting read: "Project Gutenberg EBook of Calculus Made Easy"

    This is a good book to read, free pdf version. Don’t judge the book by it’s cover. Read section 1, and 2, see if you learn something you weren’t expecting ~.^ This is the above link’s target: https://www.gutenberg.org/files/33283/33283-pdf.pdf Enjoy!
  36. malawi_glenn

    Other Collection of Free Online Math Books and Lecture Notes (part 1)

    School starts soon, and I know students are looking to get their textbooks at bargain prices 🤑 Inspired by this thread I thought that I could share some of my findings of 100% legally free textbooks and lecture notes in mathematics and mathematical physics (mostly focused on geometry) (some of...
  37. Z

    I Spivak, Ch. 20: Understanding a step in the proof of lemma

    In Chapter 20 of Spivak's Calculus is the lemma shown below (used afterward to prove Taylor's Theorem). My question is about a step in the proof of this lemma. Here is the proof as it appears in the book My question is: how do we know that ##(R')^{n+1}## is defined in ##(2)##? Let me try to...
  38. Manuel Cardoso

    Calculus book for 14 year old physics enthusiast

    Hello, I'm trying to learn calculus, what books would you recommend? For a beginner.
  39. N

    Calculus Comparing "Calculus" and "Early Transcendentals" Textbooks for MST124

    I'm going to be starting my first Mathematics module (MST124 - Essential Mathematics I, at the Open University) and I have been looking or a Calculus textbook to use as a supplementary text. I've found a couple of textbooks that I like the look of (Stewart's and Larson's), which both come in...
  40. E

    I Solving the Boat Crossing a River Problem using the Calculus of Variations

    I was hoping to explore the Calculus of Variations. How do we prove by Calculus of Variations that the minimum time for boat crossing a river (perpendicular to the current for starters) with current ##v_r##, and boat velocity in still water ##v_b## that the path will be a straight line? I...
  41. 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...
  42. junyoung

    I Open interval or Closed interval in defining convex function

    The Korean textbook standard defines the convexity of the function as an open section. Many textbooks and university calculus textbooks define the convexity of the curve as an open section. However, some textbooks define convexity as closed sections. Do you think it is right to define the...
  43. TGV320

    Calculus Are These Books Good for Self-Studying Calculus in College?

    Hi, I am going to be a 1st year college student in China this fall. I have a lot of interest for calculus and math in general, and I am wondering if the books I have in hand are good for self study. High school math here stopped the at the derivatives, didn't went further. The two books I...
  44. 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...
  45. brotherbobby

    Finding Area of Shaded Segment in Circle Using Calculus

    [FONT=times new roman]Problem Statement : [FONT=times new roman]To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP.[FONT=times new roman] Attempt 1 (without calculus): I mark some relevant lengths inside the circle...
  46. bigmike94

    James Stewart Calculus replacement

    Summary: Need a multivariable calculus textbook For calculus I’ve been using James Stewart textbook as a guide, I find it really hard to follow so I just checkout the chapter titles and then use online courses that explain the chapters, for example professor Leonard and Michel Van biezen...
  47. S

    MHB Are the inference rules of propositional calculus tautologies?

    Are the inference rules of propositional calculus tautologies ,yes or no
  48. G

    What Is the Best Angle Theta to Reduce Travel Time in Calculus?

    Summary: Trying to differentiate with respect to ## \theta ## is entangling me in cos and sec terms. A simple problem I found, while looking for calculus practice. Roads between home and main road are 30 mph, main road is 60mph: What is the optimum ## \theta ## to minimise journey time? ## t...
  49. topsquark

    MHB Chain Rule in Difference Calculus

    I'm having some problems using the chain rule and I'm not sure where the trouble lies. For example: If I'm not mistaken, if we have the composite function f(g(n)) then \Delta f(g(n)) = \dfrac{ \Delta f(g) }{ \Delta g } \dfrac{ \Delta g(n) }{ \Delta n } Let f(g(n)) = (n^2)^2. Then f(g) = g^2...
  50. N

    The connection between potential energy and force

    Hi, if the force is the derivative of potential energy, does it mean that the force is equal to mg and with a constant gravity, it will be the same at any height? But in real life, F (or mg) would be different on the Earth's surface and 400 km above it (~8 m/s^2). So, this formula is used to...
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