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Has anyone completed the Calculus courses on Khan Academy (Integral, Differential and Multivariable)? If so, would you recommend Khan Academy for Calculus inste

by neovive 11y ago

Has anyone completed the Calculus courses on Khan Academy (Integral, Differential and Multivariable)? If so, would you recommend Khan Academy for Calculus instead of the resources mentioned on Better Explained? I'm working through the Linear Algebra courses now on KA and they are very good.

jasode 11y ago
I haven't seen the Khan Academy Calculus videos that require signup but my answer assumes it is similar to (or exactly the same as) the youtube videos presented by Salman Khan. If you're already making great progress from Khan vids for Linear Algebra, you'll probably have the same success with Khan's Calculus. S Khan's approach is friendly but it's also very similar to a traditional teacher using a whiteboard in a classroom. For example, S Khan dives right into the formal concepts of "functions" and "limits" and all the associated notation. This mirrors how a professor would present it. In fact, Salman often remarks that he has his "college textbook" open as a resource to order the sequence of topics and demonstrate examples. Kalid Azad's approach with BetterExplained is to present images, pictures, and analogies that's not found in typical Calculus textbooks. This helps many learners get past the "mystery" of what the topic is about. You'll notice that the first chapters of Azad's approach does not discuss "limits"[1]. If you're already doing fine with Linear Algebra, you may not need all the pretty pictures as a gentle introduction. [1] The idea of "limits" is just one example of a philosophical difference on how to teach calculus. Isaac Newton didn't use the rigorous concept of "limits" when he invented calculus. His peers derided his approach as "ghosts of departed quantities". It was the later mathematicians who refined calculus to be based on more rigorous foundations (Real Analysis) that "limits" was retrofitted onto calculus. It's interesting that we now think the best pedagogy for high school students is to force "limits" on them to learn calculus because ... It's More Rigorous. Unfortunately, "limits" are not intuitive.
- btilly 11y ago
  Your history of Calculus is a little..muddled. Newton's approach was based on the idea that rate of change was a fundamental quantity. This lead to fluxions. His notation for Calculus was x with a dot over it. He used his method as a rough notation to figure out what the answers should be, before coming up with pure geometric constructions to use in the Principia Mathematica. Inspired by the Principia, Leibniz reinvented Calculus with the idea of infinitesmals. Infinitesmals are nonzero numbers that are smaller than any real number. He published, which started a priority dispute between him and Newton. Leibniz gave us our dy/dx notation. In the original, d is an operator meaning "look an infinitesmal away". Literally d(y) = y(x+dx)-y(x). (This converges faster if you make it y(x+dx/2)-y(x-dx/2)...) You do the calculation, then drop any infinitesmals and get your answer. Ever wondered why the second derivative is d^2y/dx^2? That's because you do d(d(y))/d(x)d(x). The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals. Far from being a peer of Newton's, he was a bishop who was demonstrating that science had leaps of faith as big as any in religion. His criticisms were accurate. The fact that math's foundations were crap occasionally inspired effort. But not much. Lagrange's contribution was to view everything as a power series, and do formal derivatives that way. To him we owe our f' notation. Then Joseph Fourier upended the whole apple cart with Fourier series. He took infinite sums of nice functions like sin and cos, and wound up with a step function. What happens if your step is an infinitesmal away? This was a crisis. Cauchy addressed this by making infinitesmals sequences that go to 0. This approach failed for the chain rule when the derivative is 0, but that's where the idea of Cauchy sequences comes from. Weirstrauss in the 1870s came up with the modern limit approach. Mathematics was (very literally) rewritten. Since then limits have proven to be a horrible pedagogical tool, but that is how we teach. However there are serious pedagogical suggestions ranging from skipping over the justification for Calculus to using infinitesmals (turns out you can make precise sense of them with modern set theory) to writing the tangent line approximation with little-o notation. (The last suggestion is from Knuth! See https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... for details.) Sorry for the history lesson, I love this topic. :-)
  - jasode 11y ago
    >The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals. It was a criticism of both Newton and Liebnez: "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" -- from https://en.wikipedia.org/wiki/The_Analyst#Content https://en.wikipedia.org/wiki/The_Analyst#Content Your history is more detailed but it also focuses more on the evolving notations and more of the intermediate intellectual steps in between Principia Mathematica and today's rigorous Real Analysis. I had the lesser ambition of pointing out that the non-intuitive "limits" is not what Newton did and it is somewhat ironic considering we force it on today's teenagers to "learn calculus".
  - Chris2048 11y ago
    Is there any forum for discussing this type of thing? I have some questions wrt issue like this, but you ask on a place like MathOverflow and you often get unintuitive answers, as if for the working mathematician, rather than explanations. For example, I've always though of dy as intuitively involving some kinf of, higher-function over another; d[y(x), a] = y(x+a) - y(x) such that; if I(x) = x so that d[I(x), a] = d[x, a] and d[x, a] = a then d[y(x), a]/d[x, a] = d[y(x), d[x, a]]/d[x, a] so we drop reference to a (or lim as a->0) and = dy(x)/dx = dy/dx But look, there's an implicit 'a' or 'dx' in that y(x), plus y(x) is often written just 'y' so that you can always switch around which the free variable is (an equation just describes the relationship, e.g. y = 2x can generate either y(x) = 2 * x or x(y) = y/2) Also, since the imagined HO function d[f, a] actually takes a function for the first arg, there needs to be a implicit identity function for dx, so really df(x)/dh(x) means "rate of change of f, for change in x, relative to (normalised) rate of change in h, for change in x". Then the identity function becomes the default such that dx is short for dI(x). Maybe this descibes the same idea: https://en.wikipedia.org/wiki/Differential_operator https://en.wikipedia.org/wiki/Differential_operator https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a_function https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a... But its a little undecipherable to me...
    - 11y ago
- neovive 11y ago
  Thanks for the advice. I actually really liked the "How To Learn Trigonometry Intuitively [1] on Better Explained as my Trig experience was limited to memorizing the formulas. It was only when I ventured into making a basic game that I learned the actual practical uses of Trigonometry. I wish math in school focused more on why and less on how, especially since all schools have access to computers and knowing where/when to apply mathematical theories and concepts is so important. [1] http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/

by neovive 11y ago

jasode 11y ago

I haven't seen the Khan Academy Calculus videos that require signup but my answer assumes it is similar to (or exactly the same as) the youtube videos presented by Salman Khan. If you're already making great progress from Khan vids for Linear Algebra, you'll probably have the same success with Khan's Calculus. S Khan's approach is friendly but it's also very similar to a traditional teacher using a whiteboard in a classroom. For example, S Khan dives right into the formal concepts of "functions" and "limits" and all the associated notation. This mirrors how a professor would present it. In fact, Salman often remarks that he has his "college textbook" open as a resource to order the sequence of topics and demonstrate examples. Kalid Azad's approach with BetterExplained is to present images, pictures, and analogies that's not found in typical Calculus textbooks. This helps many learners get past the "mystery" of what the topic is about. You'll notice that the first chapters of Azad's approach does not discuss "limits"[1]. If you're already doing fine with Linear Algebra, you may not need all the pretty pictures as a gentle introduction. [1] The idea of "limits" is just one example of a philosophical difference on how to teach calculus. Isaac Newton didn't use the rigorous concept of "limits" when he invented calculus. His peers derided his approach as "ghosts of departed quantities". It was the later mathematicians who refined calculus to be based on more rigorous foundations (Real Analysis) that "limits" was retrofitted onto calculus. It's interesting that we now think the best pedagogy for high school students is to force "limits" on them to learn calculus because ... It's More Rigorous. Unfortunately, "limits" are not intuitive.

btilly 11y ago
Your history of Calculus is a little..muddled. Newton's approach was based on the idea that rate of change was a fundamental quantity. This lead to fluxions. His notation for Calculus was x with a dot over it. He used his method as a rough notation to figure out what the answers should be, before coming up with pure geometric constructions to use in the Principia Mathematica. Inspired by the Principia, Leibniz reinvented Calculus with the idea of infinitesmals. Infinitesmals are nonzero numbers that are smaller than any real number. He published, which started a priority dispute between him and Newton. Leibniz gave us our dy/dx notation. In the original, d is an operator meaning "look an infinitesmal away". Literally d(y) = y(x+dx)-y(x). (This converges faster if you make it y(x+dx/2)-y(x-dx/2)...) You do the calculation, then drop any infinitesmals and get your answer. Ever wondered why the second derivative is d^2y/dx^2? That's because you do d(d(y))/d(x)d(x). The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals. Far from being a peer of Newton's, he was a bishop who was demonstrating that science had leaps of faith as big as any in religion. His criticisms were accurate. The fact that math's foundations were crap occasionally inspired effort. But not much. Lagrange's contribution was to view everything as a power series, and do formal derivatives that way. To him we owe our f' notation. Then Joseph Fourier upended the whole apple cart with Fourier series. He took infinite sums of nice functions like sin and cos, and wound up with a step function. What happens if your step is an infinitesmal away? This was a crisis. Cauchy addressed this by making infinitesmals sequences that go to 0. This approach failed for the chain rule when the derivative is 0, but that's where the idea of Cauchy sequences comes from. Weirstrauss in the 1870s came up with the modern limit approach. Mathematics was (very literally) rewritten. Since then limits have proven to be a horrible pedagogical tool, but that is how we teach. However there are serious pedagogical suggestions ranging from skipping over the justification for Calculus to using infinitesmals (turns out you can make precise sense of them with modern set theory) to writing the tangent line approximation with little-o notation. (The last suggestion is from Knuth! See https://micromath.wordpress.com/2008/04/14/donald-knuth-calculus-via-o-notation/ https://micromath.wordpress.com/2008/04/14/donald-knuth-calc... for details.) Sorry for the history lesson, I love this topic. :-)
- jasode 11y ago
  >The famous quote of "ghosts of departed quantities" was written by George Berkeley as a criticism of infinitesmals. It was a criticism of both Newton and Liebnez: "And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?" -- from https://en.wikipedia.org/wiki/The_Analyst#Content https://en.wikipedia.org/wiki/The_Analyst#Content Your history is more detailed but it also focuses more on the evolving notations and more of the intermediate intellectual steps in between Principia Mathematica and today's rigorous Real Analysis. I had the lesser ambition of pointing out that the non-intuitive "limits" is not what Newton did and it is somewhat ironic considering we force it on today's teenagers to "learn calculus".
- Chris2048 11y ago
  Is there any forum for discussing this type of thing? I have some questions wrt issue like this, but you ask on a place like MathOverflow and you often get unintuitive answers, as if for the working mathematician, rather than explanations. For example, I've always though of dy as intuitively involving some kinf of, higher-function over another; d[y(x), a] = y(x+a) - y(x) such that; if I(x) = x so that d[I(x), a] = d[x, a] and d[x, a] = a then d[y(x), a]/d[x, a] = d[y(x), d[x, a]]/d[x, a] so we drop reference to a (or lim as a->0) and = dy(x)/dx = dy/dx But look, there's an implicit 'a' or 'dx' in that y(x), plus y(x) is often written just 'y' so that you can always switch around which the free variable is (an equation just describes the relationship, e.g. y = 2x can generate either y(x) = 2 * x or x(y) = y/2) Also, since the imagined HO function d[f, a] actually takes a function for the first arg, there needs to be a implicit identity function for dx, so really df(x)/dh(x) means "rate of change of f, for change in x, relative to (normalised) rate of change in h, for change in x". Then the identity function becomes the default such that dx is short for dI(x). Maybe this descibes the same idea: https://en.wikipedia.org/wiki/Differential_operator https://en.wikipedia.org/wiki/Differential_operator https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a_function https://en.wikipedia.org/wiki/Derivative#The_derivative_as_a... But its a little undecipherable to me...
  - 11y ago
neovive 11y ago
Thanks for the advice. I actually really liked the "How To Learn Trigonometry Intuitively [1] on Better Explained as my Trig experience was limited to memorizing the formulas. It was only when I ventured into making a basic game that I learned the actual practical uses of Trigonometry. I wish math in school focused more on why and less on how, especially since all schools have access to computers and knowing where/when to apply mathematical theories and concepts is so important. [1] http://betterexplained.com/articles/intuitive-trigonometry/ http://betterexplained.com/articles/intuitive-trigonometry/