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  1. M

    Inexact Differentials (Distance Example)

    Heya, I am not trying to solve a particular problem, just trying to wrap my head around inexact differentials. My goal is understand the connection this has to exact ODEs.
  2. M

    Inexact Differentials (Distance Example)

    Let's start with this example of walking from Point A to Point B and back. 1) What does it even mean to talk about derivatives and differentials in one dimension? The derivative is supposed to reflect the ratio of changes between two dimensions. 2) How can df = 1dx but Δf = x_B - x_A? My...
  3. M

    Vectors as Inputs

    In this video (, it is said that the inner function in a composition must be a scalar and can't be a vector. What does this mean? It is common in multivariable calculus to consider functions with multiple inputs, which can be viewed geometrically as a single...
  4. M

    General Integrating Factor

    There is a remote possibility that this is not an intentional exercise, as my book actually states the equation as 2y dx + 3x dx = 0. I have assumed this to be a misprint. This exercise appears in the section on exact differential equations. The scalar curl of the equation is ∂(3x)/∂x -...
  5. M

    Ricatti ODE v. generalized Ricatti ODE; use of Bernouli ODE step (in video)

    This video ( introducing the Riccati ODE references a more general form than the one it covers. I presume this is a reference to the notably more daunting linear algebraic equations in the Wikipedia article on Linear-Quadradic-Gaussian Control...
  6. M

    Ricatti ODE v. generalized Ricatti ODE; use of Bernouli ODE step (in video)

    I noticed that in the Ricatti ODE introduction in this video (, the form of the general solution is stated as y = y_1 + u and the Ricatti ODE becomes a Bernoulli ODE, which eventually becomes linear, while in this video...
  7. M

    Quadratic Form Minimization Exercises?

    I am trying to internalize this video and perhaps work an example or two. I understand the professor's point that quadratic form minimization can be used to solve Ax = b from linear algebra, using gradient descent from calculus (seemingly an exercise for computers moreso than humans), but does...
  8. M

    Separation of Variables Substitution

    Textbook problem: Consider the equation y' = f(at + by + c), where a, b, and c are constants. Show that the substitution x = at + by + c changes the equation to the separable equation x' = a + bf(x). Use this method to find the general solution of the equation y' = (y + t)^2. Since nearly...
  9. M

    difference between local linearization/quadratic approx, Multivariable Taylor Approx?

    Is there a difference between local linearization/quadratic approximations and multivariable taylor approximations? They appear to be identical (for first and second degree approximations, respectively), but go under different names, and in, i.e., the Khan Academy videos on local linearization...
  10. M

    Implicit Differentiation: single-variable vs. multi-variable

    I noticed that the implicit differentiation procedure from single variable calculus, i.e., x^3 + sin(y) = y ---> (3x^2)dx + (cos(y))dy = (1)dy ---> (3x^2)dx = (1 - cos(y))dy ---> dy/dx = (3x^2)/(1 - cos(y))) (better explanation: doesn't appear to...
  11. M

    Interpreting a Quote About the Left Nullspace

    At the beginning of this video, the instructor gives an interesting explanation of Left Nullspace... "What Left Nullspace is...these are the known unknowns of Left Nullspace; the unbuildable reality. Alright, this is the stuff that, if you're thinking, and you see things out in the world, these...
  12. M

    Exact value of the summation from k = 1 to infinity of 3/(4^k)

    In particular, I'm trying to figure out what solution process I used on the exam. Unfortunately, my handwriting and organizational skills were extremely poor. Can anyone figure out how I got the correct answer of 1?
  13. M

    Second Derivative Question In the above video, is it stated that d(df) or d²f can be thought of as the difference between two consecutive output changes, each resulting from an input change of size dx. Why is a difference, rather than a quotient, the relevant expression? This is...
  14. M

    Is success realistic using u-substitution when the derivative isn't in the integrand?

    If one attempts a u-substitution that leaves x in the new integrand, it stands to reason that the x could be eliminated by inverting the substitution function and making another substitution. For example, to take the integral of sqrt(1 + x^-2) dx, we might let u = 1 + x^-2, and thus dx =...
  15. M

    Trig Substitution

    Hello, I'm having a disproportionately difficult time learning trig-substitution compared to integration by parts, u-substitution, and partial fractions (every video tutorial seems to use a slightly different process). Should assigning the legs to the right triangle be thought of as part of...
  16. M

    Why does Revenue = Units Sold * Demand? Wouldn't units sold depend on demand?

    In why does revenue = units sold * demand? I don't understand how this is determined. Wouldn't units sold depend on demand?
  17. M

    Derivative of Square Root Visual: the challenge at timestamp 12:23 There is a challenge at 12:23 asking the viewer to arrive at the formula for (d/dx) sqrt(x) by considering small changes to a square. What is the correct way to approach this? I tried to copy the technique used to find (d/dx) (x^2) at 2:25, by...
  18. M

    Fundamental Theorem of Calculus Video

    In this video, speed, defined as the distance traveled in one unit of time, is represented by rectangles on the distance-time (distance as a function of time) graph. At first, the examples stick to a constant speed per unit of time, and illustrate that within each time unit, the slope of the...