• Continuation of University of Tokyo 1S1 Mathematical Science Foundation: Differential and Integral Calculus.
  • Lecture 1: Rolle’s theorem, mean value theorem, L’Hôpital’s rule.

  • Lecture 2: Taylor’s theorem, Taylor series, specific examples.

  • Lecture 3: Applications of Taylor expansion.

  • Lecture 4: Limits, continuity, partial derivatives of functions of two variables.

  • Lecture 5: Total differentials, differentiation of composite functions.

  • Lecture 6: Higher-order partial derivatives, commutation of partial derivatives.

  • Lecture 7: Parametrically represented curves, implicit function theorem.


Lecture 1: Rolle’s theorem, mean value theorem, L’Hôpital’s rule

  • Goal: Proof of L’Hôpital’s rule
    • I thought L’Hôpital was something else, but it turned out to be L’Hôpital(blu3mo)
  • Proof:
    • Assumption: Extreme value theorem
    • Proposition 1: At maximum/minimum, f’(x) = 0
      • This can be derived from the definition of differentiation.
    • Rolle’s theorem: If a function is continuous and differentiable in a certain interval, and f(a) = f(b), then there exists at least one value of x between a and b where f’(x) = 0.
      • This is also reasonable.
      • Proof: There exist maximum/minimum values within the range (by the extreme value theorem), and they satisfy f’(x) = 0 (from Proposition 1), something like that.
        • We need to consider some more detailed cases.
    • Mean value theorem: If a function is continuous and differentiable in a certain interval, there exists at least one point between a and b where the slope of is equal to the slope of f’(x).
      • Proof:
        • Distort f(x) a bit to transform it into the form .
        • Then, it becomes equivalent to Rolle’s theorem, so the proof is complete.
          • I see, smart (blu3mo)
    • Generalized mean value theorem: If a function is continuous and differentiable in a certain interval, and has the same slope as , then there exists at least one value of x between a and b.
      • I understand the symmetry, but I can’t grasp the geometric image (blu3mo)
      • If we express the mean value theorem with g and f, and divide f’s expression by g’s expression, does this result come up?
    • L’Hôpital’s rule (goal): If we consider the case of a → b in the generalized mean value theorem, L’Hôpital’s rule emerges.
  • Addition: Considering the content of the second lecture, L’Hôpital’s rule is used to obtain the value of interest by approximating it with a linear approximation using Taylor expansion centered around the desired value.
    • Since Taylor expansion is performed around the x value of interest, the approximation can be simple and linear (blu3mo)(blu3mo)

Lecture 2:

  • How to derive Taylor’s theorem
    • Mean value theorem
    • By transforming this, we get
    • Generalizing this, we get
      • image
        • In the mean value theorem, c (an undetermined value) was used in almost all terms as a (the leftmost term)
        • Only the last term (Lagrange remainder term) has an undetermined value of c
          • It states that the position of c, as in the mean value theorem, is somewhere between a and b
          • As a result, R_n+1 indicates that it lies within this range
          • image

Lecture 3: Applications of Taylor expansion

  • Conditions for being able to perform Maclaurin expansion of a function into an infinite series
    • The function is of class , meaning it can be differentiated infinitely many times.
    • As n approaches infinity, the remainder term converges to 0.
    • These two assumptions are important (blu3mo)
  • Discussion on applications of Maclaurin expansion, where known expansions are transformed into forms that include the desired expansion
    • These are based on the assumption that image holds, but it should be noted that this is not always the case.

Lecture 4: Limits, continuity, partial derivatives of functions of two variables

  • About the differentiation of functions of two variables

  • Definition of distance -> Definition of limits -> Definition of continuity -> Definition of partial derivatives- Since it depends from left to right, it feels like defining in order.

  • If we define the distance on a plane using, for example, the Manhattan distance instead of the Euclidean distance, would it change the laws of limits, continuity, and partial derivatives? (blu3mo)

    • It might not change as much as expected. (takker)
      • If we can define even partial derivatives using only the properties of the distance space, not specific to Euclidean distance, then the form of differentiation remains invariant in any distance space.
    • It seems interesting to try it out.
  • Triangle inequality

    • The triangle inequality for various values frequently appears, doesn’t it? (blu3mo)
    • It seems to have high universality, but I still haven’t quite grasped it.
    • Is it the property expected in the definition of “distance”?
  • There are countless ways to take limits in the case of multivariable functions.

    • This is important. (blu3mo)
    • We assume that a limit exists only when the limit taken from all directions is the same. (In this course)
      • I think this generally holds true. (takker)
    • Because of this property, problems arise, and "differentiable does not necessarily mean continuous".