University of Tokyo 1S2 Differential and Integral Calculus

• 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.

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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 $\frac{f(b)-f(a)}{(b-a)}$ is equal to the slope of f’(x).
    • Proof:
      • Distort f(x) a bit to transform it into the form $\frac{f(b)-f(a)}{(b-a)}=0$.
      • 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 $\frac{f(b)-f(a)}{g(b)-g(a)}$ has the same slope as $\frac{f^{\prime }(x)}{g^{\prime }(x)}$, 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.
    • When considering this part rigorously, epsilon-delta definition becomes necessary.
• 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
    • $\frac{f(b)-f(a)}{(b-a)}=f^{\prime }(c)$
  • By transforming this, we get
    • $f(b)=f(a)+f^{\prime }(c)(b-a)$
      • If you look closely, it already looks like a second-order Taylor expansion
  • Generalizing this, we get
    • 
      • In the mean value theorem, c (an undetermined value) was used in almost all terms as a (the leftmost term)
      • Only the last term $R_{n+1}$ (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
        • 

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 $C^{\infty }$, meaning it can be differentiated infinitely many times.
  • As n approaches infinity, the remainder term $R_n(x)$ 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 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".