The logical structure of calculus, request for help.

Consider calculus of a real variable.

Is the following the logical structure for at least differential calculus, assuming knowledge of algebra, functions, set theory?

* minimal knowledge of vector spaces.
- Definition of a vector space.
- Definition of a subspace.
- Theorems involving subspace.
- Definition of a norm.

* minimal knowledge of differential calculus.
- Definition of a limit.
- Theorems involving combinations of limits.
- Definition of continuity at a point.
- Definition of continuity on an interval.
- Theorem: The intermediate value.
- Theorems involving combinations of continuous functions.
- Definition of derivative at a point.
- Definition of derivative on an interval.
- Theorem that differentiable at b implies continuous at b.
- Theorems for computing derivatives; power law, etc.
- Theorem of the chain rule.


The above should be the bare minimum and a logical structure for teaching calculus?

Note, I've left out definitions and derivatives of various functions, including the trigometric and exponential, theorems and definition (extreme value) used in max/min and optimization problems.

I've learned the above myself, but will be reviewing and improving my understanding to prepare for tutoring a friend.

Please add any definitions, theorems, logical structure, etc, that will be of an aid. It's difficult for me to tutor this person some times because of gaps in understanding. I seek to eliminate that.

Thanks, Adam.
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Relevant Pages

  • Re: The logical structure of calculus, request for help.
    ... >Consider calculus of a real variable. ... >- Theorems involving combinations of limits. ... >- Definition of continuity on an interval. ... >- Theorems for computing derivatives; ...
    (sci.math)
  • Re: Galileos Paradox
    ... Tony Orlow wrote: ... The outer measure of those two sets is exactly the same. ... theorems based on what we wish. ... continuity, differentiability) doesn't do what we want, then we come up ...
    (sci.math)
  • Re: Galileos Paradox
    ... Tony Orlow wrote: ... The outer measure of those two sets is exactly the same. ... theorems based on what we wish. ... continuity, differentiability) doesn't do what we want, then we come up ...
    (sci.math)
  • Re: Galileos Paradox
    ... Tony Orlow wrote: ... The outer measure of those two sets is exactly the same. ... theorems based on what we wish. ... continuity, differentiability) doesn't do what we want, then we come up ...
    (sci.math)
  • Re: Inequality involving e
    ... Saysero wrote: ... First of all why do you think it is a homework? ... The reason why I said "I am not supposed to use derivatives and the ... theorems that apply to their relation with functions." ...
    (sci.math)
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