Order of concepts taught in calculus 1 & 2

  • #1
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Hello everyone. I am about to start Calculus 1 and then Calculus 2. I want to get an idea of how the order of concepts of these to classes are generally laid out.


Thanks
 

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  • #2
RUber
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It has been a while since I have taken the courses, and there may be different sequences used by different profs, but what I have seen has been:
--
Limits & Continuity
Definition of a derivative as a limit
Special functions: Log, exponential, trig functions, etc.
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Integration defined as a limit / Reimann Integration.
Application and problem solving.
Multiple dimensions and parametrics.
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I am sure I have forgotten something, but that sums up what I remember from those two classes.
 
  • #3
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What about chain rule and optimization, anti-derivatives?
 
  • #4
SteamKing
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What about chain rule and optimization, anti-derivatives?
1. The chain rule is a special technique applied to finding the derivative of a function composed in terms of other functions.
2. Optimization is typically an application of derivatives to certain problems.
3. "Anti-derivative" is another name given to the indefinite integral of a function,
since integration and differentiation are inverse operations of one another, according to the Fundamental Theorem of Calculus.
 
  • #5
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Ok, chain rule and optimization fall under the category of concepts related to understanding derivatives. And anti-derivative is the same as indefinite integral. Is there any difference between an integral and indefinite integral? so new to all this.
 
  • #6
RUber
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Integrals come in definite and indefinite (sometimes improper, but don't worry about that for now). Definite integrals are more of what you might consider the "area under the curve" between two endpoints. Indefinite integrals are like you said, the anti-derivative...more of a functional form such that if F is the anti-derivative of f then
##\int_a^b f(x) dx = F(b)-F(a).##
Another example would be if ##f(x) = 2x##, ##\int f(x) dx = x^2 + C.##

Improper integrals are usually expressed over an infinite range, where a more "proper" integral would be expressed as the limit as the endpoint goes to infinity of the definite integral.
 
  • #7
jtbell
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I am about to start Calculus 1 and then Calculus 2. I want to get an idea of how the order of concepts of these to classes are generally laid out.

Have you checked out the table of contents in a textbook or two? For a basic course like that, I doubt that many instructors jump around a lot, although they may skip some secondary topics.

Also, when I Googled for "calculus syllabus" one of the first things I saw was a course page for Calculus I at Columbia University, including a syllabus with a schedule of topics covered, and homework assignments. There's probably a similar page for their Calculus II, although I didn't look.
 
  • #9
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Found this one on google. Does this look about right?



Functions and graphs. Inverse functions.

The limit of a function. Algebraic computation of limits.

Continuity.

Exponential and logarithmic functions.

An introduction to the derivative. Tangents.

Techniques of differentiation.

Derivatives of trig., exponential and log. functions.

Rates of change. Rectilinear motion.

The chain rule.

Implicit differentiation.



Related rates. Linear approximation and differentials.

Extreme values of a continuous function.

The mean value theorem.

Sketching the graph of a function.

Curve sketching with asymptotes.

l’Hopital’s rule.

Optimization in physical sciences, etc.

Antidifferentiation.



Area as the limit of a sum.

Riemann sums and the definite integral.

The fundamental theorem of calculus.

Integration by substitution.

Introduction to differential equations.

The mean value theorem for integrals.

Numerical integration.
 

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