What are some ways to deepen understanding and challenge yourself in Calculus?

[physior]

hello
I just did the first lecture on differentiation
what the derivative is, and how is it calculated
I didn't find it to be difficult at all
it's just some forms of algebraic calculations
when will the challenging stuff kick in?
what should I expect in the future?
thanks!

 

[PeroK]

hello
I just did the first lecture on differentiation
what the derivative is, and how is it calculated
I didn't find it to be difficult at all
it's just some forms of algebraic calculations
when will the challenging stuff kick in?
what should I expect in the future?
thanks!

Differentiation is easy. Integration is hard!

 

[GFauxPas]

How to calculate derivatives is easy, if sometimes a bit messy. You'll learn lots of interesting ways to use the derivative, though:

Where does a function reach its extrema (as high/low as a function ever gets, on the real line or in an interval)?
How can you graph a function knowing that you're plotting all the parts of interest? plotting points isn't sufficient if you're missing any interesting part of the graph
How can you find the derivative if you're not given y=f(x)? For example, given y=f(t), x = g(t), or f(x,y)=z
Given different points x1,x2 on a function, you can analyze what's in between them on the interval (x1..x2)
If you're given a question about a function, it's often only necessary to show the question has an answer, rather than know what the answer is. Calculus can be very useful for that. (Does f have an inverse? Does f(x)=0 have a solution?)

Here's what I did if a section of the course was too easy:
Try proving things that are told to you. Or look up the proofs online (I use wikipedia, Paul's math notes, and ProofWiki). Can you think of any other proofs? Usually there are several ways to prove something.
You might be given a definition that's informal. What's the formal definition of things like "limit", "continuous", "series (infinite sums)"?
Learn the notation for concepts that make it a lot easier to state things without words, to make note-taking easier and to allow you to understand stuff you read online.
Generalize concepts. You might be given a theorem about z=f(x,y). What about y=f(x1,x2,...,xn)? You might be dealing with real numbers. What happens if you put in complex numbers? If you're dealing with a continuous function, what can you say about a piecewise continuous function?
Ask your professor for a term project. You might get extra credit or even a scholarship.
Play around with definitions different from what your book gives. Often a concept has more than one definition, and you can find other ones online. (Make sure you don't lose points on a test for using a definition not in the book)