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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Discussion 3 -- The Derivative}
\centerline{November 10, 2004}
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\noindent
{\it Background}
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We have now developed the properties of limits and continuous functions.
Our next topic will be a consideration of the foundations of differential
calculus, starting with the concept of the derivative of a function.
As in calculus, we say that a function $f : D \to {\bf R}$
is {\it differentiable} at $x = c$ if the limit
$$\lim_{x\to c} {f(x) - f(c)\over x - c} = \lim_{h\to 0} {f(c+h) - f(c)\over h}\leqno(1)$$
exists (the second form comes by introducing $h$ as the difference
between $x$ and $c$ and writing $x = c + h$).  
If so, we call the value of the limit the {\it derivative of $f$ at $x = c$}, written
$f'(c)$.  

Note:  In calculus, you learned a number of rules (the Product, Quotient, Chain Rules, and others)
for computing derivatives.  In some of the questions below, you will need
to use these even though we have not developed them in this course.  If
the question says use the definition to compute $f'(c)$, though, you
will need to use (1).
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{\it Discussion Questions}
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A)  
\item{1)}  Show using the definition of the derivative
that for all positive integers $n$, the function $f(x) = x^n$ is 
differentiable for all $c$, and find $f'(c)$.  Note: To calculate the limit
for $f'(c)$, you will have to decide which form of the definition you want
to use.  If you use the second (the $\lim_{h\to 0}$ form), you will want to use
the binomial theorem (Exercise 10.18).  No matter which form
you use, use the Limit Theorems for function limits (Theorem 20.13);
don't go back to the $\varepsilon$-$\delta$ definition of the limit of a function.
\item{2)} Show using the definition of the derivative that
$f(x) = \sqrt{x}$ is differentiable at all $c > 0$, and find $f'(c)$.
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\noindent
B)  By considering the equation 
$$f(x) = {f(x) - f(c)\over x - c}(x - c) + f(c)$$
show that if $f$ is differentiable at $x = c$, then $f$
is continuous at $x = c$.
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\noindent
C)  
\item{1)}  Let 
$$f(x) = \cases{x\sin\left({1\over x}\right) & if $x \ne 0$\cr
                         0 & if $x = 0$\cr}$$
Show that $f$ is continuous at $c = 0$, but $f(x)$ 
is not differentiable at $c = 0$, by using the definition.
\item{2)}  Let 
$$g(x) = \cases{x^2\sin\left({1\over x}\right) & if $x \ne 0$\cr
                         0 & if $x = 0$\cr}$$
Show that $g$ is differentiable at $c = 0$.  Show using the
Product and Chain Rules from calculus that $g'(x)$ exists
for all $x \ne 0$ as well.  Is $g'(x)$ a continuous function, 
though? 
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\noindent
D)  Let 
$$f(x) = \cases{x^2 & if $x$ is a rational number\cr
                0 & if $x$ is an irrational number\cr}$$
\item{1)}  Show that $f(x)$ is continuous at $c = 0$, but $f(x)$
is not continuous at any $c \ne 0$.
\item{2)}  Is $f(x)$ differentiable at $c = 0$?  Why or why not?
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\noindent
E)  Recall that a function $f : {\bf R} \to {\bf R}$ is 
said to be {\it even} if $f(-x) = f(x)$ for all $x$, and {\it odd}
if $f(-x) = -f(x)$ for all $x$.  
\item{1)}  Assume that $f$ is differentiable for all $x$ and 
even.  What can you say about $f'(x)$?  Prove your assertion.
\item{2)}  Assume that $f$ is differentiable for all $x$ and 
odd.  What can you say about $f'(x)$?  Prove your assertion.
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{\it Assignment}
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Group write-ups due Monday, November 15. 
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