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\centerline{Mathematics 242 -- Principles of Analysis}
\centerline{Problem Set 7 -- {\bf due:} Friday, April 12}
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{\it 'A' Section}
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\noindent
1. Let $f(x) = {15 x \over x^4 + x^2 + 1}$.  Using the Intermediate
Value Theorem, 
\item{A)}  Show:  For all $k \in [-5,5]$, there exist
$c \in [-1,1]$ such that $f(c) = k$.  
\item{B)}  Show:  For all $k$ with $0 < k < 5$, there exist some $c \in (1,\infty)$
such that $f(c) = k$.  
\item{C)}  Show that if $m$ satisfies $0 < m < 15$, then 
$f(x) = mx$ has two real solutions other than $x = 0$.
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\noindent
2. Show that there are at least three real solutions of the equation
$\sin(x) + 2\cos(x) = x/2$.  Hint:  Look at the values of 
$g(x) = \sin(x) + 2\cos(x) - x/2$ at ``nice'' multiples of ${\pi\over 2}$.  
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\noindent
3.  Using the definition of the derivative, find the value of $f'(c)$, or 
say why $f$ is not differentiable at $x = c$:
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\item{A)}  $f(x) = x^3 + 2x + 1$ at $c = 2$.
\item{B)}  $f(x) = \sin(|x|)$ at $c = 0$.  Hint: Look back at Problem Set 6, B 2.   
\item{C)}  The function defined by
 $$f(x) = \cases{x^2 & if $x > 1$\cr
                          2x - 1 & if $x \le 1$\cr}$$
at $c = 1$.  
\item{D)}  The function defined by
$$f(x) = \cases{x^2 & if $x \in {\bf Q}$\cr
                            0 & if $x \in {\bf Q}^c$\cr}$$
at $c = 0$. 
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\noindent
4.  Suppose $f,g$ are differentiable functions with $f(g(x)) = {x\over x^2 + 1}$ and
that 
$$\matrix{g(1) = 1 & g(2) = 4\cr
          g'(1) = 2 & g'(2) = -1\cr}$$
Determine the equation of the tangent line to the given graph at the given point.
\item{A)}  $y = f(x)$ at $(1,f(1))$.
\item{B)}  $y = (f\circ g)(x)$ at $(2,(f\circ g)(2))$.   
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\noindent
{\it 'B' Section}
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\noindent
1.  Let $f$ be continuous on $[0,1]$ with $f(0) < 0$ and $f(1) > 1$.  
Suppose that $g$ is another continuous function on $[0,1]$ such that
$g(0) \ge 0$ and $g(1) \le 1$.  Show that there exists some $c \in (0,1)$
such that $f(x) = g(x)$.  
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\noindent
2. Let $f$ be continuous on $[a,b]$ with $f(a) < k < f(b)$.  Here
is a variation on our proof of the Intermediate Value Theorem.  
\item{A)}  Let 
$$T = \{x \in [a,b] \mid f(x) > k\}.$$
Show that $T$ has a greatest lower bound and that $f({\rm glb}(T)) = k$.
\item{B)}  Will this ${\rm glb}(T)$ always be the same as the $c$ we found
in our proof of the IVT with $f(c) = k$?  If so, prove they are the same;
if not, give a counterexample.  
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\noindent
3.  This property deals with another property of real-valued functions of a real variable
sometimes called {\it Lipschitz continuity}.  
\item{A)}  Let $f$ be a function on an interval $I$ with the property that there exists
a strictly positive constant $k$ such that $|f(x) - f(x')| \le k |x - x'|$ for all $x,x' \in I$ (this is 
the definition of Lipschitz continuity).  Show 
that $f$ is uniformly continuous on $I$.  
\item{B)}  The converse of the statement in part A is not true:  
Show that $f(x) = x^{1/3}$ is uniformly continuous on $[-1,1]$, but there is
no constant $k$ such that $|f(x) - f(x')| \le k |x - x'|$ for all $x,x' \in [-1,1]$.
Hint:  Think slopes of secant lines to the graph $y = x^{1/3}$.  
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4.  Let $f$ and $g$ be differentiable on $(a,c)$ and let $b \in (a,c)$.  Assume 
$f(b) = g(b)$.  
Define a new function by 
$$p(x) = \cases{f(x) & if $x \in (a,b)$ \cr
                g(x) & if $x \in [b,c)$\cr}$$
Show that $p$ is differentiable on $(a,c)$ if and only if $f'(b) = g'(b)$.  

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