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\centerline{MATH 135 -- Calculus 1}
\centerline{The Derivative Function -- Practice Problems}
\centerline{October 4, 2013}
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1.  Consider the function $f(x) = {x^3\over 3} - x$.  
\item{a.}  Compute $f'(x) = \lim_{h \to 0} {f(x+h) - f(x)\over h}$.  
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\item{b.}  Here is a plot of $y = f(x)$.  On the empty set of axes below plot $y = f'(x)$.  
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\item{c.}  Circle the correct answers:  On the interval $(-\infty,-1)$, the derivative $f'(x)$ is {\it positive/negative} and $f(x)$ is {\it increasing/decreasing}.   On the interval $(-1,1)$, the derivative $f'(x)$ is {\it positive/negative} and $f(x)$ is 
{\it increasing/decreasing}.
On the interval $(1,+\infty)$, the derivative $f'(x)$ is {\it positive/negative} and $f(x)$ is {\it increasing/decreasing}.  
\item{d.}  What happens on $y = f(x)$ when $f'(x) = 0$? $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
\item{e.}  (Harder -- we have not discussed this!)  What happens on the graph $y = f(x)$ then the tangent line to $y = f'(x)$ is horizontal?  $\underline{\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad}$
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2.  Given the graph $y = f(x)$ below, make a {\it qualitative} sketch of $y = f'(x)$ on the empty set of axes below.  
Show any places where $f'(x)$ changes sign and places where $f'(x)$ fails to exist.  

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