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\centerline{Mathematics 133 -- Intensive Calculus for Science 1}
\centerline{Discussion 5 -- Derivative ``Shortcuts,'' I}
\centerline{October 19, 2005}
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{\it Background}
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Last time, we saw the derivative power rule:  For all real $n$,
$${d\over dx}(x^n) = n x^{n-1}$$
and the Constant Multiple and Sum Rules:  If $f(x)$ and $g(x)$
are differentiable at $x$ and $c$ is constant, then $cf(x)$
and $f(x) + g(x)$ are also differentiable at $x$ and 
$${d\over dx}(cf(x)) = cf'(x), \qquad {d\over dx}(f(x) + g(x)) = f'(x) + g'(x).$$
Today, we want to practice using these to compute derivatives of polynomials
and make use of the general facts about derivatives we know from Chapter 2.
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{\it Discussion Questions}
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A)  Compute the derivatives of the following functions:
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\item{1)}  $f(x) = 7x^4 - 2x^{1/2} + 3 x^{-2} + 1$
\item{2)}  $g(x) = x^{4/3} - {1\over 8\sqrt{x}} + x$
\item{3)}  $h(x) = x^e + \pi x^{\sqrt{2}} - {1\over x^\pi}$
\item{4)}  
$$k(x) = {5x^2 - 2x + 3\over x^{1/3}}$$
(Hint: Split up into a sum of fractions and simplify.  Then
compute the derivative!)
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B)  Consider $f(x) = x^4 - 4x^3$.
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\item{1)}  Find $f'(x)$ and $f''(x)$ and factor each of them.
\item{2)}  Find all $x$ where $f'(x) = 0$, and determine the 
intervals where $f'(x) > 0$ and the intervals where $f'(x) < 0$.
\item{3)}  Find all $x$ where $f''(x) = 0$, and determine the 
intervals where $f''(x) > 0$ and the intervals where $f''(x) < 0$.
\item{4)}  Find the values of $f(x)$ at the $x$'s from parts 2,3.
Using those points and the other information from parts 2,3, sketch
the graph $y = x^4 - 4x^3$, indicating correct concavities.
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C)  Let $P = (a,a^2-2a+4)$ be a general point on the parabola
$y = x^2 - 2x + 4$. 
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\item{1)}  Find the equation of the {\it tangent line} to 
the parabola at $P$.  (Give your answer in slope-intercept 
form.  Both the slope and the intercept will depend on $a$, since
the tangent lines are different depending on what $a$ we pick.)
\item{2)}  How many different $a$'s are there where the tangent
line passes through $(0,0)$.  Find them all.
\item{3)}  Graph the parabola $y = x^2 - 2x + 4$ together with 
the tangent lines passing through $(0,0)$.
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