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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Trigonometric Derivatives}
\centerline{November 1, 2017}
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\noindent
{\it Background}
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In today's video, we saw how the addition formulas for $\sin(x)$ and $\cos(x)$, combined
with some trigonometric limits we saw back in Chapter 2 of the text lead
to the derivative formulas:
$$\frac{d}{dx} \sin(x) = \cos(x) \qquad {\rm and} \qquad \frac{d}{dx} \cos(x) = -\sin(x).$$
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\noindent
{\it Questions}
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\noindent
\begin{enumerate}
\item[(1)]  For each function, use the appropriate short-cut rules to find the first derivative, and then differentiate again to get the second derivative:
\begin{enumerate}
\item[(a)]  $f(x) = \displaystyle{3\sin(x) + 4\cos(x)}$. 
\item[(b)]  $g(x) = \cot(x) = \displaystyle{\frac{\cos(x)}{\sin(x)}}$.  Your life will 
be a lot easier here if you simplify the first derivative \emph{before differentiating again} 
to get $g''(x)$.  
\item[(c)]  $h(x) = \displaystyle{\sin(x) e^x}$.   Also find the third derivative $h'''(x)$ for 
this one.
\end{enumerate}
\item[(2)]  Consider the graph $y = x - \sin(x)$ on the back of this sheet.
\begin{enumerate}
\item[(a)] Do the tangent lines to this graph ever have a negative slope?  Why or why not?
\item[(b)] Do the tangent lines ever have zero slope?  Where does that happen?
\item[(c)] Where do the tangent lines have the steepest positive slope?  For which $x$
does that happen?
\item[(d)] Sketch the graph $y = x - \sin(x)$ and check your work with a graphing
calculator if you have one (or if one of your classmates can share theirs).
\end{enumerate}
\end{enumerate}
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