calculus:resources:calculus_flipped_resources:derivatives:2.4_trig_derivatives_tex [Department of Mathematics and Statistics, Binghamton University]

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[[http://www2.math.binghamton.edu/lib/exe/fetch.php/calculus/resources/calculus_flipped_resources/arc_chord_limit.png|arc_chord_limit.png]]\\ \\ 


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\begin{document}

\begin{frame}
\large 
\begin{block}{}
\begin{center}$\displaystyle\lim_{x\rightarrow 0}\dfrac{\sin(x)}{x} = 1 
\hskip 15pt \displaystyle\lim_{x\rightarrow 0}\dfrac{\cos(x)-1}{x} = 0$\end{center}
\end{block}
Evaluate the limits:
\vskip 5pt

\begin{columns}
\begin{column}{0.5\textwidth}
$$\lim_{x\rightarrow 0}\dfrac{\sin(6x)}{\sin(9x)}$$
\vskip 15pt
$$\lim_{\theta\to 0}\dfrac{\cos(7\theta)-1}{sin(9\theta)}$$
\vskip 15pt
$$\lim_{t\to 0}\dfrac{\tan(16t)}{\sin(4t)}$$
\end{column}
\begin{column}{0.5\textwidth}
$$ \lim_{x\to 0}\dfrac{\sin(x^7)}{x}$$
\vskip 15pt
$$ \lim_{x\to 2}\dfrac{\sin(x-2)}{x^2+6x-16}$$
\vskip 15pt
$$ \lim_{x\rightarrow 2}\dfrac{3-3\tan(x)}{\sin(x)-\cos(x)} $$
\end{column}
\end{columns}
\end{frame}



\begin{frame}
\LARGE
Use both the derivatives of $\sin(x)$ and $\cos(x)$ and the quotient rule to show:
$$\frac{d}{dx}\left( \tan(x)\right)=\sec^2(x)$$
and
$$\frac{d}{dx}\left( \sec(x)\right)=\sec(x)\tan(x)$$
\end{frame}

\begin{frame}
\LARGE
\begin{columns}
\begin{column}{0.5\textwidth}
Find $f'(x)$:
$$f(x)=5x^2+7\sin(x)$$
\vskip 20pt
Find $g'(\theta)$:
$$g(\theta)=\sec(\theta)\tan(\theta)$$
\end{column}

\begin{column}{0.5\textwidth}
Find $F'(x)$:
$$F(x)=\dfrac{3-\sec(x)}{\tan(x)}$$
\vskip 20pt
Evaluate:
$$\frac{d^2}{d\theta^2}\left(\theta\sin(\theta)\right)$$
\end{column}
\end{columns}
\end{frame}

\begin{frame}
\LARGE
Find the equation of the tangent line to the curve $y=14x\sin(x)$ at $x=\pi/2$.
\vskip 55pt

Find $$\frac{d^{103}}{dx^{103}}\left(\sin(x)\right) \mbox{ and } 
\frac{d^{201}}{dx^{201}}\left(\cos(x)\right)$$
\end{frame}

\begin{frame}
\emph{Step into your teacher's shoes. What is wrong (if anything)
with the following calculations? Explain any errors and correct
for them.}
\vskip 5pt

\begin{block}{}
Find all values of x in the interval $[0,4\pi)$ that satisfy the equation
$$\sin(2x) = \cos(x).$$
{\bf Solution:} Since $\sin(2x) = 2\sin(x)\cos(x)$,
$$ \sin(2x) = \cos(x) \hskip 20pt \Rightarrow \hskip 20pt 
2\sin(x)\begin{cancel}\cos(x)\end{cancel} = \begin{cancel}\cos(x)\end{cancel} $$
$$\Rightarrow \hskip 20pt 2\sin(x) = 1 \hskip 20pt \Rightarrow \hskip 20pt \sin(x) = \frac{1}{2}$$
Therefore, $x = \frac{\pi}{3}$ or $x=\frac{5\pi}{3}$.
\end{block}

\end{frame}

\begin{frame}
What does $\displaystyle\lim_{x\rightarrow 0} \frac{\sin(x)}{x} = 1$ mean? 
Explain why three of the options are false and one is true.
\begin{itemize}
\item[a)] $\frac{0}{0} = 1$.
\item[b)] The tangent to the graph of $y =\sin(x)$ at $(0,0)$ is the line $y=x$.
\item[c)] You can cancel the x's.
\item[d)] $\sin(x) = x$ for x near 0.
\end{itemize}
\end{frame}

\begin{frame}
The figure shows a circular arc of length $s$ and a chord of length $d$, 
both subtended by a central angle $\theta$. Find 
$\displaystyle\lim_{\theta\rightarrow 0^+} \frac{s}{d}$.
\begin{center}
\includegraphics[height=2in]{arc_chord_limit.png}
\end{center}
It may be helpful to review the formulas associated to arcs and isosceles triangles. 
Further, why would the limit $\displaystyle\lim_{\theta\rightarrow 0} \frac{s}{d}$ not exist?
\end{frame}

\end{document} 
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