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Module 2 Tutorial Exam Review Sheet

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\noindent {\bf A.} {\it Determine the derivative of each of the
following functions directly from the (limit) definition; i.e. the
``Three-Step Method."}

\begin{enumerate}

\item   $f(x) = 2x + 3$

\item   $f(x) = 5 - 2x$

\item   $f(x) = x^2$

\item   $f(x) = 2x^2 + 5x$

\item   $f(x) = x^3$

\item   $f(x) = \frac{1}{x}$

\item   $f(x) = \sqrt{x}$

\item   $f(x) = \frac{1}{\sqrt{x}}$

\item   $f(x) = \frac{1}{x^2}$

\item   $f(x) = \frac{5}{x^2} - 7$

\end{enumerate}

\newpage
\noindent {\bf B.} {\it Determine the derivative of each of the
following functions using the shortcuts we have studied. }

\begin{enumerate}

\item   $f(x) = 2x + 3$

\item   $C(x) = 2$

\item   $L(x) = e^\pi$

\item   $f(x) = \ln \left( \sin e^{3\pi} - \cos e^\pi \right)$

\item   $f(x) = 5 - 2x$

\item   $f(x) = \frac{1}{9} - \frac{7x}{4}$

\item   $f(x) = x^2$

\item   $f(x) = \sqrt{x}$

\item   $f(x) = \frac{1}{x}$

\item   $f(x) = \frac{4}{\sqrt{x}}$

\item   $f(x) = 2x^2 + 5x$

\item   $f(x) = x^{17}$

\item   $f(x) = \frac{8}{x}$

\item   $f(x) = \frac{-19}{x^2}$

\item   $f(x) = 4\sqrt{x} + 2$

\item   $f(x)   =   x^2 + 2x + 1$

\item   $g(x)   =   x^2 + 5x - 7$

\item   $h(x)   =   7x^9 - 12x^4$

\item   $j(t)   =   \frac{1}{t} + \frac{7}{t^2}$

\item   $f(x)   =   3\sqrt{x} - 9 \sqrt{x^7}$

\item   $f(x)   =   2^x$

\item   $L(x)   =   2^{2x}$

\item   $f(x)   =   5 \cdot 4^x$

\item   $g(x)   =   3e^x$

\item   $\Gamma (x)   =   e^{3x}$

\item   $f(t)   =   te^t$

\item   $f(x)   =   \frac{1}{e^x}$

\item   $f(x)   =   e^{1/x}$

\item   $T(x)   =   10^{3x}$

\item   $f(x)   =   \ln (5x)$

\item   $f(x)   =   \ln \sqrt{x}$

\item   $f(x)   =   \sin (3x)$

\item   $f(x)   =   x^2 \sin (3x)$

\item   $f(x)   =   3x\sin x$

\item   $f(x)   =   \cos (e^x)$

\item   $f(x)   =   \cos(x - 4) + (\sin x)(\ln x)$

\item   $f(x)   =   2\frac{\sqrt{x}}{x^4} + \frac{9}{\sqrt{x}}$

\item   $F(x)   =   (x^4 + 2x - 1)\sqrt{3x^2 - 2}$

\item   $f(t)   =   (4t^7 + 9t^2 - 11t + 5)\left( \frac{4}{t} + \frac{1}{\sqrt{t}}\right)$

\item   $K(x)   =   5\sqrt{x}(3x^2 - 4x + 2)$

\item   $f(x)   =   \frac{\sqrt{x}}{x^3} \left(x^{11} - 7x^4 \right)$

\item   $f(x)   =   e^x (5x^3 - 3x^2 + 1)$

\item   $f(x)   =   (2x^9 + 17x^2 - 14)\cos x$

\item   $f(x)   =   \sqrt{x} \cdot 6^x$

\item   $f(x)   =   11^x \cdot \frac{1}{\sqrt{x}}$

\item   $f(x)   =   (\ln x)(e^x)$

\item   $f(x)   =   e^x \cos x$

\item   $f(x)   =   (3x^4 - 15\sqrt{x})\sin (3x)$

\item   $f(x)   =   4^x \sin x$

\item   $f(x)   =   \cos e^x$

\item   $f(x)   =   \ln \sin x$

\item   $g(t)      =   \sin (\cos t)$

\item   $f(x)   =   \frac{\sin x}{\cos x}$

\item   $f(x)      = \frac{\sqrt{x}}{x^2 + 2x + 1}$

\item   $f(x)      =    \frac{e^x + \ln x}{2x + 1}$

\item   $f(x)      =   \frac{\ln x}{\sqrt{x}}$

\item   $f(x)   =  \frac{11^x}{\sqrt{x}}$

\item   $f(x)      =    \frac{x^e + 11x^2}{\sin x}$

\item   $f(x)      =    \frac{\ln x}{e^x + x^4 + 1}$

\item   $f(x)      =    \frac{x}{e^x + x^4 + 1}$

\item   $f(x)      =    (x^4 + 7x^9 + 1)^{15}$

\item   $f(x)      =    (x^4 + 7x^9 + 1)^{-15/2}$

\item   $f(x)      =    \sin (x^4 + 7x^9 + 1)$

\item   $S(t)       =   2e^{t^2 + 3t - 1}$

\item   $S(t)      =   te^{t^2 + 3t - 1}$

\item   $G(p)       =   \frac{p^4 + 3p^3 + 1}{p^5 - 2p^2 + 2} $

\item   $R(\theta)   =   \sin^2 \theta + \cos^2 \theta$.

\end{enumerate}

\newpage
\noindent {\bf C.} {\it    Find an equation of the tangent line to
each curve at the point $(a, f(a))$ for each given value of $a$.}

\begin{enumerate}

\item   $f(x)   =   \ln (\ln (x))$ with $a = 1, e, e^2$.

\item   $f(x)   =   \cos x$ with $a = 2\pi, 3\pi, 4\pi$.

\item   $f(x)   =   \sin x$ with $x = \frac{\pi}{2}, \pi, \frac{3\pi}{2}, 2\pi$.

\item   $f(x)   =   2x^4 + 3x^3 + 4x^2 + 5x + 6$ with $a = 1, 2, 5$.

\item   $f(x)   =   e^{3x}$ with $a = 0, 1, 2$.

\end{enumerate}

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