\hfill \hfill } %} \newtheorem{theorem}{Theorem} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][Proof]{\noindent\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \geometry{left=0.4in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\Large \color{blue} Lecture Notes} \chead{ \LARGE Are you ready for calculus 2?} \rhead{\large page \thepage} \lfoot{\small \copyright $\;$ Hidegkuti 2012} \rfoot{\small Last revised: January 2, 2015} \cfoot{} \textwidth 7.6in \textheight 9.6in \setlength{\headheight}{27pt} \setlength{\parindent}{0pt} \begin{document} \begin{enumerate} \item Differentiate each of the following.% %TCIMACRO{\TeXButton{3 col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion a) \ $f\left( x\right) =3x^{4}-x^{3}+4x^{2}-x+7$\medskip b) \ $f\left( x\right) =\sqrt[5]{2x^{6}+5x^{2}+1}$\medskip c) \ $f\left( x\right) =\cos x+x\sin x$\medskip d) \ $f\left( x\right) =\cos \left( 2x-\dfrac{\pi }{2}\right) $\medskip e) \ $f\left( x\right) =\left( 3x-1\right) ^{100}$\medskip f) \ $f\left( x\right) =\dfrac{-x+5}{\sqrt{x^{2}+1}}$\medskip g) \ $f\left( x\right) =\dfrac{x^{2}+1}{\sin 5x}$\medskip h) \ $f\left( x\right) =\cot \left( \dfrac{\pi }{2}x\right) $\medskip i) \ $f\left( x\right) =\sqrt{1+\sqrt{x}}$\medskip j) \ $f\left( x\right) =\tan 3x$\medskip k) \ $f\left( x\right) =\tan \left( 3x^{2}\right) $\medskip l) \ $f\left( x\right) =\sec x+\tan x$\medskip\ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item Find the exact value of each of the following. \ %TCIMACRO{\TeXButton{3 col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion a) \ $\sin \left( \dfrac{-7\pi }{3}\right) $ b) \ $\sin \left( \cos ^{-1}\left( -\dfrac{1}{2}\right) \right) $ c) \ $\tan ^{-1}\left( \tan \left( \dfrac{7\pi }{3}\right) \right) $ \ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item What is the exact value of $\sin x$ if $\tan x=-2$? \item Assume that for all real numbers $x$ and $y,$% \begin{eqnarray*} \sin ^{2}x+\cos ^{2}x &=&1\text{ \ } \\ \sin \left( x+y\right) &=&\sin x\cos y+\cos x\sin y\text{ \ \ and \ } \\ \cos \left( x+y\right) &=&\cos x\cos y-\sin x\sin y \end{eqnarray*}% Prove each of the following.% %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion a) \ $\sin \left( x-y\right) =\sin x\cos y-\cos x\sin y$ b) \ $\cos 2x=2\cos ^{2}x-1$ c) \ $\cos 2x=1-2\sin ^{2}x$ d) \ $\sin x=\pm \sqrt{\dfrac{1-\cos 2x}{2}}$ e) \ $\tan \left( x+y\right) =\dfrac{\tan x+\tan y}{1-\tan x\tan y}$ f) \ $\sec ^{2}x=1+\tan ^{2}x$ g) \ $\cos ^{2}x=\dfrac{1}{2}\left( \cos 2x+1\right) $ h) \ $\dfrac{\sin x}{1+\cos x}=\dfrac{1-\cos x}{\sin x}$ \ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item Simplify each of the following% %TCIMACRO{\TeXButton{4 col begin}{\begin{multicols}{4}}}% %BeginExpansion \begin{multicols}{4}% %EndExpansion a) \ $\sin \left( \sin ^{-1}x\right) $ b) \ $\cos \left( \sin ^{-1}x\right) $ c) \ $\sin \left( \tan ^{-1}x\right) $ d) $\tan \left( \cos ^{-1}x\right) $ \ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item Prove that the following expressions are all equivalent. %TCIMACRO{\TeXButton{5 col begin}{\begin{multicols}{5}}}% %BeginExpansion \begin{multicols}{5}% %EndExpansion $A=\sqrt{\dfrac{1+\sin x}{1-\sin x}}$ $B=\dfrac{1+\sin x}{\cos x}$ $C=\sec x+\tan x$ $D=\dfrac{\cos x}{1-\sin x}$ $E=\dfrac{1}{\sec x-\tan x}$ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion (Hint: prove that $A=B$ and that $B=C$ and that $B=D$ and then $D=E$) \item Prove each of the following.% %TCIMACRO{\TeXButton{2 col begin}{\begin{multicols}{2}}}% %BeginExpansion \begin{multicols}{2}% %EndExpansion a) \ $\log _{24}90=\dfrac{\ln 2+2\ln 3+\ln 5}{3\ln 2+\ln 3}$ b) \ $2\log _{10}\left( 2x\right) +\log _{10}\left( 25x\right) -3\log _{10}\left( 0.1x\right) =5$ c) \ $\log _{2}3\cdot \log _{3}4\cdot \log _{4}5\cdot \log _{5}6\cdot \log _{6}7\cdot \log _{7}8=3$ d) \ $\log _{3}\left\vert \tan x\right\vert =-\log _{3}\left\vert \cot x\right\vert $ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \pagebreak \item Compute each of the following limits.% %TCIMACRO{\TeXButton{4 col begin}{\begin{multicols}{4}}}% %BeginExpansion \begin{multicols}{4}% %EndExpansion a) \ $\lim\limits_{x\rightarrow \infty }\dfrac{\sin 5x}{x}$\medskip b) \ $\lim\limits_{x\rightarrow 0}\dfrac{\sin 5x}{x}$\medskip c) \ $\lim\limits_{x\rightarrow 0}\dfrac{\sqrt{9+x}-3}{x}$\medskip d) \ $\lim\limits_{x\rightarrow 2}\dfrac{\dfrac{1}{x}-\dfrac{1}{2}}{x-2}$% \medskip e) \ $\lim\limits_{x\rightarrow 2}\dfrac{1}{x^{2}-4}$\medskip f) \ $\lim\limits_{x\rightarrow 2^{-}}\dfrac{1}{x^{2}-4}$\medskip g) \ $\lim\limits_{x\rightarrow 0}\dfrac{\cos x+1}{x-\pi }$\medskip h) \ $\lim\limits_{x\rightarrow 0^{+}}\log _{3}x$\medskip i) \ \ $\lim\limits_{x\rightarrow \infty }\tan ^{-1}x$\medskip j) \ $\lim\limits_{x\rightarrow 1^{-}}\dfrac{x^{2}+x-2}{x^{2}-1}$\medskip k) \ $\lim\limits_{x\rightarrow -1^{-}}\dfrac{x^{2}+x-2}{x^{2}-1}$\medskip l) \ $\lim\limits_{x\rightarrow \infty }\dfrac{x^{2}+x-2}{x^{2}-1}$\medskip\ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item Use implicit differentiation to differentiate each of the following.% %TCIMACRO{\TeXButton{3 col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion a) $\left( x^{2}-y^{2}\right) ^{4}=2xy^{2}$ b) $\left( x+y\right) ^{3}=\sin x-\sin y$ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item Prove that $\dfrac{d\left( x^{2}-x\right) }{dx}=2x-1$, using the definition of the derivative as the limit of the differential quotient. \item Prove that if $f\left( x\right) =\sin ^{-1}x$ then $f^{\prime }\left( x\right) =\dfrac{1}{\sqrt{1-x^{2}}}$ \item Evaluate each of the following integrals.% %TCIMACRO{\TeXButton{3 col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion a) \ $\dint \dfrac{1}{x^{2}+1}~dx$\medskip b) \ $\dint \dfrac{x^{2}}{x^{2}+1}~dx$\medskip c) \ $\dint \sin 5x~dx$\medskip d) \ $\dint \dfrac{1}{-x+2}~dx$\medskip e) \ $\dint \left( 3x-1\right) ^{10}~dx$\medskip f) \ $\dint \dfrac{1}{\sqrt{1-9x^{2}}}~dx$\medskip g) \ $\dint \dfrac{1}{x-5}~dx$\medskip h) \ $\dint 3-\dfrac{2}{x-5}~dx$\medskip i) \ $\dint \dfrac{3x-17}{x-5}~dx$\medskip\ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \item Express each of the following in terms of the variable given. a) \ Let $a$ be the side of a regular triangle. \ Express its area in terms of $a$. b) \ Let $h$ be the height of a regular triangle. \ Express its area in terms of $h$. c) \ Let $A$ be the area of a square. \ Express its perimeter in terms of $% A. $ d) \ Let $c$ be the hypotenuse of an isosceles right triangle. \ We write a semicircle on each of its sides as shown on the picture below. \ Express the area of the shaded region in terms of $c$.\FRAME{dtbpF}{1.7988in}{1.7824in}{% 0pt}{}{}{pic1.bmp}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file "F";width 1.7988in;height 1.7824in;depth 0pt;original-width 0.0683in;original-height 0.0675in;cropleft "0";croptop "1.0011";cropright "1";cropbottom "0";filename 'pic1.bmp';file-properties "XNPEU";}} \end{enumerate} \pagebreak \begin{center} {\LARGE Answers\bigskip } \end{center} \begin{enumerate} \item a) \ $f^{\prime }\left( x\right) =12x^{3}-3x^{2}+8x-1$ \ \ \ \ b) \ \ $% f^{\prime }\left( x\right) =\dfrac{12x^{5}+10x}{5\left( 2x^{6}+5x^{2}+1\right) ^{4/5}}=\dfrac{1}{5}\dfrac{12x^{5}+10x}{% 2x^{6}+5x^{2}+1}\sqrt[5]{2x^{6}+5x^{2}+1}\medskip $ c) \ $f^{\prime }\left( x\right) =x\cos x$ \ \ \ \ \ \ d) \ $f^{\prime }\left( x\right) =-2\sin \left( 2x-\dfrac{\pi }{2}\right) $ \ \ \ \ \ \ \ \ e) \ $f^{\prime }\left( x\right) =300\left( 3x-1\right) ^{99}\medskip $ f) \ $f^{\prime }\left( x\right) =\dfrac{-5x-1}{\left( x^{2}+1\right) ^{3/2}} $ \ \ \ \ \ \ \ \ \ g) $\ f^{\prime }\left( x\right) =\dfrac{2x\sin 5x-5\left( x^{2}+1\right) \cos 5x}{\sin ^{2}5x}\medskip $ h) \ $f^{\prime }\left( x\right) =-\dfrac{\pi }{2}\csc ^{2}\left( \dfrac{\pi }{2}x\right) =-\dfrac{\pi }{2}\left( \cot ^{2}\dfrac{\pi }{2}x+1\right) $ \ \ \ \ \ \ \ \ \ i) \ $f^{\prime }\left( x\right) =\dfrac{1}{4\sqrt{x}\sqrt{% \sqrt{x}+1}}\medskip $ j) \ $f^{\prime }\left( x\right) =3\sec ^{2}3x=3\tan ^{2}3x+3\medskip $ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k) \ $f^{\prime }\left( x\right) =6x\left( \tan ^{2}\left( 3x^{2}\right) +1\right) =6x\sec ^{2}\left( 3x^{2}\right) $ l) \ $f^{\prime }\left( x\right) =\sec x\tan x+\sec ^{2}x=\tan ^{2}x+1+% \dfrac{\sin x}{\cos ^{2}x}$ \item a) $-\dfrac{\sqrt{3}}{2}$ \ \ \ \ \ \ b) \ $\dfrac{\sqrt{3}}{2}$ \ \ \ \ \ \ \ \ \ c) \ $\dfrac{\pi }{3}$ \item $\pm \dfrac{2\sqrt{5}}{5}$ \item see solutions \item a) \ $x$ \ \ \ \ \ b) \ $\sqrt{1-x^{2}}$ \ \ \ \ c) \ $\dfrac{x}{\sqrt{% x^{2}+1}}$ \ \ \ \ \ \ \ d) \ $\dfrac{\sqrt{1-x^{2}}}{x}$ \item see solutions \item see solutions \item a) \ $0$ \ \ \ \ b) \ $5$ \ \ \ c) \ $\dfrac{1}{6}$ \ \ \ \ d) \ $-% \dfrac{1}{4}$ \ \ \ \ e) \ $\func{undefined}$ \ \ \ \ f) \ $-\infty $ \ \ \ \ g) $\ -\dfrac{2}{\pi }$ \ \ \ \ h) $\ -\infty $ \ \ \ \ i) \ \ $\dfrac{\pi }{2}$ \ \ \ \ j) \ $\dfrac{3}{2}$ \ \ \ \ k) \ $-\infty $ \ \ \ l) \ $1$ \item a) $y^{\prime }=\dfrac{-y^{2}+4x\left( x^{2}-y^{2}\right) ^{3}}{% 2xy+4y\left( x^{2}-y^{2}\right) ^{3}}$ \ \ \ \ \ \ \ \ \ \ b) \ $y^{\prime }=% \dfrac{\cos x-3\left( x+y\right) ^{2}}{\cos y+3\left( x+y\right) ^{2}}$ \item see solutions \item see solutions \item a) \ $\tan ^{-1}x+C$ \ \ \ \ \ b) \ $x-\tan ^{-1}x+C$ \ \ \ \ \ \ \ \ \ c) \ $-\dfrac{1}{5}\cos 5x+C$ \ \ \ \ \ \ \ \ \ d) \ $-\ln \left\vert -x+2\right\vert +C$ e) \ $\dfrac{\left( 3x-1\right) ^{11}}{33}+C$ \ \ \ \ \ \ f) \ $\dfrac{1}{3}% \sin ^{-1}\left( 3x\right) +C$ \ \ \ \ \ \ \ \ \ g) \ $\ln \left\vert x-5\right\vert +C$ \ \ \ \ \ \ \ \ h) \ $3x-2\ln \left\vert x-5\right\vert +C $ i) \ $3x-2\ln \left\vert x-5\right\vert +C$ Solution for b): \ \begin{equation*} \dint \dfrac{x^{2}}{x^{2}+1}dx=\dint \dfrac{x^{2}+1-1}{x^{2}+1}dx=\dint \dfrac{x^{2}+1}{x^{2}+1}-\dfrac{1}{x^{2}+1}dx=\dint 1-\dfrac{1}{x^{2}+1}% dx=\dint 1dx-\dint \dfrac{1}{x^{2}+1}dx=x-\tan ^{-1}x+C \end{equation*} \item a) \ $\dfrac{\sqrt{3}}{4}a^{2}$ \ \ \ \ \ \ \ \ b) \ $\dfrac{\sqrt{3}}{% 3}h^{2}$ \ \ \ \ \ \ \ \ \ c) \ $4\sqrt{A}$ \ \ \ \ d) \ $\dfrac{1}{4}\pi c^{2}$ \end{enumerate} {\small %TCIMACRO{\TeXButton{\small}{\small}}% %BeginExpansion \small% %EndExpansion } \href{https://teaching.martahidegkuti.com/shared/lnotes/lecturenotes.html}{% For more documents like this, visit our page at\ https://teaching.martahidegkuti.com and click on Lecture Notes. \ E-mail questions or comments to mhidegkuti@ccc.edu.} \end{document}