\documentclass{article} \usepackage{graphicx} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{enumerate} \usepackage{ifthen} \usepackage{ulem} % for strikethrough \sout{...} command \usepackage{color} \setlength{\oddsidemargin}{-0.5in} \setlength{\evensidemargin}{-0.5in} \setlength{\topmargin}{-0.5in} \setlength{\headheight}{0.0in} \setlength{\headsep}{0.0in} \setlength{\topskip}{0.0in} \setlength{\textheight}{10in} \setlength{\textwidth}{7.5in} \usepackage{pgfplots} \pgfplotsset{compat=1.14} \usepackage{tikz} \usepgfplotslibrary{fillbetween} %\newcommand{\bdot}{\, {\bf \cdot} \,} \newcommand{\bdot}{\, {\scriptstyle{\bullet}} \,} \newcommand{\B}[1]{{\bf {#1}}} \newcommand{\la}{\langle} \newcommand{\ra}{\rangle} \newcommand{\force}{{$\mbox{\;}$}} \newcommand{\myspace}{\mbox{\parbox[c][0.4in]{0.4in}{\vfill}}} % height then width \newcommand{\varspace}[2]{\mbox{\parbox[c][#1]{#2}{\vfill}}} \begin{document} \pagestyle{empty} \noindent \parbox{2in}{\bf Math 1120 Form A} \hfill {\Large \bf Test \#2} \hfill \parbox{2in}{\bf \hfill Oct. $5^\mathrm{th}$, 2021} \vspace{0.15in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{-0.1in} \begin{itemize} \item Double angle: \qquad $\cos^2(x) = \dfrac{1}{2}\left(1+\cos(2x)\right)$, \quad $\sin^2(x) = \dfrac{1}{2}\left(1-\cos(2x)\right)$, \quad $\sin(2x)=2\sin(x)\cos(x)$ \item Angle sum and difference: \quad $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$, \quad $\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$,\\ \force \hspace{1.675in} $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$, \quad $\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$ \item Product to sum: \qquad $2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)$, \quad $2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)$,\\ \force \hspace*{1.25in} $2\sin(x)\cos(y) = \sin(x-y)+\sin(x+y)$ \end{itemize} \noindent {\bf \large 1. (25 points)} Pythagorean Theorem \begin{enumerate}[(a)] \item Write down the Pythagorean Theorem trigonometric identity relating sine and cosine. Then write down the other version relating tangent and secant. \vspace{0.25in} Identity \#1: \quad \underline{\hspace*{2.25in}} \qquad \qquad Identity \#2: \quad \underline{\hspace*{2.25in}} \vspace{0.1in} \item In each of the following integrals, use the proper (inverse) trigonometric substitution to simplify the integral. {\bf Do NOT} integrate. Just substitute and {\bf simplify}. \begin{itemize} \item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+9\strut}}$ %\item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+4\strut}}$ \vspace{1in} %\item $\displaystyle \int \dfrac{\sqrt{x^2-9\strut}}{x}\,dx$ \item $\displaystyle \int \dfrac{\sqrt{x^2-4\strut}}{x}\,dx$ \vspace{1in} \item $\displaystyle \int_0^5 \sqrt{25-x^2}\,dx$ %\item $\displaystyle \int_0^{4} \sqrt{16-x^2}\,dx$ \vspace{1in} \end{itemize} \item Given $x=3\sin(\theta)$, rewrite $7\theta-4\tan(\theta)$ in terms of $x$. %\item Given $x=5\sin(\theta)$, rewrite $4\theta-7\tan(\theta)$ in terms of $x$. \vspace{0.75in} \end{enumerate} \noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x(x+6)^3(x^2+x+3)^2}$. %\noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x^3(x+7)(x^2+x+11)^2}$. \newpage \noindent {\bf \large 3. (15 points)} Complete the Square \begin{enumerate}[(a)] \item To integrate $\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate. %\item To integrate $\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate. \vspace{0.5in} $\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}} %$\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}} \vspace{0.75in} \item Compute \quad $\displaystyle \int \dfrac{3}{x^2-4x+13}\,dx$ %\item Compute \quad $\displaystyle \int \dfrac{2}{x^2-6x+13}\,dx$ \vfill \end{enumerate} \noindent {\bf \large 4. (18 points)} Integrate! \begin{enumerate}[(a)] \item $\displaystyle \int x^2e^{4x}\,dx$ %\item $\displaystyle \int x^2e^{-x}\,dx$ \vfill %\item $\displaystyle \int_0^{\pi/2} \sin(3x)\sin(2x)\,dx$ %\item $\displaystyle \int_0^{\pi/2} \sin(4x)\sin(3x)\,dx$ %\vfill \item $\displaystyle \int \sin^3(x)\cos^4(x)\,dx$ %\item $\displaystyle \int \sin^4(x)\cos^3(x)\,dx$ \vfill \item $\displaystyle \int_0^1 \mathrm{arctan}(x)\,dx$ \vfill \end{enumerate} \newpage \noindent {\bf \large 5. (34 points)} Integrate! \begin{enumerate}[(a)] \item $\displaystyle \int e^x\sin(3x)\,dx$ %\item $\displaystyle \int e^x\cos(3x)\,dx$ \vfill \item $\displaystyle \int \tan^5(x)\sec^4(x)\,dx$ %\item $\displaystyle \int \tan^3(x)\sec^6(x)\,dx$ \vfill \item $\displaystyle \int \sqrt{4-x^2}\,dx$ %\item $\displaystyle \int \sqrt{9-x^2}\,dx$ \vfill \item $\displaystyle \int \dfrac{2x^2+7x+4}{x(x+1)^2}\,dx$ %\item $\displaystyle \int \dfrac{3x^2+1}{x(x-1)^2}\,dx$ \vfill \end{enumerate} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \newpage %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \noindent \parbox{2in}{\bf Math 1120 Form B} \hfill {\Large \bf Test \#2} \hfill \parbox{2in}{\bf \hfill Oct. $5^\mathrm{th}$, 2021} \vspace{0.15in} \noindent {\large\bf Name:} \underline{\hskip 3.0 truein} \hfill {\bf Be sure to show your work!} \vspace{-0.1in} \begin{itemize} \item Double angle: \qquad $\cos^2(x) = \dfrac{1}{2}\left(1+\cos(2x)\right)$, \quad $\sin^2(x) = \dfrac{1}{2}\left(1-\cos(2x)\right)$, \quad $\sin(2x)=2\sin(x)\cos(x)$ \item Angle sum and difference: \quad $\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$, \quad $\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$,\\ \force \hspace{1.675in} $\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$, \quad $\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$ \item Product to sum: \qquad $2\sin(x)\sin(y)=\cos(x-y)-\cos(x+y)$, \quad $2\cos(x)\cos(y)=\cos(x-y)+\cos(x+y)$,\\ \force \hspace*{1.25in} $2\sin(x)\cos(y) = \sin(x-y)+\sin(x+y)$ \end{itemize} \noindent {\bf \large 1. (25 points)} Pythagorean Theorem \begin{enumerate}[(a)] \item Write down the Pythagorean Theorem trigonometric identity relating sine and cosine. Then write down the other version relating tangent and secant. \vspace{0.25in} Identity \#1: \quad \underline{\hspace*{2.25in}} \qquad \qquad Identity \#2: \quad \underline{\hspace*{2.25in}} \vspace{0.1in} \item In each of the following integrals, use the proper (inverse) trigonometric substitution to simplify the integral. {\bf Do NOT} integrate. Just substitute and {\bf simplify}. \begin{itemize} %\item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+9\strut}}$ \item $\displaystyle \int \dfrac{dx}{x^2\sqrt{x^2+4\strut}}$ \vspace{1in} \item $\displaystyle \int \dfrac{\sqrt{x^2-9\strut}}{x}\,dx$ %\item $\displaystyle \int \dfrac{\sqrt{x^2-4\strut}}{x}\,dx$ \vspace{1in} %\item $\displaystyle \int_0^5 \sqrt{25-x^2}\,dx$ \item $\displaystyle \int_0^{4} \sqrt{16-x^2}\,dx$ \vspace{1in} \end{itemize} %\item Given $x=3\sin(\theta)$, rewrite $7\theta-4\tan(\theta)$ in terms of $x$. \item Given $x=5\sin(\theta)$, rewrite $4\theta-7\tan(\theta)$ in terms of $x$. \vspace{0.75in} \end{enumerate} %\noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x(x+6)^3(x^2+x+3)^2}$. \noindent {\bf \large 2. (8 points)} Write down the ``forms'' we would use to find the partial fraction decomposition of \quad $\displaystyle \dfrac{3x^5-x^4+9x^2+8}{x^3(x+7)(x^2+x+11)^2}$. \newpage \noindent {\bf \large 3. (15 points)} Complete the Square \begin{enumerate}[(a)] %\item To integrate $\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate. \item To integrate $\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ we would need to split the integral into a $u$-substitution integral and an integral where we complete the square in the radical. Split this integral into those two pieces. {\bf Do NOT} integrate. \vspace{0.5in} %$\displaystyle \int \dfrac{6x}{\sqrt{1+3x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}} $\displaystyle \int \dfrac{10x}{\sqrt{1+4x-x^2}}\,dx$ \quad $=$ \quad \underline{\hspace*{2in}} \quad $+$ \quad \underline{\hspace*{2in}} \vspace{0.75in} %\item Compute \quad $\displaystyle \int \dfrac{3}{x^2-4x+13}\,dx$ \item Compute \quad $\displaystyle \int \dfrac{2}{x^2-6x+13}\,dx$ \vfill \end{enumerate} \noindent {\bf \large 4. (18 points)} Integrate! \begin{enumerate}[(a)] %\item $\displaystyle \int x^2e^{4x}\,dx$ \item $\displaystyle \int x^2e^{-x}\,dx$ \vfill %\item $\displaystyle \int_0^{\pi/2} \sin(3x)\sin(2x)\,dx$ %\item $\displaystyle \int_0^{\pi/2} \sin(4x)\sin(3x)\,dx$ %\vfill %\item $\displaystyle \int \sin^3(x)\cos^4(x)\,dx$ \item $\displaystyle \int \sin^4(x)\cos^3(x)\,dx$ \vfill \item $\displaystyle \int_0^1 \mathrm{arctan}(x)\,dx$ \vfill \end{enumerate} \newpage \noindent {\bf \large 5. (34 points)} Integrate! \begin{enumerate}[(a)] %\item $\displaystyle \int e^x\sin(3x)\,dx$ \item $\displaystyle \int e^x\cos(3x)\,dx$ \vfill %\item $\displaystyle \int \tan^5(x)\sec^4(x)\,dx$ \item $\displaystyle \int \tan^3(x)\sec^6(x)\,dx$ \vfill %\item $\displaystyle \int \sqrt{4-x^2}\,dx$ \item $\displaystyle \int \sqrt{9-x^2}\,dx$ \vfill %\item $\displaystyle \int \dfrac{2x^2+7x+4}{x(x+1)^2}\,dx$ \item $\displaystyle \int \dfrac{3x^2+1}{x(x-1)^2}\,dx$ \vfill \end{enumerate} \end{document}