\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=0.5in,right=0.5in,top=0.5in,bottom=0.4in} \pagestyle{fancy} \lhead{\color{blue} Math 207} \lfoot{\small \copyright $\;$ Hidegkuti, 2013} \cfoot{} \chead{\large Antiderivatives} \rfoot{\footnotesize Last revised: October 31, 2017} \rhead{\footnotesize page \thepage } \textwidth 7.5in \textheight 9.5in \setlength{\headheight}{26pt} \setlength{\parindent}{0pt} \begin{document} Compute each of the following antiderivatives \begin{enumerate} \item %TCIMACRO{\TeXButton{3 col begin}{\begin{multicols}{3}}}% %BeginExpansion \begin{multicols}{3}% %EndExpansion $\dint x^{2}dx$ \item $\dint \left( x^{5}-2x^{4}+3\right) ~dx$ \item $\dint \sqrt{y}~dy$ \item $\dint ab^{2}x^{3}~dx$ \item $\dint ab^{2}x^{3}~da$ \item $\dint ab^{2}x^{3}~db$ \item $\dint 1~dx$ \item $\dint \sin \theta ~d\theta $ \item $\dint \cos \alpha ~d\alpha $ \item $\dint -10~dt$ \item $\dint \left( -10t+v_{0}\right) ~dt$ \item $\dint \left( \dfrac{1}{m^{2}}-\sqrt{m}\right) ~dm$ \item $\dint \left( -\dfrac{3}{\sqrt{y}}-\sqrt{y}+2xy\right) ~dx$ \item $\dint \left( -\dfrac{3}{\sqrt{y}}-\sqrt{y}+2xy\right) ~dy$ \item $\dint \left( \sin ^{2}\theta +\cos ^{2}\theta \right) ~d\theta $ \ %TCIMACRO{\TeXButton{multcol end}{\end{multicols}}}% %BeginExpansion \end{multicols}% %EndExpansion \bigskip \bigskip \item Find a function $f$ with the following properties: $f^{\prime }\left( x\right) =x^{2}-6x+1$ \ and $f\left( 0\right) =-2$. \item Find a function $f$ with the following properties: $f^{\prime }\left( x\right) =6x^{2}+10x-2$ \ and $f\left( 2\right) =27$. \item Find a function $f$ with the following properties: $f^{\prime }\left( x\right) =\dfrac{1}{x^{3}}$ \ and $f\left( -1\right) =3$. \item Find a function $f$ with the following properties: $f^{\prime }\left( x\right) =\sin x$ \ and $f\left( \pi \right) =-2$. \item The following problem is an application to physics, where things are very often functions of time, denoted by $t$. \ So use $t$ instead of $x$. a) \ We know the following about $f^{\prime \prime }$: it is a constant, with value $a.$ \ Find $f^{\prime }$ if we also know that $f^{\prime }\left( 0\right) =v_{0}$ b) \ Find $f$ if we also know that $f\left( 0\right) =s_{0}$\bigskip \end{enumerate} \pagebreak \begin{center} {\LARGE Answers}\bigskip \end{center} \begin{enumerate} \item $\dfrac{x^{3}}{3}+C\ \ \ \ \ \ $2. $\ \dfrac{1}{6}x^{6}-\dfrac{2}{5}% x^{5}+3x+C\ \ \ \ \ \ \ $3. $\ \dfrac{2}{3}y^{3/2}+C$ \ $\ \ \ \ \ $4. $\ \dfrac{1}{4}ab^{2}x^{4}+C$ \ \ \ \ \ 5. $\ \dfrac{1}{2}a^{2}b^{2}x^{3}+C$ \item[6.] $\dfrac{1}{3}ab^{3}x^{3}+C$ \ \ \ \ \ 7. \ $x+C$ \ \ \ \ 8. $\ -\cos \theta +C$ \ \ \ \ \ 9. \ $\sin \alpha +C$ \ \ \ \ \ 10. $\ -10t+C$ \ \ \ \ 11. \ $\ -5t^{2}+tv_{0}+C$ \item[12.] $-\dfrac{1}{m}-\dfrac{2}{3}m\sqrt{m}+C\ \ \ \ \ \ \ $13. $\ -% \dfrac{3}{\sqrt{y}}x-\sqrt{y}x+x^{2}y+C$ \ \ \ \ \ 14. $\ -6\sqrt{y}-\dfrac{2% }{3}y\sqrt{y}+xy^{2}+C$ \ \ \ \ \ 15. $\ \theta +C$ \item[16.] $f\left( x\right) =\dfrac{1}{3}x^{3}-3x^{2}+x-2\qquad $17. \ $% f\left( x\right) =5x^{2}-2x+2x^{3}-5\qquad $18. \ \ $f\left( x\right) =-% \dfrac{1}{2x^{2}}+\dfrac{7}{2}$ \item[19.] $f\left( x\right) =-\cos x-3\ \ \ \ \ \ \ \ $20. \ a) \ $% f^{\prime }\left( t\right) =at+v_{0}\ \ \ \ \ \ \ $b) \ $f\left( t\right) =% \dfrac{a}{2}t^{2}+v_{0}t+s_{0}$ \end{enumerate} \vspace{2.5in} \vspace{4in} \bigskip {\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}