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Math 305 \hfill Homework Set 1  \hfill Spring 2017

\begin{center}Due Monday, January 23\end{center}

{\bf I.} Draw the following graphs without the aid of a calculator or
computer. Label zeros and the $y$-intercept.

\vspace{.15in}

1. $y = 3e^{-2x}$ \hspace{1.2in} 4. $\dis y = {\frac{1}{2}} \sin (3x+\pi)$ (two cycles)

\vspace{.15in}

2. $y=e^{-x}\cos x$ \hspace{1in} 5. $\dis y = e^{2 \ln\, |x|}$

\vspace{.15in}

3. $\dis y= \frac{1}{1+e^x}$ \hspace{1.1in} 6. $y = x \sin x$

\vspace{.15in}

{\bf II.} Do the following integrals. Take the derivative to check your answers.

\vspace{.15in}

1. $\dis \int \frac{1}{x+1} \, dx$ \hspace{1.1in} 10. $\dis \int \frac{e^x+e^{-x}}{e^x - e^{-x}} \, dx$

\vspace{.15in}

2.  $\dis \int \frac{1}{\sqrt{x+1}} \, dx$ \hspace{1in} 11. $\dis \int x^2 e^x  \, dx$ 
\vspace{.15in}

3.  $\dis \int_e^{e^2} \frac{1}{x \ln x}  \, dx$  \hspace{1in} 12.  $\dis \int_0^4 \frac{5}{3x+1}  \, dx$ 
\vspace{.15in}

4. $\dis \int \frac{1}{1+x^2} \, dx$ \hspace{1in} 13.  $\dis \int \frac{1}{x+x^2} \, dx$


5.  $\dis \int \frac{x}{1+x^2} \, dx$  \hspace{1in} 14.  $\dis \int \frac{1}{1-x^2} \, dx$

6. $\dis \int \frac{x}{1+x} \, dx$ \hspace{1in} 15.  $\dis \int x \sin x\, dx$

7. $\dis \int \frac{t^2}{1+t^2} \, dt$ \hspace{1in} 16.  $\dis \int \sin^3 x \cos^2 x \, dx$ 

8.  $\dis \int \sec^2 x \, dx$ \hspace{1in} 17.  $\dis \int \cot x \, dx$

9.  $\dis \int \sec x \, dx$ \hspace{1in} 18.  $\dis \int \tan^2 x \, dx$



{\bf III.} Find all functions $y(x)$ that satisfy the following conditions.

\vspace{.15in}

1. $\dis y' = 5x/y$ \hspace{1.2in} 5. $\dis y'= \frac{3}{4} \sqrt{x}$ \mbox{and} $y(0)=10$

\vspace{.15in}

2. $\dis y' = \frac{\sqrt{x}}{2y}$ \hspace{1.3in} 6.   $\dis y' = 3y$ \mbox{and} $y(0)=10$

\vspace{.15in}
3. $y' = 3y$ \hspace{1.4in} 7. $\dis y'' = g$,  $y(0) = 4$  \mbox{and} $y(10)=12$\\ \hspace*{2.6in}($g$ is a constant)

\vspace{.15in}
4. $\dis y' = x(1+y)$ \hspace{1in} 8. $y''=\sin x$, $y(0)=1$  \mbox{and} $y'(0) = 7$

\vspace{.15in}

{\bf IV.} Answer the following.

\vspace{.15in}

1. If radium has a half-life of 1620 years, what percentage of a sample will be left after 500 years?

\vspace{.1in}

2. If a sample of radioactive material decays at a rate of 10\% per day, what is its half-life?

\vspace{.1in}

{\bf V.} Read about {\em Taylor series} if you need to then do the following.

\vspace{.15in}

1. Find the Taylor series of $y = e^{x^2}$ centered about $x=0$. What is the radius of convergence?

\vspace{.1in}

2. Find the Taylor series of $y=1/x$ centered about $x=1$. What is the radius of convergence? 

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