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\u{\LARGE {MA2160 \hs Sections 6 and 8 \hs Exam 3}}\\
\u{\Large {April 21, 2004}}\\
\u{\Large {Do all problems}}\\
\u{\Large {Show all work}}\\
\end{center}

{\large {\n 1. Evaluate the following series exactly. 

\vso

(a) $\sum_{n=5}^{\infty}\left ({1\over 3}\right)^n$

\vso

(b) $1+3+{{3^2}\over {2!}}+{{3^3}\over {3!}}+{{3^4}\over {4!}}+\cdots $

\vs

\n 2. Find the first four non-zero terms in the Taylor expansion of 
$f(x)=cosx$ about $x=\pi/4$.

\vs

\n 3. (a) Find the first four non-zero terms in the Taylor expansion of 
$f(x)={1\over{1-x}}$ about $x=0$.

\vso

(b) Integrate the formula you found in part 3(a) to find the first four non-zero 
terms in the Taylor expansion of $f(x)=ln(1-x)$ about $x=0$.

\vso

(c) Use the formula you found in part 3(b) to find an approximation of 
$ln(.9)$.

\vs

\n 4. Using the slope field below for the differential equation $y'=1+xy$, 
plot the solution that passes through the point $(1,1)$.

\pagebreak

\n 5. Solve the initial value problem $y'=-x^2/y^2, y(0)=1$ exactly.

\vs

\n 6. (a) Write down the formula that is used in the Euler approximation to the 
solution of the initial value problem $y'=f(x,y), y(x_0)=y_0$.

\vso

(b) Consider the initial value problem $y'=x+1, y(0)=1$. Use Euler's method 
with two steps to find an approximation to the solution of the problem at 
$x=0.2$  

\vso

(c)Find the exact solution of the problem in part (b). What is $y(.2)$? What is 
the error in the approximation of part (b) (that is, $|y(x_2)-y_2|$)?

\vs

\n 7. At time $t=0$, a bottle of juice at $80^{\circ}F$ is placed in a mountain 
stream whose temperature is $50^{\circ}F$. After $4$ minutes, its temperature 
is $72^{\circ}F$. Let $H(t)$ denote the temperature of the juice at time $t$,
in minutes.

\vso

(a) Write a differential equation for $H(t)$, using Newton's Law of Cooling.

\vso

(b) Solve the differential equation, showing the steps.

\vso

(c) When will the temperature of the juice reach $60^{\circ}F$?

}}

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