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University of The Outer Rim \\

{\bf Tuesday, November 9, 3:00 - 4:00 PM} \\

{\bf MAT 186F TERM TEST} \\

Calculus I \\

Duration: 60 minutes \\

No Study Aids and No Calculators Allowed

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{\bf TOTAL MARKS: 30}



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{\bf NAME:} \hfill \underline{\hspace{3.5in}}



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{\bf STUDENT NUMBER:} \hfill \underline{\hspace{3.5in}}



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{\bf SIGNATURE:} \hfill \underline{\hspace{3.5in}}



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{\bf TUTORIAL:} \underline{\hspace{1in}} \hfill {\bf TUTOR:} \underline{\hspace{
2in}}

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{\bf Instructions:} Present in ink your solutions to the following questions in
the spaces provided; make sure that your method of finding the solutions is clear. You may use the back of each page for any rough
work that you need to do.

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\begin{enumerate}

\item Using Newton's method
to approximate a solution of the equation $\cos x = x$ with the
initial guess  taken to be $x_0 = \pi/2,$ what will be the next value, $x_1,$ generated by the formula?



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\item The closed curve $\displaystyle{x^4+y^4=1}$ looks like a slightly squashed circle and surrounds a region $R.$ Find the rectangle of largest area  inscribed in the region $R.$

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\item Given that $\displaystyle{F(x) = \int_0^{x^2+2x} \sqrt{1 + t^{43}} dt},$ find $F^\prime (x).$




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\item The function $\displaystyle{f(x)=\frac{x^2-2x-2}{x-3} }$ is asymptotic to a line $\displaystyle{y=mx+c}.$ Find $m,c$ and sketch $f$ over the range
$[0,5],$ indicating all local extrema.



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\item Sketch the curve $\displaystyle{y=|(x-1)^3|}$ and find the area under it, over the interval $[0,2].$



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\item Given that $\displaystyle{f(x)= x^3\sqrt{1-x^4}},$
\begin{enumerate}
\item evaluate the integral of $f$ over the interval $[0,1]$
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\item write down the {\em Mathematica} input required to compute this integral.
\end{enumerate}


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\item A vehicle travelling at $60 \ m/s$ applies constant deceleration $a$ and comes to a stop after travelling a further distance of $100 \ m.$ What was the constant deceleration?



\end{enumerate}
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