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{\bf date:} the $16^{\rm th}$ of October, 1998 

{\bf place:} room 221 Hayes

{\bf time:} 8:30-9:20

{\bf 111 - Exam II}

This exam contains 10 problems worth 10 points each. You may use only a
blank sheet of paper, a pencil, a rubber gum, a ruler and a small
calculator. You can use your calculator only to add, substract,
multiply or divide two numbers. This exam is taken under the honor
code. \\

{\bf Name:}
\\

{\bf Recommendation}

{\it Never give a "solitary" answer without justifying it by previous
calculations or reasoning. } \newpage {\bf Problems} \\

{\bf 1.} Calculate $$\lim_{x\to\infty}\frac{2x^3-x^2+x-1}{x^3-x+1}\ .$$
Give a geometrical interpretation for this limit.

{\bf 2.} Calculate the first derivative of the function
$$f(x)=\frac{x^3+1}{x^2+1}$$ and bring it to the simplest form you
can.

{\bf 3.} Calculate $f'(2)$ where $f(x)=(x^2+x+1)^2$ . ({\it Hint:
First, calculate $f'(x)$.})

{\bf 4.} Calculate the second derivative of the function
$f(t)=t^3-t^2+t-1$ .

{\bf 5.} Calculate the limit $$\lim_{x\to-2}\frac{x^2+5x+6}{2x+4}\ .$$

{\bf 6.} Sketch the graph of a function $f(x)$ knowing that $f(1)=-1$,
$f'(1)=0$, $f''(1)=1$, $f(-2)=2$, $f'(-2)=0$, $f''(-2)=-2$, $f(0)=0$.
Specify a minimum point and a maximum point for this function.

{\bf 7.} Sketch as accurately as possible the graph of the function
$f(x)=-x^2+2x+3$. Clearly plot any $x$-intercept, $y$-intercept, and
extreme point.

{\bf 8.} Show that the function $f(x)=x^5+x^3+x$ is increasing for all
$x$. ({\it Hint: Observe that $1$, $x^2$, and $x^4$ are never negative
quantities.})

{\bf 9.} Find the inflection point of the function
$f(x)=\frac{x^3}{3}+\frac{5x^2}{2}+6x$ and state where the function is
concave up and where concave down.

{\bf 10.} A rectangular box is made of three kinds of material. The top
face costs \$ 5 per square foot, the bottom face \$ 10 per square foot,
and each lateral face \$ 6 per square foot. Suppose that the bottom
(top) face is a square and the volume of the box is 5 square feet. Is
\$ 44 enough to construct such a box? Justify your answer. ({\it Hint:
First, find the minimum cost of the box.) \\

{\bf Good luck!}


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