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\preface
\magnification=1200

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\def\lamina{\scaledpicture .5in by 0in (lamina scaled 700)}
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\title
\noindent
{\bf Math 120: Calculus}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\hfill
Tutorial Instructor:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\sl April 25, 1995}\hfill
Tutorial Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Calculators are {\it not} allowed on this exam.  Hand in this answer page only. 
Record your answers  to the multiple choice problems by  placing an $\times$
through one letter for each problem on this answer sheet. There are  19 multiple
choice  questions, worth 5 points each.  An additional 5 points will be given for
your correct tutorial section number.    \vskip .25cm \noindent {\bf You are
taking this exam under the honor code.}

\def\answerskip{\vskip0.25in}

\def\problemskip{\vskip1in}



\note %1
\problem
What is the sixth degree Taylor polynomial $P_6 (x)$ for $f(x) = \sin x$?  (Hint:
You can use the fact that 
$\displaystyle P_{2n+1} (x) = \sum_{k=0}^{n} {{(-1)^k x^{2k+1}} \over {(2k+1)!}}$
and
$\displaystyle P_{2n+2} (x) = P_{2n+1} (x)$, or else just do it from scratch.)
\correct $\displaystyle x - {{x^3} \over {3!}} + {{x^5} \over {5!}}$
\wrong $\displaystyle 1 - {{x^2} \over {2!}} + {{x^4} \over {4!}} - {{x^6} \over
{6!}}$
\wrong $\displaystyle 1 + {{x^2} \over {2!}} + {{x^4} \over {4!}} +{{x^6} \over
{6!}}$
\wrong $\displaystyle x  - {{x^3} \over {3!}} + {{x^5} \over {5!}} - {{x^7} \over
{7!}} + {{x^9}
\over {9!}} - {{x^{11}} \over {11!}} + {{x^{13}} \over {13!}}$
\wrong $\displaystyle -x + {{x^3} \over {3!}} - {{x^5} \over {5!}}$

\note %2
\problem
Find the third degree Taylor polynomial for the function $f(x) = e^{-x}$.
\correct
$\displaystyle 1 - x + {{x^2} \over {2!}} - {{x^3} \over {3!}}$
\wrong
$\displaystyle 1 + x + {{x^2} \over {2!}} + {{x^3} \over {3!}}$
\wrong
$\displaystyle -1 + x - {{x^2} \over {2!}} + {{x^3} \over {3!}}$
\wrong 
$\displaystyle -  1 - x - {{x^2} \over {2!}} - {{x^3} \over {3!}}$
\wrong
$\displaystyle -1 - x - x^2 - x^3 $

\note %3
\problem 
Find the $k^{\rm th}$-degree Taylor coefficient of the function
$\displaystyle f(x) \ =\ {1\over{x+1}}$
\correct
$\displaystyle (-1)^k$
\wrong
$\displaystyle 1$
\wrong
$\displaystyle {{1} \over {k!}}$
\wrong
$\displaystyle {{(-1)^k} \over {k!}}$
\wrong
$\displaystyle -1$

\note %4
\problem
Which of the following is equal to the binomial coefficient 
$\displaystyle {8\choose3}$?
\correct $56$
\wrong $28$
\wrong $112$
\wrong $8$
\wrong $-28$

\note %5
\problem 
A certain kind of die has four sides, labelled 1 to 4. When the die is rolled, 
each side has a probability of $1/4$ that it will appear on the bottom.  If the die
is rolled 5 times, what is the probability that the side labelled ``3'' appears on
the bottom in exactly two of the five rolls?
\correct
$\displaystyle 10 \cdot \left ( {1 \over 4} \right )^2 \cdot \left ( {3 \over 4}
\right )^3$
\wrong $\displaystyle 5 \cdot \left ( {1 \over 4} \right )^3 \cdot \left ( {3
\over 4}
\right )^2$
\wrong $\displaystyle 10 \cdot \left ( {1 \over 3} \right )^2 \cdot \left ( {2
\over 3}
\right )^3$
\wrong $\displaystyle 10 \cdot \left ( {1 \over 4} \right )^3 \cdot \left ( {3
\over 4}
\right )^2$
\wrong $\displaystyle \left ( {2 \over 5} \right ) \left ( {1 \over 4} \right )$

\note %6
\problem Recall Simpson's rule:
$$\int_a^b f(x)\,dx \approx {{b-a}\over{3n}}\,
   \big[\,f(x_0) + 4f(x_1) + 2f(x_2) + 4 f(x_3 ) + \dots + 2 f(x_{n-2} )
+ 4f(x_{n-1}) + f(x_n)\,\big]$$
Which of the following do you get by applying Simpson's
rule with $n=4$ to the integral
$$\int_1^3 {1\over x}\,dx$$
\correct $\displaystyle {1 \over 6} \cdot \left [ 1 + {8 \over 3} + 1 + {8 \over 5}
+ {1 \over 3} \right ]$
\wrong $\displaystyle {1 \over 6} \cdot \left [ 1 + {2 \over 3} + {1 \over 2} + {2
\over 5} + {1 \over 3} \right ]$
\wrong $\displaystyle {1 \over 6} \cdot \left [ 1 + {3 \over 2} + {2} + {5
\over 2} + {3} \right ]$
\wrong $\displaystyle {1 \over 6} \cdot \left [ 1 + {12 \over 2} + {4} + {20
\over 2} + {3} \right ]$
\wrong $\displaystyle {1 \over 2} \cdot \left [ 1 + {4 \over 3} + 1 + {4 \over 5}
+ {1 \over 3} \right ]$

\note %7
\problem Recall that associated to the trapezoid rule for approximating
$\displaystyle
\int_a^b f(x) \ dx$ is the error \break estimate 
$$|\hbox{error}| \le {{M(b-a)^3}\over{12n^2}}$$
where $M$ is the maximum value of $|f''(x)|$ for $a\le x\le b$.  
Suppose you
used the trapezoid rule with $n=4$ to estimate $$ \int_1^3 {1 \over
x}\ dx$$  What bound on the error is guaranteed by the error estimate formula? 
(Note that for this problem you  don't need to remember what the trapezoid
rule is; you just need the error estimate given above.)
\correct  $\displaystyle {1 \over 12}$
\wrong $\displaystyle {1 \over 24}$
\wrong $\displaystyle {1 \over 216}$
\wrong $\displaystyle {1 \over 324}$
\wrong $\displaystyle {2}$

\note %8
\problem Find the length of the curve $\displaystyle y = {2 \over 3} x^{3 \over
2}$ from $x=0$ to $x=3$.
\correct $\displaystyle {{14} \over 3}$
\wrong $\displaystyle {{2} \over 3}$
\wrong $\displaystyle {16 \over 3}$
\wrong $\displaystyle {{9} \over 2}$
\wrong $\displaystyle {{\sqrt{2}} \over 3}$

\note %9 
\problem Three objects are placed on the $x$-axis.  The first, having mass 5,
is placed at $x=1$.  The second, having mass 8, is placed at $x=2$. The third,
having mass $3$, is placed at $x=3$.  Where is the center of mass of the system?

\vskip .6in
\hskip 1.5in \xcoord

\correct At $\displaystyle x= {{15} \over 8}$
\wrong At $\displaystyle x= 2$
\wrong At $\displaystyle x= {{15} \over {16}}$
\wrong At $\displaystyle x= {7 \over 4}$
\wrong At $\displaystyle x= {{21} \over {16}}$

\note %10
\problem Four objects are placed in the $(x,y)$-plane.  They all have the same
mass.  They are placed at the points $(1,2)$, $(2,3)$, $(0,4)$ and $(1,7)$. 
Where is the center of mass of the system?
\correct At the point $(1,4)$
\wrong It cannot be computed from the given information
\wrong At the point $(1,6)$
\wrong At the point $\displaystyle (1, {{7} \over 2})$
\wrong At the point $\displaystyle ({{3} \over 2}, {7 \over 2})$

\note %11
\problem A lamina ($=$ region of the plane) has the shape of a right isosceles
triangle with sides of \break length 4:

\vskip 1.3in \hskip 1.6in
\lamina

\noindent Which of the following represents the $x$-coordinate of the center of
mass (i.e.\ the centroid) of the lamina?
\correct $\displaystyle {1 \over 8} \int_0^4 (4x - x^2 ) \ dx$
\wrong $\displaystyle {1 \over {16}} \int_0^4 (4x - x^2 ) \ dx$
\wrong $\displaystyle {1 \over {16}} \int_0^4 (4-x) \ dx$
\wrong $\displaystyle {1 \over {8}} \int_0^4 (4 - x ) \ dx$
\wrong $\displaystyle {1 \over {8}} \int_0^4 (4-x)^2 \ dx$

\note %12
\problem
Set up, but do not evaluate, an integral for the length of the arc of the curve
$\displaystyle y^2 = 4\sin x$ from the point $(0,0)$ to the point $\displaystyle
({{\pi} \over 2}, 2)$.  (Note that the equation is $\displaystyle y^2 = 4\sin x$,
not $\displaystyle y = 4\sin x$.)
\correct $\displaystyle \int_0^{{\pi} \over 2} \sqrt{1 + {{\cos^2 x} \over {\sin
x}}} \ dx$
\wrong $\displaystyle \int_0^{{\pi} \over 2} \sqrt{1 + 16 \cos^2 x} \ dx$
\wrong $\displaystyle \int_0^{\sqrt{{\pi} \over 2}} \sqrt{1 +  16 \cos^2 x} \ dx$
\wrong $\displaystyle \int_0^{{\pi} \over 2} \sqrt{1 + {{\cos x} \over {\sqrt{\sin
x}}}} \ dx$
\wrong $\displaystyle \int_0^{{\pi} \over 2} \sqrt{1 + {1 \over {\sin
x}}} \ dx$

\note %13
\problem In the expansion of $\displaystyle (x+y)^{15}$, what is the coefficient
of $xy^{14}$?
\correct 15
\wrong 14
\wrong 105
\wrong 1
\wrong 91

\note %14
\problem A bloodbank has 10 units of type A$^+$ blood available; but, unknown to
them, 5 of the units are contaminated with serum hepatitis.  Suppose that 2 units
are chosen at random.  What is the probability that both units chosen will be
contaminated?
\correct $\displaystyle {2 \over 9}$
\wrong $\displaystyle {1 \over 2}$
\wrong $\displaystyle {2 \over 5}$
\wrong $\displaystyle {1 \over 5}$
\wrong $\displaystyle {1 \over 9}$

\note %15
\problem Set up, but do not evaluate, an integral for the length of the curve $y
= x^2$, $3 \leq x \leq 5$.
\correct $\displaystyle \int_3^5 \sqrt{1 + 4x^2} \ dx$
\wrong $\displaystyle \int_3^5 \sqrt{1 + 2x} \ dx$
\wrong $\displaystyle \int_3^5 \sqrt{1 + x^2} \ dx$
\wrong $\displaystyle \int_9^{25} \sqrt{1 + 4x^2} \ dx$
\wrong $\displaystyle \int_9^{25} \sqrt{1 + 2x} \ dx$

\note %16
\problem Recall that for the trapezoid rule to approximate $\displaystyle \int_a^b
f(x) \ dx$, the error estimate is 
$$
|\hbox{error}| \le {{M(b-a)^3}\over{12n^2}}
$$
where $M$ is the maximum value of $|f''(x)|$ for $a\le x\le b$.  
Using this, how
large do we have to choose $n$ so that the trapezoid rule approximation to the
integral $$ \int_1^3 6 \ln x \ dx$$ is accurate to within $10^{-4}$? 
(Note that for this problem you don't need to remember what the trapezoid rule is;
you just need the error estimate given above.  The numbers in this problem were
chosen so that you do {\it not} need a calculator.)
\correct $n > 200$
\wrong $n > 100$
\wrong $n > 50$
\wrong $n > 10$
\wrong $n > 5$

\note %17
\problem
Which of the following is a row of Pascal's triangle?
\correct $1 \ \ 6 \ \ 15 \ \ 20 \ \ 15 \ \ 6 \ \ 1$
\wrong $1 \ \ 6 \ \ 10 \ \ 15 \ \ 10 \ \ 6 \ \ 1$
\wrong $1 \ \ 6 \ \ 20 \ \ 35 \ \ 20 \ \ 6 \ \ 1$
\wrong $1 \ \ 6 \ \ 15 \ \ 15  \ \ 6 \ \ 1$
\wrong $1 \ \ 6 \ \ 10 \ \  10 \ \ 6 \ \ 1$

\note %18 
\problem A certain partial deck of cards contains 6 red cards and 4 black
cards.  Two cards are chosen at random.  What is
the probability that both cards are black?  
\correct $\displaystyle {6 \over {45}}$
\wrong $\displaystyle {6 \over {90}}$
\wrong $\displaystyle {16 \over {100}}$
\wrong $\displaystyle {6 \over {15}}$
\wrong $\displaystyle {4 \over {10}}$

\note %19 
\problem Let $f(x) = 3x^5 - x^2 + 1$.  Find the degree 4 Taylor polynomial $P_4
(x)$ for $f(x)$.
\correct $P_4 (x) = -x^2 + 1$
\wrong $P_4 (x) = 3x^5 - x^2 + 1$
\wrong $P_4 (x) = 1$
\wrong $P_4 (x) = -x^2$
\wrong $P_4 (x) = x^4 - x^2 + 1$

\appendix
%That's all folks!

\end
