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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Exam 1 -- Things To Know}
\centerline{February 10, 2006}
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\noindent
{\it General Information}
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As announced in the course syllabus, the first full-period
midterm exam of the semester will be given in class on Friday, 
February 17.  The format will be similar to that of the midterm exams
last semester.  The exam will cover the material since the start
of the semester (Quizzes 1,2,3), including the following material
from Chapters 5, 6, and section 1 of Chapter 7.
\item{1)}  Riemann sums and the definite integral
\item{2)}  Applications to computing areas, average values, total
change of a function
\item{3)}  The Fundamental Theorem of Calculus:  If $F(x)$
is an antiderivative of a continuous function $f(x)$ on $[a,b]$, then 
$$\int_a^b f(x)\ dx = F(b) - F(a)$$
Also know the second part of the fundamental theorem:  If $f$
is continuous on $a \le x \le b$, then $F(x) = \int_a^x f(t)\ dt$
is differentiable and 
$${d\over dx} \int_a^x f(t)\ dt = f(x)$$
\item{4)}  Antiderivatives graphically and numerically
\item{5)}  The power, sum, and constant multiple rules for antiderivatives
\item{6)}  Differential equations ${dy\over dx} = f(x)$
\item{7)}  Integrals by substitution
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\noindent
There will be a review for the exam in class on Wednesday or 
Thursday next week (February 15 or 16).  Rosie Arcuri
will be available on Thursday evening at the usual
time for other, ``last minute'' questions.
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\noindent
{\it Sample Exam}
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\noindent
I.  This is the graph 
$$y = f(x) = \cases{1 & if $-2 \le x \le -1$\cr
                    -x & if $-1 < x \le 0$\cr
                   \sqrt{4 - x^2} & if $0 < x \le 2$\cr}$$
(made up of straight line segments and a quarter circle):
\vglue 2.0truein
\item{A)} (15) Use left- and right-hand sums with $\Delta x = 1$ to 
estimate $\int_{-2}^2 f(x)\ dx$.
\item{B)} (10) Using the relation between the integral and area,
find the {\it exact} value of $\int_{-5}^5 f(x)\ dx$.
\item{C)} (10) Let $F$ be an antiderivative of $f$ with $F(-2) = 1$.
Use the Fundamental Theorem of Calculus to find the values 
$F(0)$ and $F(2)$.
\item{D)} (5) $F$ is the same antiderivative of $f$ as in part C.
Is $y = F(x)$ concave up or down for $0 < x \le 2$?  Explain how
you can tell.
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\noindent
II.  Compute each of the following integrals.
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\item{A)} (10)  $\int {\root 3\of x} - {4\over x} + \cos(x)\ dx$
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\item{B)} (15)  $\int x \sin(12 - 3x^2)\ dx$
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\item{C)} (15)  $\int_0^2 {t^2\over 7 + 2 t^3}\ dt$
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\noindent
III. The temperature $Q(t)$ of a cup of coffee is changing at the rate
$${dQ\over dt} = -3.5e^{-.05t}\leqno(1)$$
degrees Celsius per minute $t$ minutes after it is poured. 
the temperature at time $t = 0$ was $Q(0) = 90$.
\item{A)} (15)  Solve the differential equation (1) with the initial
condition $Q(0) = 90$ to find a formula for $Q(t)$.  
\item{B)} (5) What was the {\it average temperature} of the coffee 
between $t = 0$ and $t = 5$?
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\noindent
Look over Problem Sets 1,2,3, the last three quizzes, and see
the Review Problems from the ends of Chapters 5,6,7 in
the text for additional practice problems.
\bye
