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\centerline{Mathematics 134 -- Intensive Calculus for Science 2}
\centerline{Exam 1 -- Things To Know}
\centerline{February 8, 2002}
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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 15.  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.
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\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 
$$\int_a^b f(x)\ dx = F(b) - F(a)$$
The second part of the fundamental theorem:
$${d\over dx} \int_a^x f(t)\ dt = f(x)$$
\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 Thursday, February 14.  
I can also try to set up an evening session with Kathryn Blaisdell
on Thursday for other questions.
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\noindent
{\it Sample Exam}
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\noindent
I.  The following graph shows $y = f(x) = 
\cases{1 & if $-2\le x \le 0$\cr
\sqrt{1-x^2} & if $0 \le x \le 1$\cr
0 & if $1 \le x \le 2$\cr}$
                                                   
\vglue 2.0truein
\item{A)}  Use the information from the graph to compute the 
left- and right-hand sums for $f(x)$ on $[-2,2]$ with $\Delta x = 1/2$.
\item{B)}  Should the value of your {\it left-hand sum} be
smaller than or larger than the actual value of the integral of this function?  
Explain how you can tell.
\item{C)}  The part of the graph from $x = 0$ to $x = 1$
is part of a circle.  Use this to determine the exact average value 
of $f(x)$ on $[-2,2]$.
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\noindent
II.  The following graph shows $y = F'(x)$, the {\it derivative}
of $F$.  Sketch the graphs $y = F(x)$ in the cases $F(0) = 0$ and 
$F(0) = 1$.
\vglue 1.0truein
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\noindent
III.  A child throws a baseball vertically upward with velocity 12 ft/sec 
from an initial height of 4 feet.
The acceleration of the baseball due to gravity is -32 ft/sec${}^2$, so
the velocity function $v$ satisfies the differential equation 
$${dv\over dt} = -32$$
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\item{A)} Find formula for the velocity as a function of time.
\item{B)}  When does the baseball reach its highest point?  When
does it hit the ground?
\item{C)}  Find the greatest height the ball reaches.
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\noindent
IV.  Compute the following integrals.  
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\item{A)}  $\int 5x^4 - 3\sqrt{x} + e^x + {2\over x}\ dx$
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\item{B)}  $\int x(4x^2 - 3)^{3/5}\ dx$
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\item{C)}  $\int_1^2 e^{\sin(\pi x)}\cos(\pi x)\ dx$
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\item{D)}  $$\int {t^2 + 1\over t^3 + 3t + 3}\ dt$$
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\noindent
Also see Problem Sets 1,2,3, the last three quizzes, and 
the Review Problems from the ends of Chapters 5,6,7 in
the text for other practice problems.
\bye
