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\centerline{Mathematics 134 -- Calculus With Fundamentals 2}
\centerline{Exam 1 -- Review Sheet}
\centerline{February 9, 2018}
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\noindent
{\it General Information}
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As announced in the course syllabus, the first 
midterm exam of the semester will be given in class on Friday, February 16.  
There will be four or five questions (each one with several parts), similar in
format to the sample exam questions below.
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\item{$\bullet$}  You may use a scientific calculator for the exam (graphing calculators
OK). 
\item{$\bullet$}  Use of phones, tablets, I-pods, and 
all other electronic devices besides a basic calculator
{\it is not allowed} during the exam.  Please leave such devices in your room 
or put them away in your backpack (make sure cell phones are turned off).
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\noindent
{\it What will be covered}
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The exam will cover the material since the start
of the semester -- Problem Sets 1, 2, including the following material
from sections 5.1 through 5.5, 5.7 of Rogawski and Adams.  Note that there
{\it will be} a $u$-substitution problem on this exam, even though that
topic was not covered in the first two problem sets.  See practice questions below
for the type of question I might ask.  
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\item{1)}  Riemann sums and the definition of the definite integral. 
\item{2)}  The Fundamental Theorem of Calculus:  Part II:
If $f(t)$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t)\ dt$,
then $F'(x) = f(x)$.   Part I: (the ``Evaluation Theorem'') If $G(x)$
is an antiderivative of a continuous function $f(x)$ on $[a,b]$, then 
$$\int_a^b f(x)\ dx = G(b) - G(a)$$
\item{} {\it Recall, though:} our discussion of these topics in class
was slightly different from the way they are presented in the text.
I will not ask you about the way Rogawski and Adams prove the ``Evaluation
Theorem'' in their Section 5.4, but I might ask you about the technique
we used to prove the second part of the Fundamental Theorem (see p. 295-296).  
\item{3)}  Antiderivatives (indefinite integrals) and basic antiderivative rules: 
All rules coming from
basic derivative formulas:  Know $\int x^n\ dx$,
$\int e^x\ dx$, $\int \sin(x)\ dx$, $\int \cos(x)\ dx$,
$\int {1\over x^2 + 1}\ dx$, $\int {1\over \sqrt{1 - x^2}}\ dx$,
and so forth, plus the
sum, and constant multiple rules 
\item{4)}  Integrals by $u$-substitution
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\noindent
There will be a review for the exam in class on Thursday, February 15.  Tori
will also be doing her usual evening session that night.  
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\noindent
{\it Review Problems}
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The Review Problems 1--30, and 39--118 at the end of Chapter 5 are good
for preparation for this exam.  It's not necessary to work
all of them.  But you should try a good selection and practice choosing 
a method at least for most of the integrals in problems 17--30 and 43--75.
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it Note:  The actual exam will be {\it definitely} shorter than the following
list of questions.  The purpose here is just to give an idea of the range
of different topics that will be covered and how questions might be posed.}
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\noindent
I.  
\item{A)}  Evaluate the $L_4$, $R_4$, and $M_4$ 
Riemann sums for $f(x) = x^2 + 2x$ for $0 \le x \le 1$
(Note: you are using $N = 4$ each time).
\item{B)}  In part A), one of your values is definitely larger than 
the actual value of $\int_0^1 x^2 + 2x\ dx$ and one is definitely smaller
than the integral.  Which is which?  (Answer without calculating the
value of the integral and then check your work.)
\item{C)}  The following limit represents a definite integral:
$$\lim_{N\to\infty} \sum_{j=1}^N {\cos\left({j\pi\over N}\right)\over 1+ \left({j\pi\over N}\right)^2} 
{\pi\over N}$$
What is the integral?  (Note:  This is just asking for the definite integral, not for
the number that integral would compute.)
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\noindent
II.  Let 
$$f(x) = \cases{1 & if $0 \le x \le 3$\cr
                x - 2 & if $3\le x \le 5$\cr
                13 - 2x & if $5 \le x \le 8$\cr}$$
\item{A)}  Sketch the graph $y = f(x)$.
\noindent
In the rest of the parts, $F(x) = \int_0^x f(t)\ dt$, where $f$ is the 
function from part A.  
\item{B)}  Compute $F(1), F(2), F(3), F(4), F(5), F(6), F(7), F(8)$ given the information in the graph of $f$.
\item{C)}  Are there any critical points of $F$ (places where $F'(x) = 0$)?  If so, 
find them and say whether they are local maxima, local minima, or neither.  If not, say why not.
\item{D)}  How is the graph of $F(x)$ related to the graph of 
$$G(x) = \int_2^x f(t)\ dt?$$
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\noindent
III.  Find the {\it derivatives} of the following functions:
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\item{A)}  $f(x) = \int_0^x \sin(t)/t\, dt$.
\item{B)}  $g(x) = \int_5^{x^3} \tan^4(t)\ dt$.
\item{C)}  $h(x) = \int_{-3x}^{5x} e^{t^2} \sin(t)\ dt$. 
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\noindent
IV. 
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\item{A)}  Compute $\int 5x^4 - 3\sqrt{x} + e^x + {2\over x}\ dx$
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\item{B)}  Apply a $u$-substitution to compute $\int x(4x^2 - 3)^{3/5}\ dx$
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\item{C)} Apply a $u$-substitution to compute $\int_1^2 e^{\sin(\pi x)}\cos(\pi x)\ dx$
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\item{D)}  Compute 
$$\int_0^1 {t^2 + 1\over t^3 + 3t + 3}\ dt$$
(Hint:  How does the bottom relate to the top?)
\item{E)} Compute $\int e^{-7x}\ dx$. 
\item{F)} $\int \cos(2x) \sin(2x) \ dx$
\bye
