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\centerline{Mathematics 136 -- Calculus 2}
\centerline{Exam 1 -- Review Sheet}
\centerline{February 14, 2020}
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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 21.  
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 which 
may have graphing capabilities. 
\item{$\bullet$}  Use of cell phones, I-pods, and all other electronic devices
besides a 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, 3, including the following material
from sections 5.1 through 5.7 and 7.1 through 7.3 of Rogawski and Adams:
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\item{1)}  Riemann sums and the definition of the definite integral. 
\item{2)}  The Fundamental Theorem of Calculus:  Part 1:
If $f(t)$ is continuous on $[a,b]$ and $F(x) = \int_a^x f(t)\ dt$,
then $F'(x) = f(x)$.   Part 2: (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 first part of the Fundamental Theorem (what Rogawski
and Adams call the second part -- 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)}  Net change, the formula $\int_a^b f'(x)\ dx = f(b) - f(a)$ and its
consequences and uses.  
\item{5)}  Integrals by $u$-substitution
\item{6)}  Integration by parts and reduction formulas
\item{7)}  Trigonometric integrals, using the short table of reduction formulas
passed out in class on February 11.  (Note:  You do {\it not} need to memorize all
of them; a copy of the formula sheet or a subset will be provided for your use
during the exam.)
\item{8)}  Integrals by trigonometric substitution.
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\noindent
There will be a review for the exam in class on Wednesday, February 19.
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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 
and 3--11 and 13--29 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.  The odd-numbered problems have answers given in the back
of the book so you can check your work.  
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it Note:  The actual exam will be {\bf considerably} 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:
$$\lim_{N\to\infty} \sum_{j=1}^N {\cos\left({j\pi\over N}\right)\over \left({j\pi\over N}\right)^2 + 1} \cdot {\pi\over N}$$
What is the integral?
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\noindent
II.  
\item{A)}  A particle moves along a straight-line path with velocity given by 
$v(t) = \sqrt{t + 5}$ (meters per second) at time $t$ (in seconds).  Estimate 
the total distance traveled between $t = 0$ and $t = 10$ by using the $L_5$ (left-hand
sum) approximation.
\item{B)}  Express the exact value of the total distance traveled as a definite
integral and evaluate it using the Fundamental Theorem of Calculus.  
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\noindent
III.  
\item{A)}
Compute $\lim_{N\to\infty} \sum_{i=1}^N \left({5i\over N}\right)^2 {5 \over N}$ using 
the power sum formulas from page 265.  (Note:  If I ask a question like this, I would give those
formulas -- you don't need to memorize them.)
\item{B)} The limit in part (A) computes a definite integral.  Using the Fundamental Theorem
of Calculus (part I), check your work in part (A).
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\noindent
IV.  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
V.  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
VI. Note:  I will probably ask some questions in this form, indicating a method to use.
But you should also be prepared for others as in the next question where you need
to choose which method applies.  
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\item{A)}  Compute $\int 5x^4 - 3\sqrt{x} + e^x + {2\over x}\ dx$
using basic integration formulas.
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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)} Integrate by parts to compute $\int x^2 e^{6x}\ dx$
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\item{D)}  Integrate with an appropriate $u$-substitution:
$$\int {t^2 + 1\over t^3 + 3t + 3}\ dt$$
(Hint:  How does the bottom relate to the top?)
\item{E)} Compute $\int x^4 \ln(3x)\ dx$ using integration by parts. 
\item{F)} Find $\int \sin^4(3x)\ dx$ using our trigonometric reduction formulas.
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\noindent
VII.  Compute each of the integrals below.  You must show all work
for full credit.  
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\item{A)}  $$\int {e^{\sqrt{\sin(x)}} \cos(x) \over \sqrt{\sin(x)}}\ dx$$
\item{B)}  $$\int \cos^2(2x) \sin^2(2x) \ dx$$
\item{C)}  $$\int {x^2\ dx \over \sqrt{16 - x^2}}$$
\item{D)}  $$\int {x+2\over x^2 + 1}\ dx$$
(Split into two fractions and integrate each separately.) 
\bye
