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\centerline{Mathematics 136 -- Calculus 2}
\centerline{Exam 1 -- Review Sheet}
\centerline{February 14, 2014}
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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.  
The format will be similar to that of the midterm exams
last semester.  
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\item{$\bullet$}  You may use a scientific calculator for the exam which 
does NOT have graphing capabilities (a TI-30 or equivalent).  {\it No 
calculators will be provided from this point on since our old departmental
supply is defunct and will not be replaced.} 
\item{$\bullet$}  Use of cell phones, 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, 3, including the following material
from sections 5.1 through 5.6 of Stewart:
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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 $F(x)$
is an antiderivative of a continuous function $f(x)$ on $[a,b]$, then 
$$\int_a^b f(x)\ dx = F(b) - F(a)$$
\item{3)}  Antiderivatives graphically and numerically
\item{4)}  Basic antiderivative rules: All rules coming from
basic derivative formulas:  Know $\int x^n\ dx$,
$\int a^x\ dx$, $\int \sin(x)\ dx$, $\int \cos(x)\ dx$,
$\int {1\over x^2 + 1}\ dx$, $\int {1\over \sqrt{1 - x^2}}\ dx$,
and so forth, plus the
sum, and constant multiple rules 
\item{5)}  Integrals by substitution
\item{6)}  Integrals by parts: Selecting appropriate
$u$ and $dv$, computing $du$ and $v$, using the parts formula
$\int u\ dv = uv - \int v\ du$, then finishing the integral on 
the right.  Recall that this might involve using parts again,
or another method such as substitution.
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\noindent
Note:  Some problems may ask you to carry out a particular
integration method on a problem.  Others may leave the choice
up to you.  {\it Be prepared for both types of questions!}
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 - 46 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 9 - 46.
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\noindent
{\it Sample Exam Questions}
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\noindent
{\it Note:  The actual exam will be 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 Riemann sums for $f(x) = x^2 - 2x$ for $0 \le x \le 1$
using $n = 4$ and $x_i^* = $ right endpoints, then repeat with left endpoints, 
then repeat with midpoints.  
\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:
$$\lim_{n\to\infty} \sum_{i=1}^n {\cos\left({i\pi\over n}\right)\over {i\pi\over n}} 
{\pi\over n}$$
What is the integral?
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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 $y = f(x)$.
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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 $F$?  If so, find them and say whether 
they are local maxima, local minima, or neither.  If not, say why not.
\item{D)}  Sketch the graph $y = F(x)$.  What happens to the graph if the 
definition of $F(x)$ is changed to 
$$F(x) = \int_2^x f(t)\ dt?$$
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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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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 
$$\int {t^2 + 1\over t^3 + 3t + 3}\ dt$$
(Hint:  How does the bottom relate to the top?)
\item{E)} Apply integration by parts to compute $\int x^2 e^{-2x}\ dx$
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\noindent
V.  Compute each of the integrals below using some combination of
basic rules, substitution, integration by parts.  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)}  $$\int e^x \sin(2x) \ dx$$
\item{C)}  $$\int_0^1 \tan^{-1}(x)\ dx$$
\item{D)}  Use integration by parts to show this reduction formula:
If $n$ is a positive integer, then
$$\int x^n e^{ax}\ dx = {x^n e^{ax}\over a} - {n\over a} \int x^{n-1} e^{ax}\ dx$$
\item{E)}  Apply the result from part (D) to compute $\int x^4 e^{-2x}\ dx$.  
\bye
