Calc II - Exams
Note: More information, with possible changes,
will be added as the exam dates near.
First Midterm Exam: Fri Oct 2 (in class)
The first exam will cover Chapters 4 and 5 of the text. In other words,
this exam will cover topics from Homeworks 1-5 plus Sections 5.3 and 5.3.
At the least, you should be comfortable with the following:
- Riemann sums You should be able to approximate
areas under curves with a specific Riemann sum (e.g., n=4 using the right
endpoint rule) and compute exact areas using Riemann sums for simple
functions like y=x or y=x^2.
- Integrals You should know the basic properties of definite
and indefinite integrals, and understand the difference and relation
between definite and indefinite integrals. You should know and be able to
state the Fundamental Theorem of Calculus (part II is more important), and
use the FTC to compute definite and indefinite integrals for simple functions
as in the homework (polynomials, roots, simple trigonometric functions)
and be able to use the technique of substitution (Sec 4.5) to compute
integrals. For substitution, you should be comfortable with both the form
for indefinite integrals as well as the form for definite integrals.
- Area and Volume You should be able to compute areas between curves
as in Sec 5.1, and volumes of special types of objects such as solids of
revolution, both by the cross-section method (Sec 5.2) and the method of
cylindrical shells (Sec 5.3).
- Other applications Understand the interpretation of the FTC
as the net change theorem. For instance, given an acceleration function,
initial position and initial velocity, determine the position and velocity
functions. You should also understand the relation of average values of
functions to integrals (Sec 5.5), and be able to compute things like average
velocity.
Suggestions for preparation:
I suggest you begin by reviewing your old homeworks and lecture notes, in
particular the examples/problems from lecture/discussion. In addition,
you may want to do extra practice problems from the book in Sections 5.3 and
5.5 as these sections were not covered in homework assignments.
One you feel moderately comfortable with your ability, you should try doing
the practice problems below, treating them as a sort of mock exam. This
means, after preparing, you should try to do these on your own (i.e., try
them by yourself without looking at notes/HWs/etc or
the book, except to get the statment of the book problems).
I cannot overemphasize the importance
of making sure you can do these on your own
After attempting the practice problems on your own, look at your notes/text
and/or work with others to figure out how to do what you couldn't.
Then check your answers against the solutions to make sure you have understood
things correctly. Afterwards, go over the problems you struggled with
and make sure you can do those (or similar) problems on your own.
Practice Problems (Suggested time limit:
2~2.5 hours. If you spot any errors, please let me know so I can fix them.)
Practice Problems with Solutions:
(If you spot any errors, please let me know so I can fix them.)
[Update 9/29: corrected solutions to 19 and 25.]
Second Midterm Exam: Fri Nov 20 (in class)
The second exam covers most of Chapters 6 and 7, focusing on topics covered in
HW 6-10. Specifically, the exam will Sections 6.1-6.4, 6.6, 6.8, 7.1-7.5
and 7.8.
At the least, you should be comfortable with the following:
- Know basic integrals From the table on p. 519, you should definitely
know off the top of your head 1-3, 5-7, 9, 11, 13, 17, 18. You should
also be able to either quickly work out or remember 4, 8, 10, 19, 20.
- Inverse functions Know what inverse functions mean
and how they are related, how to compute the derivative of an inverse function.
Know the basic inverse trig functions, how to find their domain and range,
and derivatives.
- Exponentials and logarithms Know the properties of exponentials
and logarithmic functions, how they are related, their derivatives and integrals.
- Indetermiminant forms Be able to compute limits using l'Hospitals
rule when necessary.
- Integration techniques Be able to use, and identify when to use,
substitution, integration by parts, trig substitution and partial fractions.
In particular, you should be able to integrate products of trigonometric
functions, compute the area of an ellipse and integrate rational functions.
For the trigonometric integrals, you should know the trig identities
relating
sin^{2} and cos^{2}, relating tan^{2} and sec^{2}, and the half-angle identities relating sin^{2}(x) and cos^{2}(x) with cos(2x).
- Improper integrals Be able to determine if improper integrals
converge or diverge, and compute convergent integrals.
From the above list, main focus (probably at least half of the exam)
will be focused on the topic of integration techniques.
Suggestions for preparation:
As with the first exam, I highly recommend you prepare by first reviewing
the material/your homework and working out some problems on your own (e.g., the
review problems for Chapters 6 and 7, as well as problems from 7.5).
Then try out the practice problems below like a mock exam and check
your solutions. Work towards being able to do the problems on your own.
For this, you may find it helpful to try additional problems from the book
to test out your abilities. Bring questions you have to office hours,
the Math Center, your discussion section, and/or lecture.
Practice Problems (Suggested time limit:
2~2.5 hours. If you spot any errors, please let me know so I can fix them.)
Practice Problems with Solutions
(If you spot any errors, please let me know so I can fix them.)
Final Exam: Wed Dec 16, 10:30am-12:30am
The final exam will be cumulative.
The three main things I want you to get out of the course are:
- Find areas, volumes, etc - Be able to compute areas of basic geometric
objects, like ellipses, regions bounded by lines and conic sections or
trigonometric functions, etc. Similarly for volumes, arc lengths and surface
areas of basic geometric objects like circles, spheres, solids of
revolution, etc. (This involves being able to compute various types
of integrals using the techniques of integration we studied.)
- Understand the relation between area and integration - Be
able to explain the main ideas for computing areas of regions and how
it leads to integrals. Be able to approximate areas using Riemann sums
and compute simple integrals from the definition (as a limit of Riemann
sums).
- Understand the Fundamental Theorem of Calculus - Be able to
state it, and
explain how it relates differentiation and integration, as well as its role
in calculation of integrals.
These topics will make up a large portion of the final exam, and if you
can do these satisfactorily, you should pass the final exam and also the class.
While you should focus first on the above three points, here is a more
complete list of the main topics for the final exam:
- Topics from Exam 1 - read through the bullet points listed above
for Exam 1 (these are contained in #1 above, except for the "other
applications" part)
- Topics from Exam 2 - go through the bullet points listed above
for Exam 2
- Understand functions involving exponentials and logarithms -
be able to use the Hospital's rule to compute limits of indeterminate form
and use them to graph functions involving exponentials and logarithms (e.g.,
x^{2}e^{x})
- Arc length, surface area - be able to compute arc lengths and
surface areas for some basic curves and surfaces/solids (Sec 8.1, 8.2)
- Differential equations -
know examples of modeling problems that lead to differential equations
(e.g., population growth);
know what a differential equation is
and the role of initial conditions; be able to solve the equation y' = cy
with an initial condition, as well as first-order separable differential
equations using separation of variables (online notes; see also Sec 9.1, 9.3).
Suggestions for preparation: Do lots of practice problems, and
make sure you can do them on your own. Specifically focus on HW 11 and 12
(which contain many review problems),
Exams 1 and 2, and the practice problems for the exams.
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