Test 2 Review SheetāMath 122
Spring 2005, Prof. Frank
This exam will cover sections 7.4, 7.7, 8.2, 8.3, 8.4, 8.5, and 10.1 (seven total).
General principles to guide your study:
(1) Reread each section in the book, along with your notes for that section and the
comments in the study guide for that section. Make a note of every important fact,
definition, and theorem from that section that you feel you should memorize.
(2) Go over your HW assignments and make sure that you understand all of the problems,
as well as the related problems in the text. Pay close attention to the ones you missed
the first time around.
(3) Think about the questions listed below only after you have completed your review.
Try and figure them out without referring to your notes.
(4) This review sheet is kind of āshortā because such a large component of the exam is
just integrals that you need to practice.
Note: many of the following questions are more vague and open-ended than what will be on
your exam. They are intended as food for thought, to help you explore the concepts.
I will post a list of review sheet hints on my website this weekend. Try the
problems on your own first!!!
(1) What is the definition of the exponential function? What are its domain and range,
and why?
(2) What equations govern the relationship between the exponential function and the
natural logarithm function?
(3) What is the precise definition of the number e? (For fun, and unrelated to the exam,
do you happen to know a precise definition of the number Ļ?)
(4) Draw a graph with both the exponential and natural logarithm functions on it. Label
at least two points on both graphs.
(5) Prove that the derivative of exis ex.
(6) We focused on the inverses of three trig functions: cos,sin,and tan. For each of the
three functions do the following:
(a) Give the restricted domain we take so that the function is invertible. Prove that
it is in fact invertible on this domain.
(b) Sketch the trig function on its restricted domain, and sketch the inverse trig
function on the same set of axes. For each graph label at least three points.
(c) Prove the derivative formula for the inverse trig function.
(7) True or false? sin(sinā1(.5)) = .5. Give two different justifications for your answer.
(8) True or false? sinā1(sin(3Ļ/4)) = 3Ļ/4. Justify your answer.
(9) True or false? cosā1(cos(3Ļ/4)) = 3Ļ/4. Justify your answer.
(10) True or false? tanā1(tan(3Ļ/4)) = 3Ļ/4. Justify your answer.
(11) What derivative rule does integration by parts come from? Use that rule to derive
the integration by parts formula Zudv =uv āZvdu.
