Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

3 Pages

mid22127sol

Course: MATH 211, Spring 2006
School: Wisconsin
Rating:

Word Count: 818

Document Preview

221, Math Fall 2006, Lecture 7, Solutions to Midterm 2. Try to do first the problems that look simplest to you. You don't have to follow the order. Avoid spending too much time on one question with others not done yet. Good luck! 1. Define what is the inverse function and what is the condition for its existence. State the formula for the derivative of inverse function. Evaluate cos(sin-1 (1/2) - cos-1 (1/2)). The...

Register Now

Unformatted Document Excerpt

Coursehero >> Wisconsin >> Wisconsin >> MATH 211

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
221, Math Fall 2006, Lecture 7, Solutions to Midterm 2. Try to do first the problems that look simplest to you. You don't have to follow the order. Avoid spending too much time on one question with others not done yet. Good luck! 1. Define what is the inverse function and what is the condition for its existence. State the formula for the derivative of inverse function. Evaluate cos(sin-1 (1/2) - cos-1 (1/2)). The inverse function f -1 exists if and only if f is one to one. If b is in the range of f and f (a) = b, then f -1 (b) = a. Recall that cos( - ) = cos cos - sin sin . Also, cos(sin-1 1/2) = 1 - (sin(sin-1 1/2))2 = 1 - (1/2)2 = 3/2. Similarly, sin(cos-1 1/2) = 3/2. Then cos(sin-1 (1/2) - cos-1 (1/2)) = cos(sin-1 (1/2)) cos(cos-1 (1/2))+ sin(sin-1 (1/2)) sin(cos-1 (1/2)) = ( 3/2)(1/2) + (1/2)( 3/2) = ( 3/2). 2. A particle moves along the hyperbola y = 1/x in the first quadrant in such a way that its x-coordinate (measured in meters) increases at a steady 3m/sec. How fast is the angle of inclination of the line joining the particle to the origin changing when x = 1m? Observe that tan (t) = y(t)/x(t). Notice also that since the particle moves over hyperbola, y(t) = 1/x(t). Differentiating, we get 1 d 1 2x (t) (t) = =- . 2 2 (cos((t)) dt x(t) x(t)3 When x = 1, we also have y = 1/x = 1, and so tan = 1, = /4. Plugging in all the numbers, we get 2 (t) = -6, so (t) = -3 radians per second. 3. Assume 0 < t < /2. Find an equation for the line tangent to the curve x(t) = tan t, y(t) = 1 + cos t at a point corresponding to t = /6. Can you describe the curve above in terms of the Cartesian equation? Compute x (t) = 1/(cos t)2 , y (t) = 1 - sin t. Then dy dx dy = / = (1 - sin t)(cos t)2 . dx dt dt dy Putting in t = /6, we arrive at dx = (1 - 1/2) 3 = 3/8. 4 Also when t = /6, x = 1/ 3 and y = 1 + 3/2. Therefore the equation of tangent is L(x) = 1 + 3/2 + 3/8(x - 1/ 3). 1 2 Finally, if 0 t , then tan t = 1 - (cos t)2 cos t. Thus x = gives a Cartesian expression for the curve. 1 - (y - 1)2 /(y -1) dy dx 4. Assume an implicit function y(x) is defined by ln y + xy = x Find point (1, 1). Let us differentiate, using chain rule: y -1 y + y + xy = 1, so y = In particular, y (1) = 0. Next, y = (y - 2yy )(1 + xy) - (y - y 2 )(xy + y) . (1 + xy)2 y - y2 . 1 + xy and d2 dx2 at y a Putting in y = 0, x = y = 1 leads to y = 0. 5. State and prove Rolle's theorem. You may assume that if the function f is differentiable and x0 is its local maximum or minimum, then f (x0 ) = 0. Rolle's Theorem: Assume that f (x) is defined on [a, b], differentiable on (a, b) and continuous on [a, b]. Assume that f (a) = f (b). Then there exists c in (a, b) such that f (c) = 0. Proof: If f (x) is constant, then the derivative is zero at every point, and theorem holds. So we assume that f is not constant. In that case there should exist a point c in (a, b) where f takes its maximum or minimum value different from f (a) and f (b). We know that a differentiable function has derivative equal to zero at the point where it takes maximum (or minimum) value. This completes the argument. 1 6. Investigate how many zeroes the function f (x) = sec x - x3 + 5 has on the interval [0, /2]. Explain your reasoning. Observe f becomes very large negative when x approaches 0 because of -1/x3 term. Also, f becomes very large positive as x approaches /2 because of sec x term. By the Intermediate Value Theorem, there must be at least one zero. On the other hand, 3 f (x) = sec x tan x + 4 > 0 x for any x in our interval. If there were more than one zero of f in our interval, by Rolle's Theorem we could find a point x0 between them where f (x0 ) = 0. But the derivative is positive, so such point cannot exist, and therefore there cannot be more than one zero. 7. Find absolute maximum and minimum of the following functions: a. f (x) = 2x - 2 sin 2x on [-/2, /2]; 3 b. f (x) = 2x3 - 8x2 + 8x - 7 on [0, 3]. a. Find the derivative: f (x) = 2 - 2 2 cos 2x. The function is odd, so it suffices to look at x 0. The derivative vanishes if cos 2x = 1/ 2, 2x = /4. Thus or x = /8 are the critical points. Now f (/8) = /4 - 2/ 2 = /4 - 1. Also, f (-/8) = -/4 + 1. We have to check the endpoints: f (/2) = - 2 sin = and f (pi/2) = -. Clearly, f (/2) is the absolute maximum and f (-/2) is the absolute minimum. b. f (x) = 6x2 - 16x + 8 = 2(3x2 - 8x + 4). Finding the roots of the quadratic 8 gives critical points x = 2, 2/3. Observe that f (2) = -7 and f (2/3) = -4 27 . Now at the endpoints f (0) = -7 and f (3) = -1. Clearly, x = 3 is the absolute maximum, while x = 0 and 2 are two absolute minima.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

Wisconsin - MATH - 211
Math 221, Lecture 7, Fall 2006, Midterm 1. Try to do first the problems that look simplest to you. You don't have to follow the order. Avoid spending too much time on one question with others not done yet. Good luck! 1. (15) Consider the function f (
Wisconsin - CS - 302
CS302 Fall 2006: Practice Exam 11. Assume that x, y, and z are all int variables. Consider the following code segment: if (x = 0) { if (y = 1) z += 2; } else { z += 4; } System.out.print(z); What is printed if x, y, and z are all equal to zero befo
Wisconsin - CS - 302
Multiple-choice Answers 1. A 2. C 3. E 4. A 5. C 6. E 7. B 8. E 9. A 10. D 11. D 12. D 13. B 14. E 15. D 16. B 17. A 18. B 19. A 20. B 21. B Coding Question Answers Question 1 part (a) public void printName() { System.out.println(lastName + &quot;,&quot; + fir
Wisconsin - CS - 367
0 AVL Treesa. Underneath each of the diagrams below, indicate which of the following kinds of structures the diagram might reasonably represent by writing the corresponding letters: If the diagram represents a tree, indicate its height if not indica
Wisconsin - CS - 367
What you need to know to take Calculus 221August 16, 20001Arithmetic1. Here are some things which should be easy for you. (If they are not, you may not be ready for calculus.) 1. Factor x2 - 6x + 8. 2. Find the values of x which satisfy x2 - 7
Wisconsin - MATH - 211
Wisconsin - CS - 302
Wisconsin - CS - 302
Wisconsin - CS - 302
Wisconsin - CS - 302
Wisconsin - CS - 302
Wisconsin - CS - 302
Wisconsin - CS - 367
RecursionPage 1 of 10RECURSIONContentsIntroduction Test Yourself #1 How Recursion Really Works Test Yourself #2 Recursion vs Iteration Factorial Fibonacci Test Yourself #3 Recursive Data Structures Test Yourself #4 Analyzing Runtime for Recurs
Wisconsin - CS - 367
Recursion AnswersPage 1 of 3Answers to Self-Study QuestionsTest Yourself #1The printInt method does obey recursion rule 1; it does have a base case, namely:if (k = 0) return;It obeys recursion rule 2 in a sense; the code:printInt( k - 1 );
Wisconsin - CS - 367
SearchingPage 1 of 3SEARCHINGContentsIntroduction Searching in an Array The Comparable InterfaceIntroductionSearching is a very common problem, both in life and in computer applications. It is so important that many data structures and algo
Wisconsin - CS - 367
Stacks and QueuesPage 1 of 15STACKS AND QUEUESContentsIntroduction Stacks Queues Implementing Stacks Array Implementation Test Yourself #1 Test Yourself #2 Test Yourself #3 Linked-list Implementation Test Yourself #4 Test Yourself #5 Implement
Wisconsin - CS - 367
Stacks and Queues AnswersPage 1 of 1Answers to Self-Study QuestionsTest Yourself #1public Stack() { items = new E[INITSIZE]; numItems = 0; }Test Yourself #2public E pop() throws EmptyStackException { if (numItems = 0) throw new EmptyStackExc
Wisconsin - CS - 367
P4 Model Design UML Diagram*Java File I/O Classes Java Console I/O ClassesGameApp Window Board BadgerWindowPainterGameTickListenerKeyboardListenerGameObjectOpponentRowDependency Inheritance Interface Aggregation * not all relationsh
Wisconsin - THEATER AN - 120
Exam 1 Study Guide: Fires in the Mirror: Plot: interviews with the people and their takes on the incident and race and identity in general. The plot is related to the story in that is recalling the Crown Heights crisis. 26 interviewed subjects with v
Wisconsin - COM DIS - 110
12 March 200812 March 2008 Characteristics of normal phonological development Gradual process the child needs to learn3/12/2008 2:25:00 PMspeech sounds: (stops: p,b,t,d,k,g; nasals: m,n, &quot;ng&quot;; glides: w, &quot;y&quot;; fricatives: h,f,v,&quot;th&quot; (think), &quot;th&quot;
Wisconsin - COM DIS - 110
24 March 200824 March 20083/24/2008 12:49:00 PMStuttering Questions Audio and Videos Stuttering in life a 2-minutes/100-word speech sample from a client What is stuttering? A speech pattern involving audible sound/syllable repetitions and/or pr
Wisconsin - COM DIS - 110
3 March 20083 March 2008 Reading : Chap 2 (pp. 52-57) Phonological Development &amp; Disorders3/3/2008 2:20:00 PMSpeech Perception (stuff from last lecture) Bottom-up Theories Words are recognized when the acoustic cues for their component phonemes
Wisconsin - COM DIS - 110
CD110 INRODUCTION TO COMMUNICATIVE DISORDERS Neurological Bases of Communication Disorders March 5, Fall 2008I.General Neuroanatomy Relevant to Communicative FunctionA. Cerebral Hemispheres (Text, Figs. 3.19, 3.20, 3.21) 1. Four lobes of brain
Wisconsin - COM DIS - 110
27 February 200827 February 2008 Speech Anatomy, Physiology, and Acoustics Chapter 3 (pg. 59-81)2/27/2008 2:21:00 PMSpeech Mechanism defined: It includes: respiratory system, larynx, upper articulators, nasal passageways Larynx- includes structu
TAMU Corp. Chr. - BIO - 1407
Introductory Biology, Biology 1406. TAMU-CC. 3rd lecture exam10 November 2006. Learning community sections 1. If an organism has allele 1 and allele 3, which are found on question 21, which allele is likely to be the dominant allele? a. allele 1 b.
TAMU Corp. Chr. - BIO - 1407
&lt;P&gt;One of the biggest parts of Evolution is Natural Selection. It allows nature to flourish, having only the organisms with traits most beneficial to the environment to survive. David Reznick and John Endler used this as an evolutionary tool when tes
TAMU Corp. Chr. - BIO - 1407
Green algae (live) um i think its Ulva group Chlorophyta (she had a press that she did behind the microscope to help us out but we didnt even notice) know the groups - and algae are in the kingdom protista Archaea are prokaryotic moss - live - youll
TAMU Corp. Chr. - BIO - 1407
1st Lecture exam: February 2nd. 2nd WebCT assignment due: 8:00 January 25th (Thursday) Before reading Ch.22: Survey the chapter Fertilizer X null hypothesis: If fertilizer X does indeed increase fruit production, then the control group will not have
TAMU Corp. Chr. - BIO - 1407
Introductory Biology, Biology 1407. TAMU-CC. 2nd lecture exam2 March 2007 section 001, 1-20 are 4 points each 1. Production of secondary compounds such as flavanols in Theobroma cacao occurs because: a. it happens as a side effect of making sugars,
TAMU Corp. Chr. - BIO - 1407
The idea that there is specific pairing of nitrogenous bases in DNA was the flash of inspiration that led Watson and Crick to the correct double helix and at the same time they saw the functional significance of the base-pairing rules. Prior to the d
TAMU Corp. Chr. - BIO - 1407
&lt;p&gt; Scientists found from Charles Darwin and Gregor Mendel's work that allele frequencies in a population are not expected to be altered if certain conditions remain present in a population, and that there are also conditions that will allow these al
TAMU Corp. Chr. - BIO - 1407
&lt;P&gt; Plants and animals both gather resources from the environment but in different ways. Plants use sunlight as a resource. Plants can gain energy from the nutrients in the soil and their roots bring water from the ground. Humans also need water but
TAMU Corp. Chr. - BIO - 1407
Public SpeakingAudience AdaptationIn my mind there are many beliefs, ideas, or opinions with which I could form a persuasive speech around. In a persuasive speech the objective is to draw the audiences beliefs closer to yours. If the majority of
TAMU Corp. Chr. - BIO - 1407
C3, C4, and CAMDifferent environments in different regions can not support the needs of all types of plants. Certain plants can not tolerate higher temperatures and others survive better in cooler climates. These plants are seperated into three grou
TAMU Corp. Chr. - BIO - 1407
Three Types of ArchaeaThere are three different types of Archaea. These types are Halophiles, Methanogens, and Extreme Thermophiles. All of these are organisms that are able to live in the Earth's most extreme environments. These three groups are ea
TAMU Corp. Chr. - BIO - 1407
Introductory Biology, Biology 1406. TAMU-CC. final exam8 December 2006. MWF 1:00 1. Antibiotic resistant bacteria are a major public health problem in the US. As a result of antibiotic resistant bacteria, many antibiotics can no longer be used to tr
TAMU Corp. Chr. - BIO - 1407
Memory Through PhotosWhat topic to choose a) Fire Arms. 1. the strength of guns 2. how a gun works b) Automotive. 1. mechanics of internal combustion engine. c) Photography The importance of personal photos. a) take pics for granted. 1. we rely on p
TAMU Corp. Chr. - BIO - 1407
Memory Through PhotographySo I sat on the edge of my bed last night just thinking of what kind of topic I could talk about that my audience would enjoy. I thought of many subjects of which i'm a genius. I thought of presenting a speech on the mechan
TAMU Corp. Chr. - BIO - 1407
TAMU Corp. Chr. - BIO - 1407
A monk named Gregor Mendel studied the reproduction and genetic inheritances of pea plants in an abbey garden. He documented his theories on genotypes, and organism's genetic make-up, and phenotypes, an organism's physical traits. Mendel discovered t
TAMU Corp. Chr. - BIO - 1407
Assignment goals In this assignment you will: Read concept 16.2 in your text book. You will learn how DNA is replicated and repaired. Write a summary of concept 16.2 of your text using the guiding questions. Pay particular attention to the guiding qu
TAMU Corp. Chr. - BIO - 1407
&lt;P&gt; Researchers have been been able to map out the comlete human genome due to the increase in DNA technology over the past few years. DNA contains a list of the human nucleotide sequence that allows us to study more in depth about genetic diseases a
TAMU Corp. Chr. - BIO - 1407
Adaptation EvidenceSo I'm an adult. I have new responsibilities and life is keeping me very busy. There is no down time, no relaxing, no goofing off or passing the time. The time passes quite quickly on it's own, sometimes a little too fast. In &quot;the
TAMU Corp. Chr. - BIO - 1407
U.S. History from 1865 Review Questions for Exam I 1. Explain the effort to ?reconstruct? the south following the Civil War. Who wanted this and who did not? Who won out? Why? How did African Americans come out of the deal? This discussion really is
TAMU Corp. Chr. - BIO - 1407
US History since 1877 Exam III Identification TermsSome number of the terms below will appear on the final examThe Geneva Conference The Geneva Conference what to do with indo china. May, 1954. lest laos and Cambodia go. Take the fighting parties
TAMU Corp. Chr. - BIO - 1407
INDIVIDUAL MITOSIS LAB HOMEWORKThe following relate to the Cell Cycle and the Mitosis lab you completed in biology lab this week. Use the lab manual and the textbook only for these exercises. Neatness and legibility will be a part of the grade if i
TAMU Corp. Chr. - BIO - 1407
&lt;P&gt;Natural selection plays a huge part in evolution. It keeps environments strong by only allowing the organisms with traits most beneficial to the environment to survive. When testing the diversities of guppy populations David Reznick and John Endle
TAMU Corp. Chr. - BIO - 1407
Mitosis Vocabulary Mitosiso Cell cycle- is the orderly sequence of events that accomplish cell reproduction. o Interphase- is a time of synthetic activity of the cell; this is when protein synthesis and DNA replication occur. o Chromatin- during int
TAMU Corp. Chr. - BIO - 1407
Literature CitedAhloowalia, B.S. and Malusynski. Induced mutations-A new paradigm in plant breeding. Euphytica. 118: 167-173.Mohr, H and P, Schopfer. 1994. Plant Physiology. Springer, New York. 464 pp.Searle, S.S. 1973. Environment and Plant Lif
TAMU Corp. Chr. - BIO - 1407
My GoalsShort Term o This semester I have kept up with all school work. o This week I finished all small chores. Laundry Clean Room Groceries Update Calendar o This semester I want to make higher than a 3.0 GPA o I need to remember to call my gr
TAMU Corp. Chr. - BIO - 1407
What I Have LearnedAbout two months ago I was hanging out with some freinds in an on campus appartment. During this time of simply hanging out and watching TV, one of my freinds asked me if I wanted to go throw fireworks off the balcony. We did and
TAMU Corp. Chr. - BIO - 1407
My StoryAn average eight pounds, six ounces was my initial weight at birth. Since that day on May 24th, 1988 I have gained a few pounds, but not only have I grown in size and weight, but I have grown mentally. I have learned to adapt to the ways of
TAMU Corp. Chr. - BIO - 1407
PersonalityAccording to Sigmund Freud there are three parts that make up the structure of personality. One of which was &quot;id&quot;, which is the unconscious portion of the personality that houses the most basic drives and stores repressed memories. Anothe
TAMU Corp. Chr. - BIO - 1407
My GoalsShort Term o This month I would like to finish all assignments way before the assigned due date in order to prevent stress. o This week I want to have all small chores finished. Laundry Clean Room Groceries Make a Calendar o This semeste
TAMU Corp. Chr. - BIO - 1407
Summarizing My Career and Looking AheadToday is May 1, 2007. My second semester of college is coming to an end and I now know what is expected of me and how exactly to tweak my assignments to fit each professor's idea of excellence. I have finally f
TAMU Corp. Chr. - BIO - 1407
What Are My OptionsToday is March 29, 2007. I am well into my second semester of college and now know what is expected of me and how exactly to tweak my assignments to fit each professor's idea of excellence. I have finally found, through nearly eig
TAMU Corp. Chr. - BIO - 1407
AdaptationThe biggest change of my life has just occurred. I have graduated from highschool and moved on to the college life. In my highschool days i slept late, watched alot of TV, played video games, played sports, and pretty much just sat around
TAMU Corp. Chr. - BIO - 1407
Classical vs. Operant ConditioningSince the beginning of time people have been studying behavioral patterns and what causes certain reactions. Learning can be described as a permanent alteration in behavior resulting from experience. These behaviora
TAMU Corp. Chr. - BIO - 1407
Paper #2 InstructionsDUE DATE: April 30thFormat:1. You will need a header in the upper right hand corner that includes your name, which reaction paper you are submitting (# 2) and the date. 2. Use 1 inch margins around all edges of the page. 3. U
TAMU Corp. Chr. - BIO - 1407
Red TideRed tide is a simple name describing a massive number of dinoflagellates invading coastal waters. These dinoflagellates are a group of Alveolata that live in both saltwater and freshwater. They are known for the characteristic of having two
TAMU Corp. Chr. - BIO - 1407
Research Topic SynopsisUltraviolet, visible and near infrared radiation leads to electronic excitation in absorbing molecules. In UV radiation, ionization sometimes occurs in short-wave range. The effects of ionization are taken very seriously. Ther