When you finish this lesson, you will complete the Essay Question and Discussion requirement below.
At the end of this session you will be able to
1. In your textbook, review continuity on pages 124 through 126 of Section 2.6 and on pages 154 and 155 in Section 3.1.
2. Review the notes on CONTINUITY in the Session 6 notes.
3. In your textbook, study Section 4.1, pages 244 through the middle of page 246, practicing Example 1 on a separate sheet of paper without looking at the solution in the textbook. Insure that you learn what the definition on absolute extreme value states on page 244 and insure that you know the hypothesis and conclusion of the extreme-value theorem on page 246.
4. If you have not done it, now study the notes on ABSOLUTE (GLOBAL) EXTREME VALUES OF A FUNCTION in the Session 12 notes.
At the end of this session you will be able to
1. In your textbook, review absolute extrema definition on page 244 of Section 4.1
2. In your textbook, study Section 4.1 from the bottom of page 246 through the top of page 252, practicing Examples 2 through 5 on a separate sheet of paper without looking at the solution in the textbook. Insure that you learn what the definition on local extreme value states on page 247.
3. If you have not done it, now study the notes on LOCAL (RELATIVE) EXTREMA VALUES OF A FUNCTION in the Session 12 notes.
4. On pages 252 - 253 of your textbook, practice 1, 3, 5, 7, 9, 11, 13, 15, 19, 25, 35, 39, 41 (any vertical lines are not part of the graph, they are asymptotes), 43, 45 (factor x^(-1/3) to simplify), 49, 53 (a), 39 (b), 39 (c) and 59.
At the end of this session you will be able to
1. In your textbook, review the "test" for continuity in Section 2.6, page 12.
2. Review the notes on slope found in THE EQUATION OF A TANGENT LINE in the Session 6 notes.
3. In your textbook, study Section 4.2 from the middle of page 255 through the top of page 257.
4. If you have not done it, now study the notes on ROLLE'S THEOREM in the Session 12 notes.
5. Essay Question: Verify that Rolle's Theorem is satisfied by f(x) = 2 x ^ 2 - x - 3 on the closed interval from -1 to 1.5. That means first to show that three conditions of the hypotheses is satisfied for this function. Then show the value, if any, of x that makes the conclusion true. In other words, try to find value of x that makes the derivative equal to zero.
To answer this question, you are to create your answer as a word processing document titled "Rolle / your name" and save it as an Rich Text Format (RTF). Submit your document to the instructor's Digital Drop Box:
If you are at MyMathLab (CourseCompass), then close this window. If you are not at MyMathLab, then go to coursecompass.com. Then click on the (Tools) button at the left and then click on the Digital Drop Box.
6. On page 260 of your textbook, practice 9. (Hint: This question requires you to think about the hypothesis of Rolle's Theorem.)
At the end of this session you will be able to
1. In your textbook, review average rate of change on page 75 of Section 2.1 and instantaneous rate of change on page 171 of Section 3.3.
2. In your textbook, study Section 4.2 from the top of page 257 through the bottom of page 258, practicing Examples 3 and 4 on a separate sheet of paper without looking at the solution in the textbook..
3. If you have not done it, now study the notes on MEAN VALUE THEOREM in the Session 12 notes.
4. Essay Question: Verify that the Theorem of the Mean is satisfied by f(x) = x ^3 on the closed interval from -1 to 2. That means first to show that both conditions of the hypotheses is satisfied for this function. Then show the value, if any, of x that makes the conclusion true. In other words, try to find value of x that makes the derivative equal to (f(b) - f(a)) / (b - a). Of course, to show the conclusion is true, you will need to evaluate the function at both endpoints to find the values f(b) and f(a).
To answer this question, you are to create your answer as a word processing document titled "Mean Value / your name" and save it as an Rich Text Format (RTF). Submit your document to the instructor's Digital Drop Box:
If you are at MyMathLab (CourseCompass), then close this window. If you are not at MyMathLab, then go to coursecompass.com. Then click on the (Tools) button at the left and then click on the Digital Drop Box.
5. On page 260 of your textbook, practice 1, and 3, 5, 7, 15.
At the end of this session you will be able to
1. In your textbook, study Section 4.2 from the bottom of page 258 to the bottom of page 259, practicing Example 5 on a separate sheet of paper without looking at the solution in the textbook.
2. If you still need a help with extreme values and these theorems and if you are at MyMathLab, then close this window and click on "4.1 Extreme Values of Functions" or "4.2 The Mean Value Theorem". The video lectures give a nice overviews of extreme values and the mean value. If you are not at MyMathLab.com (CourseCompass), go to login MyMathLab.com and click on "Chapter Contents" in the left margin. Then click on "Unit III: Sessions 12, 13, 14, 15, and 16: Chapter 4: Applications of Derivatives" and then on "4.1 Extreme Values of Functions" or "4.2 The Mean Value Theorem".
3. If you have not done it, now study the notes on CONSEQUENCES OF THE MEAN VALUE THEOREM in the Session 12 notes.
At the end of this session you will be able to
1. In your textbook, review derivatives of trigonometric functions in Section 3.4, pages 183 through 187.
2. In your textbook, review higher-ordered derivatives of power functions on page 168 of Section 3.2.
3. In your textbook, review velocity and acceleration on pages 172 through 175 of Section 3.3
4. In your textbook, study Section 4.2 from the bottom of page 259 through the top of page 260, practicing Example 5 on a separate sheet of paper without looking at the solution in the textbook. You will notice that there are several parts and steps to Example 5.
5. If you have not done it, now study the notes on DIFFERENTIAL EQUATION in the Session 12 notes.
6. On page 261 of your textbook, numbers 23, 27, 29 (write with negative exponent), (For the remaining practice, you must calculate "C") 33, 35, 37, 41, and 45.
ESSAY QUESTION:
In your own words, answer the following questions:
What is an absolute maximum and absolute minimum value?
What conditions must be true to guarantee that you will have both absolute extema?
What is a local maximum and local minimum value?
What is a critical value?
Why is the critical value important?
You may give examples to illustrate your statements. Use your own words as if you are explaining to someone who is meeting the extrema concept for the first time. DO NOT COPY DEFINITIONS AND EXPLANATIONS FROM THE TEXTBOOK OR FROM THE COURSE DOCUMENTS.
To answer these questions, you are to create a paragraph as a word processing document titled "Extreme / your name" and save it as an Rich Text Format (RTF). Submit your document to the instructor's Digital Drop Box:
If you are at MyMathLab (CourseCompass), then close this window. If you are not at MyMathLab, then go to coursecompass.com. Then click on the (Tools) button at the left and then click on the Digital Drop Box.
DISCUSSION BOARD POSTING
If you are at MyMathLab (CourseCompass), then close this window. If you are not at MyMathLab, then go to coursecompass.com.
Click on the (Communications) button at the left, click on the Groups link, click the underlined group, click on Group Discussion Board, and then click on the "Solutions" discussion forum. Click on the "Extrema" thread and post your solution to two of the problems listed below. Select two problems another student has not yet answered. In your posting, give a complete solution and/ or explanation of how you obtained your answer. In the subject line of your posting, give the page and problem number.
You are expected to locate errors in any of the other postings. If you are the first to see the error, then you are to help your classmate correct his/her posting. Class participation and helping others is part of this course. Therefore, you who consistently are quick to help your classmates correct errors will receive extra credit for class participation.
On page 252 of your textbook, numbers 2, 4, 6, 8, 10, 12, 14, 16, 22, 26, 36, 40, 50, 54, and 60.
On page 260 of your textbook, numbers 2, 4,6, 8, 16, 24, 28, 30, 34 (Use negative exponent for 1/x), 36, 38, 42, and 50.