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1873_solutions - An Interactive Introduction to...

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An Interactive Introduction to Mathematical Analysis: Instructor’s Manual This printable form of the instructor’s manual is a companion to the on-screen instructors manual for my book. To install the on-screen instructor’s manual, you need to convert your installation of the book from a student version to an instructor’s version. After installing the book, you should run the executable file lewin-analysis-book-instructors-manual.exe which can be found at the address http://science.kennesaw.edu/~jlewin/analysis/instructors-manual/lewin-analysis-book-instructors-manual.exe http://science.kennesaw.edu/~jlewin/CUP/lewin-analysis-instructor’s-manual.exe The running of this executable file requires a password that can be obtained by bona fide instructors at [email protected] In both the student version and the instructor’s version of the text, each group of exercises comes with a link to a solutions document. In the instructors version, the link takes you to a solutions document that shows the solutions provided to students in blue and shows the solutions provided only to instructors in green. If you make a monochrome print of this printable form of the manual, then the solutions provided to students will appear in a bold italics font and the solutions provided only to instructors will appear in an upright sans serif font. Jonathan Lewin 2 Mathematical Grammar Exercises on Use of Quantifiers Except in Exercise 2, decide whether the the sentence that appears in the exercise is meaningful or meaningless. If the sentence is meaningful, say whether what it says is true or false. 1. a. x 2 ± x . Solution: This statement is meaningless because x is unquantified. 1
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b. For every real number x we have x 2 ± x . Solution: This statement is meaningful but false because the equation x 2 ± x is false whenever x is negative. c. For every positive number x we have x 2 ± x . Solution: This statement is true. 2. a. Point at the expression x 2 andclickonthe Evaluate button . b. Point at the expression x 2 Simplify button . c. Point at the equation x 2 ± x , open the Maple menu and click on Check Equality. d. Point at the equation x ± ± 2 and click on the button to supply the definition x ± ± 2to Scientific Notebook . Then try a Check Equality on the equation x 2 ± x . 3. For every number x and every number y there is a number z such that z ± x ² y . Solution: This statement is true. 4. For every number x there is a number z such that for every number y we have z ± x ² y . Solution: This statement is false. 5. For every number x and every number z there is a number y such that z ± x ² y . Solution: This statement is true. 6. sin 2 x ² cos 2 x ± 1. Solution: This statement is meaningless because x is unquantified.
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