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7 Pages

### Homework_4_Solutions

Course: MATH 1110 1110, Fall 2011
School: Cornell
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Word Count: 700

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1110 Name: Homework Math 4 Due 9/22 or 9/23 in class Please print out these pages. Write your answers to the text exercises on separate paper and staple it to these pages. You should include computational details. These problems will be assessed for completeness. Always write neatly and legibly. Please answer the presentation problems in the spaces provided. Include full explanations and write your answers in...

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1110 Name: Homework Math 4 Due 9/22 or 9/23 in class Please print out these pages. Write your answers to the text exercises on separate paper and staple it to these pages. You should include computational details. These problems will be assessed for completeness. Always write neatly and legibly. Please answer the presentation problems in the spaces provided. Include full explanations and write your answers in complete, mathematically and grammatically correct sentences. Your answers will be assessed for style and accuracy, and you will be given written feedback on these problems. GRADES Text exercises / 20 Pres probs / 20 Staple Text exercises. Please do the following problems from the book. 2.5 # 4, 6, 7, 18, 20, 32, 33, 40, 44, 56, 60 2.6 #2, 14, 29, 38, 47, 50, 54, 66, 70, 73, 84, 100 /1 Math 1110 (Fall 2011) HW4 Presentation Problems 2 Question 1. (10 points) On a recent trip to the Department of Motor Vehicles (DMV), Professor Holm noticed a sign describing the ne for writing the DMV a bad check: one that is subsequently returned for insufcient funds in the bank account. The sign indicated that if the bad check was for \$200.00 or less, there would be a \$35.00 ne; and if the bad check was in excess of \$200.00, the ne would be 15% of the total amount of the check. Let f(x) be the function whose value at x is the ne (in dollars) imposed on a bad check of x dollars. (a) Complete the equation of this piece-wise dened function: 35 0 < x < 200; f(x) = x = 200; and 35 .15x x > 200. Note that the units of f(x) are dollars. (b) Is this function continuous at x = 200? Why or why not? We need to check that limx 200 f(x) = f(200). This is the same as checking that limx 200+ f(x) = limx 200 f(x) = f(200). Note that limx 200+ f(x) = limx 200+ .15x = 30, but f(200) = 35. We conclude that f is not continuous at x = 200. (c) Is this function non-decreasing for 0 < x < 300? Why or why not? We need to determine if for 0 < < x1 x2 < 300 it is always the case that f(x1) f(x2). Note that f(100) = 35 > 30.15 = f(201), but 100 < 201. So the function is not non-decreasing for 0 < x < 300. I nstructor Math 1 110 Name: Sample T rue/False To d emonstrate i n c lass Presentation p roblem 1 . D etermine w hether t he f ollowing s tatements a re ( always) t rue o r ( at l east sometimes) f alse, a nd c ircle y our r esponse. P leaseg ive a b rief e xplanation ( in c omplete s entences!) (b) I f f Math 1110 (Fall 2011) HW4 Presentation Problems( x) a nd g (x) b oth o ne-to-one f un 3 - a r eason w hy i t's t rue, o r a n e xample w here i t f ails. Tnun I (x - 4) Z (a) I f f (x) i s a (10 venf unction,t hen swhether :the 'following+statements are true or false, F alss Question 2. n e points) Determine o i s 9 (x) 2 f and circle your response. Please give a brief explanation (in a complete sentence!) a reason why its true, or an example where it fails. (a) Suppose f(x) is a function dened for all real numbers, and that x2 f(x) x2 for every x. Then f(0) = 0 and f(x) is continuous at x = 0. Evaluating the expression x2 f(x) x2 at x = 0, we get 02 f(0) 02. In particular, 0 f(0) 0, if f(0) is dened. Since we are given that f is dened for all real numbers, we may conclude that f(0) = 0. Further, note that lim x2 = 0 = lim x2. We may conclude by the sandwich theorem that x 0 x 0 lim f(x) = 0. In particular, lim f(x) = 0 = f(0), so f(x) is continuous at x = 0. o t ( x) (b) I f f x 0a nd g (x) b oth o ne-to-one 0unctions d efined o n a ll o f l R., hen f x f g i s a lso o ne-to-one. Tnus I F elss (b) If lim+ f(x) = lim f(x), then f(x) is continuous at x = a. x a x a Consider the example: f(x) = 0 x < 0; 1 x = 0; and 0 x > 0. We have lim f(x) = 0 = lim f(x), and so lim f(x) = 0. The function is not continuous at + x 0 x 0 x = 0, however, because f(0) = 1. x 0 Homework 4 Book Problem Answers Section 2..5: Section 2.6:
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