Homework_2_Solutions - Math 1110 Name: Homework 2 Due 9/8...

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Unformatted text preview: Math 1110 Name: Homework 2 Due 9/8 or 9/9 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 /1 Text exercises. Please do the following problems from the book. §1.5 #2, 3, 18, 22, 30, 35 §1.6 #4, 6, 12, 22, 28, 39ade, 44, 52, 64, 66, 69 §2.1 #6, 10, 14, 16, 22 Question 1. (6 points) (Thomas, §1.5 #34) Tripling your money. Determine how much time is required for an investment to triple in value if interest is earned at an annual rate of 5.75%, compounded continuously. [Your answer may contain a logarithm; you do not need to simplify.] We solve this by using the equation y = Pert where P denotes the initial investment, r denotes the annual interest rate, t denotes the number of years elapsed since the initial investment was made, and y denotes the final value of the investment. In our case, we want to determine the time required for the initial investment to triple in value, so y = 3P. We are told the annual interest rate is 5.75%, so r = .0575. We solve for t: 3P = Pe.0575t 3 = e.0575t ln 3 = .0575t ln 3 = t. .0575 ln 3 Remember that the units for .0575 are “per year,” and hence t = .0575 years . ￿ ￿ If we plug in to a calculator, we find that this is t=19.1063007... years . I nstructor Math 1 110 Name: Sample T rue/False To d emonstrate i n c lass Math 1110 (Fall 2011) HW2 Presentation Problems 2 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 ( x) a nd g (x) b oth o ne-to-one f un - a Question 2. (14 points) Determine whetheri tthe following statements are (always) true or (at least r eason w hy i t's t rue, o r a n e xample w here f ails. sometimes) false, and circle your response. Please give a brief explanation (in complete sentences!) Tnun I F alss (a)aI freasonswhy it’s true, or antexample where it fails. - 4 ) + Z f (x) i a n e venf unction, hen s o i s 9 (x) : 2 ' f ( x – (a) If f(x) is a one-to-one function and is never zero, then the function h ( x) = 1 f ( x) is also one-to-one. We need to show that if there were two x-values x1 and x2 such that h(x1 ) = h(x2 ), then x1 = x2 . So we begin by assuming that there are x1 and x2 such that h(x1 ) = h(x2 ). Then 1 1 substituting, f(x1 ) = f(x2 ) and accordingly, f(x1 ) = f(x2 ). Since f is a one-to-one function, this means that x1 = x2 , and we conclude h is one-to-one. t (b) I f f ( x) a nd g (x) b oth o ne-to-one f unctions d efined o n a ll o f l R., hen f o g i s a lso o ne-to-one. Tnus I F elss (b) If f(x) and g(x) are two functions, then f ◦ g ( x) = g ◦ f ( x) whenever both sides of the equation are defined. Consider f(x) = x2 and g(x) = x + 1. Then f ◦ g(x) = (x + 1)2 = x2 + 2x + 1 whereas g ◦ f ( x) = x 2 + 1 Note f ◦ g(1) = 4, which does not equal g ◦ f(1) = 2. In particular, f ◦ g(x) ￿= g ◦ f(x) as functions. Homework 2 Book Problem Answers: Section 1.5: Section 1.6: Section 2.1: Answers will vary in Exercise 22. ...
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