Homework_11_Solutions_Revised - Math 1110 Name: Homework 11...

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Unformatted text preview: Math 1110 Name: Homework 11 Due 11/21 or 11/22 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. §4.8 #3, 10, 15, 21, 28, 41, 46, 53, 60, 66, 72, 74, 83, 86, 92, 104, 113, 116, 121 §5.1 #1, 6, 10, 13, 18, 19, 21 §5.2 #1, 4, 6, 7, 8, 10, 13, 16, 18, 19, 22, 28, 36, 41, 44 §5.3 #2, 5, 8, 9, 12, 15, 18, 23, 26, 30, 39, 45, 48, 54, 61, 71, 75, 78, 81 §5.4 #6, 9, 16, 27, 30, 36, 39, 42, 48, 50, 53, 59, 62, 66, 69, 72, 74, 83 /1 Math 1110 (Fall 2011) HW5 Presentation Problems 2 Question 1. Please solve the following two problems. (a) Find the function f(x) that satisfies f ￿￿ (x) = −3, f ￿ (0) = 3 and f(2) = 0. Taking the antiderivative of f"(x) = −3, we see that f ￿ (x) = −3x + C1, where C1 is a constant. To determine C1 we compute f ￿ (0) = −3 · 0 + C1 = C1, and we are given that f ￿ (0) = 3. We conclude that C1 = 3. Taking the antiderivative of f ￿ (x) we see that f(x) = −3 x2 + 3x + C2, 2 where C2 is a constant. To determine C2 we compute f(2) = −3 · 4 + 3 · 2 + C2 = C2, and we 2 are given that f(2) = 0. We conclude that C2 = 0 and that f(x) = −3 x2 + 3x. 2 (b) Please write the sum 7 + 1 + 7 + 2 + 7 + 3 + 7 + 4 + 7 + 5 + · · · + 7 + (n − 1) + 7 + n in “sigma notation” and find a closed form expression for it. What is the value of the sum when n = 10? The expression is equal to n ￿ (7 + n) = k=1 n ￿ k=1 10 · 11 For n = 10, we have 7 · 10 + = 125. 2 7+ n ￿ k=1 k = 7n + n(n + 1) . 2 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 Presentation (Problems s entences!) 3 Math 1110 (Fall 2011) HW5 e xplanation in c omplete (b) I f f ( x) a nd g (x) b oth o ne-to-one f un w here i t f ails. - a r eason w hy i t's t rue, o r a n e xample Tnun I F alss Question 2. n e venf unction,t hen s following 2 ' f ( x - 4 + Z (a) I f f (x) i s aDetermine whether the o i s 9 (x) : statements) are (always) true or (at least sometimes) false, and circle your response. Please give a brief explanation (in complete sentences!) – a reason why it’s true, or an example where it fails. (a) The function f(x) = sin(cos(x)) + bers a and b. x3 + 7 is integrable on the interval [a, b] for any real numex The function f is continuous on the entire real line, so it is integrable on the interval [a, b] for any real numbers a and b. 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) The area of the region bounded by sin x and the x-axis, between x = −π and x = π is computed by ￿π sin x dx. −π The expression sin x is negative for x between −π and 0, so the area of the region bounded by sin x and the x-axis between x = −π and x = π is ￿π −π However, | sin x| dx = ￿0 − sin x dx + −π ￿π 0 ￿π −π ￿0 ￿π sin x dx = cos t + (− cos t) = 2 −π ￿π sin x dx = cos t =0 −π 0 Homework 11 Book Problem Answers: Section 4.8: Section 5.1: Section 5.2: Section 5.3: Section 5.4: ...
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This note was uploaded on 03/01/2012 for the course MATH 1110 at Cornell University (Engineering School).

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