Homework_1_Solutions - Math 1110 Name: HW 1 Solutions Due...

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Unformatted text preview: Math 1110 Name: HW 1 Solutions Due 9/1 or 9/2 in class Please print out these pages. 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; you will be given written feedback on these problems. Please 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. Tara Holm GRADES Text exercises 20 / 20 Pres probs 20 / 20 Staple 1 /1 Text exercises. Please do the following problems from the book. §1.1 # 4, 8, 10, 11, 24, 28, 32, 34, 40, 54, 66, 72 §1.2 #4, 6, 8, 11abc, 24, 26, 42, 47, 48, 59 §1.3 #6, 7, 18, 43, 50, 55 The following problems are recommended if you would like to practice with your graphing calculator, or play with one online (here, for example). You should not turn these in. §1.4 #2. 3. 20, 26, 33 Presentation problem 1. (6 points) (Thomas, §1.3 #52) For 0 ≤ θ ≤ 2π, find the values of θ that satisfy sin2 θ = cos2 θ. We know that sin2 θ + cos2 θ = 1, so if sin2 θ = cos2 θ, we must have sin2 θ = cos2 θ = 1 . That 2 1 means we are looking for values of θ where sin θ = ± √2 . For those θ, it will also be true that 1 cos θ = ± √2 . We can find such a θ using the right triangle drawn in Figure 1. This gives us an answer θ = π . We also get the same triangle, reflected 4 π π π around R2 , when θ = 34 , θ = 54 , or θ = 74 . Thus, our final answer is θ= π 3π 5π 4, 4 , 4 or 7π 4. ! θ ! Figure 1 Alternatively, we could look at the unit circle and see where it intersects the lines Figure 2 y = x and y = −x. Because points on the unit circle have (x, y)-coordinates equal to (sin θ, cos θ), the intersection points shown Figure 2 tell us the angles where sin2 θ = cos2 θ. These are the same four values of θ given above. Math 1110 (Fall 2011) HW1 2 Presentation problem 2. (6 points) (Thomas, §1.1 #64) In the figure shown below, there is a rectangle inscribed in an isosceles right triangle whose hypotenuse is on the x-axis and is 2 units long. (a) Express the y-coordinate of P in terms of x. (You might start by writing an equation of the line that joins A and B.) Let’s call the horizontal axis the t–axis. The line joining A and B is described by the equation $ y = −t + 1. %&'&()&*&++, The point P is on that line, so it has coordinates (x, −x + 1). In particular, the y–coordinate of the point P is −x + 1 . # !" - ) " (b) Express the area of the rectangle in terms of x. The shaded rectangle has width 2x, because it goes along the t–axis from −x to x. As we figured out above, it has height −x + 1. Thus, the area of the rectangle is A = (width) · (height) = 2x(−x + 1) = 2 − 2x2 . Presentation problem 3. (8 points) The graph of f(x) is shown in the figure below. It has ￿ domain ￿ [−4, 2] and range [0, 4]. Please sketch the graph of the function g(x) = − 1 f(x − 1) + 3 on the 2 same set of axes, and indicate its domain and range in the space provided. !"#$%& $'( The graph of g(x) is shown on the graph to the left in red. It has Domain(g(x)) = [−3, 3] !"#$%& !'( $"#$%& !"'$%& Range(g(x)) = [−5, −3] Math 1110 Fall 2011 Homework 1 Book Problem Answers: Section 1.1 Section 1.2: Section 1.3: Section 1.4 Optional Problems: ...
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