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Math 1170 MOCK FINAL 1 Page 2 of 15 pages 1. [5 marks] Let: f(x) = 2x 1− 3x , evaluate and simplify as much as possible the expression f(x+ h)− f(x) h where h 6= 0. Math 1170 MOCK FINAL 1 Page 3 of 15 pages 2. [5 marks] Find an expression for the polynomial f(x) of degree 4 whose graph is given below: Math 1170 MOCK FINAL 1 Page 4 of 15 pages 3. [9 marks] A restaurant charges $10 per person for a buffet. The average number of customers is 120 per day. From previous experience, the owner estimates that for every $0.50 increase in price, the number of customers decreases by 4 per day. (a) Let x be the price per person. Express the revenue per day R as a function of x. (b) If the restaurant pays an overhead of $100 per day, and the food cost is $4 per customer, express the profit per day P as a function of x. (c) Based on your answer in (b), what price should the restaurant set to maximize its profit? Math 1170 MOCK FINAL 1 Page 5 of 15 pages 4. [7 marks] A drug is eliminated from the body through the kidney in such a way that over each hour, 25% of the amount present at the beginning of the hour is eliminated. Let t be the amount of time (in hours) since the drug was first taken, and A0 the initial amount of the drug in the body. The amount of drug present in the body after t hours is given by: A(t) = A0e kt, where k is some constant. (a) Find the value of k. (b) Using the value of k from part (a), how long does it take before the amount of drug in the body is half of the initial amount? Math 1170 MOCK FINAL 1 Page 6 of 15 pages 5. [15 marks] Consider the rational function: f(x) = x3 − x2 − 2x 2x2 − 2x− 12 . (a) Find the factored form of f(x). Double-check your factoring! [3 marks] Using the factored form in part (a), identify the following features of the graph y = f(x) in part (b)-(f) (if any): (b) x-intercept(s): [3 marks] (c) y-intercept: [1 mark] (d) hole(s): [1 mark] Math 1170 MOCK FINAL 1 Page 7 of 15 pages (e) equation(s) of vertical asymptote(s): [3 marks] (f) equation of horizontal/slant asymptote: [2 marks] (g) Does the inverse function f−1 exist? Why or why not? [2 marks] Math 1170 MOCK FINAL 1 Page 8 of 15 pages [20 marks] 6. Solve the following equations: Give exact answers as well as approximate answers to two decimal places. (a) 2 log(x+ 1)− log(−x)− 1 = 0 [5 marks] (b) sin(2x) + √ 2 cos(x) = 0 where x ∈ [0, 2π) [5 marks] Math 1170 MOCK FINAL 1 Page 9 of 15 pages (c) 2x − 6(2−x) = 6 [5 marks] (d) x = arccos ( − √ 2 2 ) − cos ( arcsin (√ 5 3 )) Your answer must be completely simplified. [5 marks] Math 1170 MOCK FINAL 1 Page 10 of 15 pages [8 marks] 7. A satellite circles the earth in such a way that t minutes after launch, its distance D (in km north or south of the equator, altitude not considered) is modelled by the equation: D(t) = 4800 cos ( π 45 (t− 10) ) . The graph of D during the first 100 minutes after launch is shown below. During the first 100 minutes after launch, when is the satellite within 2400 km of the equator? (Hint: It might help to think about the proportions of its cycle when the cosine curve is less than half its amplitude.) Your answer must be exact. Do not simply use the graph to approximate. Math 1170 MOCK FINAL 1 Page 11 of 15 pages (BLANK PAGE TO SHOW WORK FOR QUESTION 7) Math 1170 MOCK FINAL 1 Page 12 of 15 pages 8. [9 marks] The body temperature varies according to a rhythm that repeats itself every 24 hours. It is highest around 5 pm, and is lowest around 5 am. Let T (t) be the body temperature (in ◦F ) at time t, where t = 0 corresponds to midnight. The low and high body temperatures are 98.3◦F and 98.9◦F respectively. (a) Find an equation of the form T (t) = a sin(bt+ c) + d that fits this information. [5 marks] (b) Sketch a rough graph of y = T (t) on [0, 24]. [4 marks] Math 1170 MOCK FINAL 1 Page 13 of 15 pages 9. [9 marks] A straight road makes an angle of 15◦ with the horizontal. When the angle of elevation of the sun is 57◦, a vertical pole at the side of the road casts a 75 feet long shadow directly down the road. Approximate the length of the pole to the nearest foot. Math 1170 MOCK FINAL 1 Page 14 of 15 pages 10. [8 marks] A triangular plot of land has sides of lengths 80 ft, 70 ft, and 60 ft. Approximate, to the nearest tenth of a degree, the largest angle between the sides. You must provide a rough diagram. Math 1170 MOCK FINAL 1 Page 15 of 15 pages 11. [6 marks] For each of the following statements, either prove that it is an identity OR provide a value x (and y) for which it is false. (a) sin(x+ y) = sin(x) + sin(y). [3 marks] (b) ln(cos(2x) + 2 sin2(x)) = 0. [3 marks] The End ANSWER KEY 1. 2 (1− 3x)(1− 3h− 3h) 2. f(x) = −1 3 (x+ 3)(x− 1)2(x+ 1) 3. (a) R = x(200− 8x) (b) P = −8x2 + 232x− 900 (c) $14.50 4. (a) k = ln(0.75) (b) ln(0.5) ln(0.75) ≈ 2.41 5. (a) f(x) = x(x+ 1)(x− 2) 2(x+ 2)(x− 3) (b) x = 0,−1, 2 (c) y = 0 (d) none (e) x = −2, x = 3 (f) y = 12x (g) f−1 does not exist because f is not one-to-one (f(2) = f(−1) = 0). 6. (a) x = √ 35− 6 ≈ −0.08 (b) x = π 2 , 3π 2 , 5π 4 , 7π 4 (c) x = log2(3 + √ 15) ≈ 2.78 (d) x = 3π 4 − 2 3 ≈ 1.69 7. [25, 40] ∪ [70, 85] 8. (a) T (t) = 0.3 sin ( π 12 t− 11π 12 ) + 98.6 (b) 9. 92 feet 10. 75.5◦ 11. (a) Not an identity. For x = y = π4 , sin(x + y) = sin ( π 2 ) = 1, which is not equal to sin(x) + sin(y) = sin ( π 4 ) + sin ( π 4 ) = √ 2; (b) Identity.