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Math 121, Quiz Two. 4 points each; score is out of 25. 1. What best describes the path of a particle whose position is descried by the following parametric equations: x(t ) cos( 2t ), y sin( 2t ),0 t 2 a) Travels in a counterclockwise direction around a circle twice, starting at (1,0). b) Travels in a counterclockwise direction around a circle once, starting at (1,0). c) Travels in a clockwise direction around a circle once, starting at (0,1). d) Travels in a counterclockwise direction half-way around a circle, starting at (1,0). (answer: A; we go around once as “t” goes from 0 to 1 because the “t” in the argument is multiplied by 2 ). 2. Under ideal conditions a certain bacteria population is known to triple every 4 hours. Suppose that there are initially 100 bacteria. How many bacteria are there in 9 hours? 9 a) b) 100 * 2 4 100 * 2 4 9 3 9 c) 100 * 2 d) 100 * 3 4 e) 100 * 3 9 9 4 t Answer: D; the formula for the population at time t is P(t ) 100 * 3 4 as one can readily check by plugging in 0 for t (the initial population) and by plugging in “4” for t (the time to triple). 0 a 1 . Which best describes the graph of y a x ? 3. Suppose a) b) c) d) The graph is increasing everywhere and has neither origin nor y-axis symmetry The graph is increasing everywhere and has y-axis symmetry The graph is decreasing everywhere and has origin symmetry The graph is decreasing everywhere and has neither origin nor y-axis symmetry Answer: D; because a is less than one the graph decreases (example: if a < 1, then a a ) and since 2 a x is always positive, origin symmetry is not possible and because a x a x graph cannot have y-axis symmetry. 3 1 (unless a = 1) the a 4. Let f be the inverse function for the function described in problem 3. Note that f ( x) log 1 ( x). 2 Which of the following graphs is the graph of f ? Circle the correct letter Answer: the graph for f has to be decreasing and negative for x > 1. So the upper left hand graph (A) is correct. 5. a) b) If f ( x ) 3 x 1 , find the inverse function x 1 3 1 f 1 ( x) 3x 1 f 1 ( x) f 1 c) f 1 ( x) d) f 1 x 1 3 does not exist because f is not a one-to-one function. Answer: (a) again! A straight forward algebra calculation. 6. a) b) c) d) Which function most accurately describes the following graph? y 5 * (3 x ) y 3 * (2 x ) y 3 * (2 x ) y 2 * (3 x ) Answer: (b). (c) and (d) are eliminated as the graphs of these functions decrease. (a) is eliminated as the y-axis intercept is less than 5. 7. (a challenge) What best describes the path of a particle whose position is described by the following parametric equations: x(t ) 2 cos( 2(t a) )), y (t ) 3 sin( 2(t )) , 0 t 2 2 2 Travels twice in a counterclockwise direction around a circle centered at the origin starting at the point (2,0) . b) Travels twice around in a clockwise direction around an ellipse centered at the origin starting at the point (0,3) . c) Travels twice around in a counterclockwise direction around an ellipse centered at the origin starting at the point (2,0) . d) Travels halfway in a counterclockwise direction around a circle centered at the origin starting at the point (2,0) . x 2 y 3 Answer: (c) : note that ( ) ( ) 1 so the equation describes an ellipse centered at the origin. The 2 2 start point is easy to compute (plug in t = 0) and, because t is multiplied by 2, we go around twice.