OC userin Mathematics·9 Jun 20183. (a) Assume that a function f(x) is concave down on its whole domain, and f(0) = -2 and f'(0) = -1. Circle each statement that must be true: • The function f(2) has no inflection points. • The derivative f'(2) is always negative. • The derivative satisfies f'(3) < -1. • The derivative satisfies f'(-3) < -1. • The function f(x) 0. (b) Sketch a graph of a function f() satisfying all of the following conditions: • The derivative function f'(x) is decreasing when x < 1 and when r > 3. • The derivative function f'(2) is increasing when 1
OC userin Mathematics·7 Jun 20181. a. Complete the definition of a simple random sample below by filling in each of the three blanks with a single word. A simple random sample (SRS) of size n consists of n individuals chosen from a population in such a way that set of n individuals has a/an of being the sample that is actually selected.
OC userin Mathematics·9 Jun 20189. Prove that if w is an nth root of unity, then is also an nth root of unity. (Recall: The nth roots of unity are the nth roots of 1.) (4 marks]
OC userin Mathematics·9 Jun 201810. The integra 1210-3dre A) 4112 B) 3 (4-3in 2) CO D ) E) none of these
OC userin Mathematics·7 Jun 20184. You want to buy a three scoop cone at Baskin-Robbins (with the three scoops stacked one at the bottom, one in the middle, and one at the top, as usual). How many different choices do you have if you do care about the order of the flavors? (That is, Choco- late/Vanilla/Chocolate is different from Chocolate/Chocolate/Vanilla. Recall that there are 31 different flavors!) (A) C(31,3) (B) P(31,3) (C) 93 (D) 313 (E) 331 (F) none of the above
OC userin Mathematics·7 Jun 20183. (8 marks) A function y(1) obeys this relationship: y' = ry² + ar?, where a is a constant. (a) Determine the value of a such that the tangent line to the graph of y(«) at (1,1) passes through (2,5). (b) For this a value, determine the concavity of the function at r= 1.
OC userin Mathematics·6 Jun 201822. A professor always gives either a fast-paced lecture or a moderately-paced lecture. Also, she has coffee at breakfast, on the average, 3 out of 4 days that she gives a lecture. If she has coffee, her lecture is fast-paced with probability 0.8. However, on momings with no coffee, her lecture is moderately-paced with probability 0.4. Suppose we know that she has given a moderately-paced lecture. What is the probability that she did not have coffee? (A) (B) (C) (D) (E)
OC userin Mathematics·7 Jun 20186. The Post Office accepts a cylindrical package only if the height plus the girth of the cylinder (h + 2nr) is not more than 42. What are the height and base radius of the cylindrical package of largest volume that can be mailed? V = ar?h; (h + 2nr) = 42 h = 42 - 27 V = ar? (42 – 29r) = 42 r2 – 27²,3 V' = 82nr – 67²p2 = 67r[14 – Tr] = 0 =r=14 h = 42 – 21(14) = 42 - 28 = 14
OC userin Mathematics·6 Jun 20182. There are 30 guinea pigs introduced into an ecosystem, and their population grows at a rate proportional to its size. In 2 years, the population grows to 85 guinea pigs. (a) Find a formula for the function P(t) that gives the population of guinea pigs at any time t measured in years. [5] [4] (b) Using your answer to part (a), determine how long it would take the guinea pig population to quadruple.
OC userin Mathematics·5 Jun 201819. The results obtained from a test of significance can always be trusted as long as you use a random sample. A) True B) False
OC userin Mathematics·8 Jun 20182. (a) If $4,000 dollars is invested in a bank account at an interest rate of 6% per year, compounded continuously. How long will it take, in years, for the balance to reach $20,000 dollars? Answer:
OC userin Mathematics·5 Jun 20183. Consider the system of equations represented as the matrix equation, AT = b[12 marks]. (a) Use the method of inverses to solve the system of equations, and clearly show your answer: 5.0 – 2y = 1 -4.x + 2y = 2 (b) If A-1 = [14-11 11 2 O 3 , and 5 = | 2 ), then evaluate . Clearly show your answer. [ 7 2 1] (c) If 4-1 - 2 31 1 2 , and b then what is the ORIGINAL system of equa- tions?
OC userin Mathematics·6 Jun 2018if x>1 have? 7. How many points of discontinuity (if any) does the function f(x)= x 2 ifxsi A) 2 B) 1 C) D) infinitely many E) none of these