OC userin Mathematics·19 Dec 20177. Let p(x) = 3x4 – 2r3 + x – 5. (C) Find the quotient ? 2 + . State the quotient and residue. 1
OC userin Mathematics·19 Dec 20177. Suppose a function f has a local maximum at r=0 with f(0) = 4, f"(x) > 0 for 2 € (-1,0) and f"(x) > 0 for x = (0,2). What can you say about f'(0)? Hint: It might help to try to sketch the graph of f(x) near x = 0.
OC userin Mathematics·18 Dec 2017(4) A spherical balloon is inflated at a rate of 5cm3/s. At what rate is its radius increasing, when the balloon is 30 cm in diameter? Hint: The volume of a sphere is given by V =
OC userin Mathematics·20 Dec 20174. A computer randomly prints a three-digit string of numbers. The string consists only of the digits 0, 1, 2, or 3 and satisfies the extra condition that the sum of its digits is 3. As an example, 021 is one such string. e. Use your answers from part d. to find the probability P(G) of selecting a 0 digit from any randomly printed string.
OC userin Mathematics·17 Dec 20175. Evaluate the following expressions. (e) p(-3), where p(x) = 24 – 3x2 + 8x – 2, using the remainder theorem.
OC userin Mathematics·19 Dec 20172-- if x > 0 1. Let f(x) = log2(2), g(x) = sin (52) and h(z) = { -322 - 42 +1 if a so Evaluate the following: (a) (hof)(4)
OC userin Mathematics·19 Dec 20175. The Infinite House of Pancakes (IHOP) offers 35 dif- ferent kinds of pancakes. You would like to order a stack of three pancakes. How many different choices do you have, considering that you care about the or- der in which the pancakes are stacked? (In other words, pecan/blueberry/pecan is a different choice from pecan/pecan/blueberry.) (a) P(35,3) (b) 75 (c) 35% (d) 335 (e) C(35,3) (f) none of the above
OC userin Mathematics·18 Dec 20175. The function f(x) = 2x5 - 5x4 - 10x3 +6 has a local maximum at x =
OC userin Mathematics·18 Dec 20171. (10 marks/ The top and bottom margins of a poster are each 4 cm and the side margins are each 2 cm. If the area of printed material on the poster is fixed at 50 cm”, find the dimensions of the poster with the smallest area. Check that the value obtained is indeed a minimum.
OC userin Mathematics·17 Dec 20173 21. There are 3 white balls and 4 black balls in an urn. Balls are drawn from the urn at marks random, one by one without replacement, until a black ball is drawn. (a) Draw a probability tree which models this stochastic process. Be sure to put appro- priate probabilities on all branches. (b) What is the probability that eractly 3 balls will be drawn? DO NOT SIMPLIFY YOUR ANSWER.
OC userin Mathematics·17 Dec 2017[1] 1. (a) Solve 2 – 31 < 7. Solution: We have – 7 0 on (-0, 1] U [2,). Next, we need to see where Vr2 – 3r +2 -1 > 0. If r2 - 3.c + 2 – 1 > 0, then V.22 – 3.0 +2 > . Observe that this is definitely true if r < 0. If r > 0, then squar- ing both sides we get 22 - 3.+2 > 1 2 > 3. V A VICO Thus, the domain of f is (-00, ). [2] (c) Find the exact value of tan(sec-14). Solution: Let y = sec-14. Then, sec y = 4. From the diagram we get that tan(sec-? 4) = tan y = 15 [2] (d) Find all r such that sin(2.x) = COS I. Solution: We have 2 sin r cos r = COS I, SO 0 = 2 sin Cos I -Cos I = cos .r (2 sin c - 1). Hence, sin(2x) = cost when cos x = 0, or when sin r = Thus, r= +ik, kez, I= +2nk, ke Z, or r= +2nk, kez [2] (e) State the precise mathematical definition of lim f(x) = 0. Solution: For every e > 0 there exists a 8 >0 such that if 0 < x-al
OC userin Mathematics·17 Dec 20175. Find the equation of the circle with center at (2, -1) and going through the point (5, 4).