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1. In some town there were To = 10" trucks So = 3. 10" SUV and Co = 5.10' cars. Each year the number of trucks T SUV - S and cars C changes according the directed graph S & 13 Let Thy Sn, C. are the numbers of trucks, SUV and cars in noth year. 1.1 Write equations between Tn Sn, Cn and 1. 2 Put Un= \\ Su N = 0, 1 , 2 ... L Ck and rewrite equations from problem I.1 in vector form : Unti = Aun . Express Un in terms of A and we. Can we find an orthonomal basis of engenvectors of matrix A ? Why ? 1.4 Find eigenvalues of matrix A. 15. Find orthonomal basis of eigenvectors of matrix A.1.6 Find orthogonal matrix of such that A = Q DQ ., where D is diagonal matrix To 1.2 Write rector was so as a liveer combination LCo of e ezey , where Ae. = Me. jz4 2 3 leees- orthonomad lanes of expenvectors from problem 1.5. 1.8. Write up as a linear combination of eeze and find like unMATH 1111 College Algebra Name Homework 2 As it is always the case that correct answers without sufficient mathematical justification may not receive full credit, make sure that you show all your work. Please circle, draw a box around, highlight. or otherwise clearly indicate your final answer for each question. By signing your name above, you attest to the fact that the work you are presenting is wholly your own. Attach all work to this coversheet. 1] Use the graph of the function f, given to the right, to answer the following questions. a) Is f decreasing on the interval (-4, 2)? b] Is f increasing on the interval (5, 6)? c) List any intervals on which f is decreasing. (5.3) d) Does the graph of f contain a local maximum? A local minimum? If so, list the ordered pair(s) (-4,2) for the extrema. 2) Plot the point (4, -2). Then, plot the point that is (-2, 1) symmetric to it with respect to (4,0) a] the x-axis, (0.0) (6,0) b] the y-axis, and c) the origin. 3) Algebraically, ie. without plotting points or looking (2. -2) at the graph, determine if the graphs of following equations exhibit any type of symmetry. List all types of symmetry, if any, that are present. List whether each function is even, odd, or neither. a] y= x - 2x - 8 bj y = = x-4 4) The last player in the NBA to use an underhand foul shot was Hall of Fame forward Rick Barry, who retired in 1980. Barry believes that current NBA players could increase their free-throw percentage if they were to use an underhand shot. Since underhand shots are released from a lower position, the angle of the shot must be increased. If a player shoots an underhand foul shot, releasing the ball at a 70-degree angle from a position 3.5-feet above the floor, then the path of the ball can be modeled by the function h(x) = _ 13axl + 2.7x + 3.5, where h is the height of the ball above the floor, x is the forward distance of the ball in front of the foul line, and v is the initial velocity with which the ball is shot in feet per second. a] The center of the hoop is 10 feet above the floor and 15 feet in front of the foul line. Determine the initial velocity with which the ball must be shot in order for the ball to go through the hoop. b] Write the function for the path of the ball using the velocity found in part (a)- c] Determine h(9). What does this value represent? 5) Find the average rate of change of the function f(x) = -x3 + 1 a) from 0 to 2. b) from 1 to 3. c] from -1 to 1. 6) Let f (x) = 1 if - 2 5 x