Final Exam MATH 1148, Spring 2019; Time: 1hr 45 min Name: Show all work for full credit and use appropriate mathematical terminology and notation to justify as needed (4pts) 1. Find an equation of a line that goes through the point (1, −4) and is parrallel to the line 3x − y = 2. (6pts) 2. Solve the following system of equations, If it has a unique solution(provide it), if it has infinetely many solutions(express them in a general form) or if it has no solution justify your answer. x + y − 2z = 1 y − 2z = 2 x − y + 2z = −3 (10pts) 3. Let f (x) = x x−7 and g(x) = 2x + 1 (a) The domain of f (x) is: (b) (f ◦ g)(x) = (c) Domain of (f ◦ g)(x) is: (d) Evaluate (f ◦ g)(−4) = (10 pts) 4. Answer the following: (a) Write as a single logarithm the expression: log(12x4 + 18x2 ) + logx−2 − log 3. (b) Expand as much as possible log3  27(3x+2)4 √ 2x+7  (15pts) 5. Solve the following equations . (a) log3 (x + 6) = −log3 (x) + 3 (b) e2x + ex = 12 (c) x3 ex = 9xex (8pts) 6. Let f (x) = −x2 + 4x + 8 . (a) Find the average rate of change of f (x) on the interval [0, 2] (b) Write f (x) in vertex form, i.e. f (x) = a(x − h)2 + k and determine the maximum or minimum value and indicate what it is. (8pts) 7. Malika bought a lead-lamp at the start of 2019. Suppose that every six months the light intensity drops to a half of a current value (that is the half-life of the lamp is 6months). (a) Write an expression of the form I = I0 ert , where r represents the rate of decline of light intensity per month and t the months that have passsed. (b) How long will it take for a lamp with a current intensity of 823 units to drop to 138 units? Use the model you developed in part (a). (5pts) 8. Evaluate: (3x4 − 12x2 + 4) ÷ (x − 2) (6pts) 9. Write an expression of a degree 5 polynomial P (x) with the following zeros: x = 0 (multiplicity 2), x = 1 (multiplicity 2), x = −4 (multiplicity 1), also given that P (−1) = 24. (10pts) 10. Find the inverse of the each of the following functions. √ (a) f (x) = 3 2x − 7 (b) g(x) = 12 − log3 (5x + 4) (8pts) 11. Let f (x) = ex . Suppose the graph of f(x) undergoes transformations in the given order of the steps provided. Write a single formula, g(x) for the graph of the resulting function when all the steps are completed. (i) A vertical stretch by a factor of 4. (ii) A horizontal left translation of 3 units (iii) A reflection about X-axis (iv) A vertical upward translation of 5 units. (v) Provide a sketch of the graph, indicate on it some key points. (10pts) 12. Given f (x) = x2 (x + 5)(x − 4). Answer the following. (a) Sketch the graph of f (x), clearly show all the intercepts and end behaviour (b) Solve the inequality x2 (x + 5)(x − 4) ≤ 0 using your answer in (a) or directly using a sign chart. (c) Interpret the relationship, if any, between the answers to parts (a) and (b). BLANK PAGE

Final Exam Math Spring 2019

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