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New York City College Of Technology Department Of Mathematics MAT 1375 – Final Examination Review Sheet Revised by Professor S. Singh (2009), A. Mukhin (2010), G. Niezgoda (2011) 1. Solve the following inequalities and express your answers in interval notation. a. x 6 x 5 0 2 u ln x 2. Let a) ln and x y2 c. x 9 0 b. x 2 3 x 2 2 d. x5 0 x 2 2x 3 v ln y . Write the following expressions in terms of u and v. b) ln 3 x5 y 4 3. For each of the following functions, (a) state the domain of the function, (b) find x- and y-intercepts (if any), (c) find vertical and horizontal asymptotes (if any) of the graph, (d) sketch the graph. 2x 3 ii. g ( x) , x 1 3x 6 i. f ( x) 2 , x 6x 8 4. Let 3x 2 x 2 iii. h( x) 2 . x x2 q( x) ax 2 bx c , where a, b and c are non-zero constants. Evaluate the following expressions: ii. q ( 2 x ) i. q ( x ) iii. q( x 2 ) iv. q ( x h) q ( x ) , h ≠ 0. h 5. Solve the absolute value inequalities a. | 4 5 x | 4 b. | 5 2 x | 7 6. Find all roots of f ( x) x 3 x 2 5 x 5 exactly. Sketch a complete graph of f(x) and label all roots clearly. 7. Let v = 5 3,5 . Find the magnitude and direction angle of v. 2 2 p = 2(cos( ) i sin( )) , and q = 5(cos( ) i sin( )) . Write the following in standard ( a bi ) 2 2 3 3 p form: a. b. pq q 8. Let c. Find the product and write the result in a+bi form: [4(cos 45 + i sin 45 )][3(cos 15 + i sin 15 )] 9. For the functions below, find the amplitude, period, and phase shift. Draw the graph over a one-period interval. Label all maxima, minima and intercepts. a. b. y = 3cos(2x) y = sin ( 2x ) 3 c. y = -3sin ( 2 x ) 3 10. Use graphical methods to find all real solutions of the equations, approximating to the nearest tenth if necessary: a. x 4 5x 2 x 2 0 b. x 4 2 x 3 8 x 2 10 x 15 0 11. Find all exact solutions of trigonometric equations. Use radian measure of angles. a. 2 cos x 1 0 b. tan 2 x 1 c. 4 sin 2 x 4 sin x 1 0 12. The initial population of a colony is 10,000 and is decreasing exponentially at 1.5% per year. a. What is the size of the colony in 5 years? b. How long will it take for the population to be half of its initial amount? 13. Use the Binomial Theorem, write in simplest form: 8 y a. The first four terms of the expansion of 2 x . 2 8 y b. The 7 term in the expansion of 2 x . 2 th 14. a. Find the sum of the first 65 terms in the arithmetic sequence below: -6, -2, 2, 6, 10, . . . b. Find the sum of the infinite geometric sequence: 24, -12, 6, -3, . . . MAT 1375 Review. Answer Sheet. 1. a. (−∞, 1] ∪ [5, +∞); b. [1, 2]; c. (−∞, -3) ∪ (3,+∞); d. (−5, −1) ∪ (3, +∞); 2. a. 1 u 2v 2 b. 5 u 4v 3 3. i. (a) (−∞, -4) ∪ (-4, -2) ∪ (- 2, +∞); (b) x-intercept: none, (с) vertical asymptotes: x 4 and x 2 y-intercept (0, 3 ) 4 , horizontal asymptote: y0 (d) Graph ii. (a) (−∞, -1) ∪ (-1, +∞); (b) x- intercept: ( (с) vertical asymptote: 3 , 0 ), y-intercept (0, 3) 2 x 1 , horizontal asymptote: y 2 (d) Graph iii. (a) (−∞, -1) ∪ (-1, 2) ∪ (2, +∞); (b) x- intercepts: ( (с) vertical asymptotes: (d) Graph x 1 and x 2 ; 2 , 0 ) and (1, 0 ) ; y-intercept (0, 1) 3 horizontal asymptote: y3 ax 2 bx c 8 a. 0 x 5 4. i. 5. ii. ax 2 (4a b) x (4a 2b c) x 1 b. or iii. ax 4 bx 2 c iv. x6 x 1, x 5 ; Graph : 6. 7. | v | = 10, θ= 11 . 6 3 1 i 5 5 8. a. b. 5 3 5i c. 6 6 3i A 3, T , 0 9. a. Graph c. Graph A 3, T , Graph: 6 10. a. x = -2.3, x = 0.6, x = 0.8, x = 2 11. a. x b. x c. x (1) n 2 2 n, and 3 4 12. a. 9272; 13. a. A 1, T 3 , 0 b. n, and 6 x x n, where 4 b. x = -2.2, x = -1, x = 2.2, x = 3 2 2 n , where 3 n , where n 0,1,2,3,... n 0,1,2,3,... b. Approximately 45.9 years 256 x 8 512 x 7 y 448x 6 y 2 224 x 5 y 3 14. a. 7930; n 0,1,2,3,... b. 16 ; b. 7 2 6 x y 4 2ax ah b