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STUDY GUIDE FOR FINAL EXAMINATION MATH 1113 1. 2. 3. 4. 5. Let f ( x) x 2 2 x . Compute the following. (a) f (3) (c) f (a) (b) (d) (e) (f) f (x + h) f (0) f (a + 2) f ( x h) f ( x ) , h0 h Find the domain of the following functions. (a) f ( x) x 2 2 x (b) f ( x) (c) f ( x) (d) f ( x) (e) (g) (i) (k) f ( x) e f ( x) sin x f ( x) csc x x3 x 1 x3 1 x3 f ( x) ln x f ( x) tan x (f) (h) (j) f ( x) sin 1 x (b) f ( x) 4 x 2 2 x 8; g ( x) 2 x 3 (b) f ( x) x 3 3 f ( x) tan 1 x Find a simplify ( f ◦ g)(x). 3 2 ; g ( x) (a) f ( x) x2 3x Find the inverse of the following functions. 1 (a) f ( x) 3 3 x Sketch the graph of the inverse of the following function. 6. 2 Put the following equations of conic sections in standard graphing form by completing the square. Then graph the conic sections. Give the following information for each conic section. Parabola: vertex, axis of symmetry, focus, and directrix. Ellipse: center, vertices, covertices, major axis, minor axis, foci, and eccentricity. Hyperbola: center, vertices, transverse axis, conjugate axis, foci, and eccentricity. (a) 4 x 2 y 2 16 x 2 y 13 0 (b) x 2 y 2 2 y 3 (c) 9 x 2 4 y 2 36 x 8 y 4 0 7. Find the equation, in standard form, of the parabola with vertex (7, −9) and focus (5, −9). 8. Find all the asymptotes of the graph of R ( x) 9. Consider the function R( x) 5x 2 . 4x 2 7x 3x 2 5 x 2 . The graph of R will have a vertical asymptote at x2 4 _______ and will have a missing point at x = __________. x 2 17 x 72 10. Find the oblique asymptote of the graph of R( x) . x3 11. Suppose a radioactive substance Barnesvillium has a half-life of 77 years. If 258 grams are present initially, how much will be left after 212 years? 12. What is the half-life of a radioactive element that decays to 33% of the original amount present in 77 years? 13. Find the balance when$19,000 is invested at an annual rate of 6% for 5 years if the interest is compounded (a) daily (b) continuously. 14. How long will it take money to double at 17% interest compounded monthly? 15. (a) (b) The angle of depression from a tower to a landmark is 19° and the landmark is 400 feet below the tower. How far away is the landmark? Suppose the angle of elevation of the sun is 23.4°. Find the length of the shadow cast by Cindy Newman who is 5.75 feet tall. 3 4 and , find sec θ and cot θ. 5 2 16. If cos 17. Find where the graphs of the following functions have vertical asymptotes or horizontal asymptotes or state there are no asymptotes. (a) y sin x (b) y tan x (c) y csc x (d) y cos 1 x (e) y tan 1 x 18. Find the exact value of the following. 3 (a) sin 1 2 19. (b) (c) tan 1 1 (d) (e) 1 tan tan 1 (f) Solve the following trigonometric equations. (a) 2 sin x 1 0 1 (c) cos 3 x 2 2 cos 1 2 5 sin 1 sin 4 cos 1 cos 3 (b) 2 cos 2 x sin x 1 0 (d) tan 3 x tan x 0 20. Approximate the solutions for each equation on the interval 0 ≤ x ≤ 2π. Round your answer to four decimal places. (a) sin x = −0.739 (b) cos x = 0.4201 (c) tan x = −5.17 21. Establish the following trigonometric identities. tan x sec x (a) (b) sin x sin x cos x 1 cot x (c) (d) sin x cos x 1 cot x csc x (e) sin x 22. Solve the following oblique triangles. (a) A = 30°, b = 12, c = 16 (c) B = 62.3°, C = 51.9°, c = 11.3 (e) A = 22°, a = 10, c = 15 (b) (d) sin 2 t cos 2 t cot t tan t 1 cos cos 2 cos 1 tan 2 a = 3.55, b = 4.08, c = 2.83 C = 57.47°, b = 8.841, c = 0.777 4 23. Find the area of the triangle ABC given a = 8.5, B = 102°, and C = 27°. 24. Convert the following polar equations into rectangular equations. (a) r 7 (b) r 4 sin 0 (c) r 2 tan 25. Convert the following rectangular equations into polar equations. (a) x + y = 1 (b) y 2 2 x x 2 (c) The circle centered at (0, −4) with radius 4. 26. (a) (b) 3 Convert 8, into rectangular coordinates. 4 Convert 5 3 , 5 into polar coordinates. ANSWERS 1. (a) (c) (e) 3 a2 – 2a x2 + 2xh + h2 – 2x – 2h (b) (d) (f) 0 a2 + 2a 2x + h – 2 2. (a) (c) (e) (g) (, ) [3, ) (, ) (, ) (b) (d) (f) (, 3) (3, ) (3, ) (0, ) (h) all real numbers except (2n 1) where n is an integer; that is, all real numbers except the 2 2 all real numbers except 2nπ where n is an integer; that is, all real numbers except the even multiples of π [−1, 1] (k) (, ) odd multiples of (i) (j) 3. (a) 9x 2 6x 4. (a) f 1 ( x) (b) 1 3 x3 (b) 16 x 2 44 x 38 f 1 ( x) 3 x 3 5 5. 6. (a) ( y 1) 2 ellipse; ( x 2) 1 ; center: (2, −1); vertices: (2, 1), (2, −3); covertices: (1, −1), 4 3 (3, −1); foci: 2, 1 3 , 2, 1 3 ; maj. axis: x = 2; min. axis: y = −1; e 2 2 2 (b) (c) 1 1 1 5 1 5 parabola; y x ; vertex: , ; axis of symm: y ; focus: 2 2 2 2 2 2 19 1 21 , ; directrix: x 8 2 8 ( x 2) 2 ( y 1) 2 1 ; center: (−2, 1); vertices: (0, 1), (−4, 1); foci: 4 9 13 2 13, 1 , 2 13, 1 ; trans. axis: y = 1; conj. axis: x = −2; e 2 hyperbola; 7. ( y 9) 2 8( x 7) 8. vertical asymptotes: x = 0, x 9. vertical asymptote at x = −2, missing point at x = 2 10. y = x − 20 11. 38.3 grams 12. 48.14 years 13. (a) (b) $25,647.32 14. 4.12 years 15. (a) (b) 13.29 feet 16. 5 4 sec ; cot 4 3 17. (a) no asymptotes (b) vertical asymptotes at x $25,646.69 1161.68 feet 7 ; horizontal asymptote: y = 0 4 (2n 1) where n is an integer 2 18. (c) 6 vertical asymptotes at x = 2nπ where n is an integer (d) no asymptotes (a) (c) (e) 19. (a) (b) (c) (d) 4 1 3 horizontal asymptotes: y (b) 3 4 (d) (f) 3 2 , y 4 11 7 k 2 , k 2 , where k is an integer 6 6 11 7 k 2 , k 2 , k 2 , where k is an integer 6 2 6 2 k 2 4 k 2 , , where k is an integer 3 9 3 9 3 5 7 k 2 , k 2 , k 2 , where k is an integer k , k 2 , 4 4 4 4 20. (a) (c) 21. The answer is in working the problems. 22. (a) (c) (e) 23. ≈ 20.6 square units 24. (a) (c) 25. (a) 26. (e) 3.9731, 5.4516 1.7619, 4.9036 (b) 1.1372, 5.1459 B ≈ 46.94°, C ≈ 103.06°, a ≈ 8.21 (b) A ≈ 58.54°, B ≈ 78.62°, C ≈ 42.84° A ≈ 65.8°, a ≈ 13, b ≈ 13 (d) no triangle B1 ≈ 12.19°, C1 ≈ 145.81°, b1 ≈ 5.636; B2 ≈ 123.81°, C2 ≈ 34.19°, b2 ≈ 22.180 x2 y2 7 x4 x2 y2 4y2 0 (c) 1 cos sin r 8 sin (a) 4 r 2, 4 2 (b) x 2 ( y 2) 2 4 (b) r 2 cos (b) 5 10, , 10, , etc 6 6 2