Pre-Calculus Final Exam Review Name: May June Use the following schedule to complete the final exam review.

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1 Pre-Calculus Final Exam Review Name: May June 2015 Use the following schedule to complete the final exam review. Homework will be checked in every day. Late work will NOT be accepted. Homework answers will be provided at the beginning of each class period for you to check your work from the previous night. FINAL EXAM SCHEDULE: Friday, May 29 th Monday, June 1 st Tuesday, June 2 nd Wednesday, June 3 rd The first exam each day begins promptly at 8:00 a.m. 6 th & 7 th hour exams [90 min each] 4 th & 5 th hour exams [90 min each] 2 nd & 3 rd hour exams [90 min each] 1 st hour exam [90 min] Day Date Assignment Completed Tues. May 19, 2015 Chapter 7 Wed. May 20, 2015 Chapter 8 Thurs. May 21, 2015 Chapter 10 Fri. May 22, 2015 Chapter 9 Tues. May 26, 2015 Chapter 11 Wed. May 27, 2015 Chapter 12 Thurs. May 28, 2015 Finish Studying/Finish Notecard 1

2 Chapter 7 Analytic Trigonometry Simplify the expression. 1. cot θ csc θ sin θ 2. cos u sec u tan u 3. sin B + cos B cot B 4. sin 2 α + cos 2 α + tan 2 α 5. sin θ (cot θ + tan θ) 6. sin 2x 1+cos 2x Verify the identity. 7. sin x tan x cos x + = sin x + cos x 8. cot x cos x 1 sin x + = 2 sec x 1 sin x cos x 9. cos2 x sin 2 x 1 tan 2 x = cos 2 x 10. tan2 x sec x = sec x cos x 2

3 Use an addition or subtraction formula to find the EXACT value of the expression. 11. sin cos tan 7π 12 Find sin 2x, cos 2x, and tan 2x from the given information. 14. cos x = 4 ; csc x < csc x = 4; x in quadrant II 5 Use a half-angle formula to find the EXACT value of the expression. 16. tan cos π 12 Find the EXACT value of each expression, if it is defined. 18. sin cos 1 ( 3 2 ) 20. tan tan (sin ) 22. sin (tan 1 ) 23. csc 2 5 (cos 1 7 ) 25 3

4 Find all solutions of the equation. Work and answers must be in radians cos x 1 = csc 2 x 4 = cos x sin x 2 cos x = tan 2 x cos x tan 2 x = 0 Find all solutions of the equation in the interval [0, 2π) cos 2 x + sin x = cos 2x + 1 = 0 4

5 Chapter 8 Polar Coordinates & Vectors Graph each point and label them accordingly. Then find the rectangular coordinates of each point. 1. (1, 5π 4 ) 2. (3, 2π 3 ) 0 3. ( 2, 7π 6 ) 4. ( 4, 7π 2 ) A point P(r, θ) is given in polar coordinates. Give two other polar representations of the point, one with r < 0 and one with r > (5, 5π ) 6. ( 3, 6π) 4 Convert the rectangular coordinates to polar coordinates with r > 0 and 0 θ 2π. 7. ( 3, 3 3) 8. ( 2, 2) 5

6 A complex number is given. Find the modulus and then write the complex number in polar form i 10. 7i i 3 Find the product z 1 z 2 and the quotient z 1 z 2. Express your answer in polar form. 12. z 1 = 7 (cos 9π 8 + i sin 9π 8 ) ; z 2 = 2 (cos 2π 3 + i sin 2π 3 ) Find the indicated power using DeMoivre s Theorem. Write your answer in complex number (standard) form. 13. ( i) ( 3 i) 4 6

7 Express the vector with initial point P and terminal point Q in component form. 15. P(1,1); Q(9, 9) 16. P( 1,3); Q( 6, 1) Find u + v, -3u + 5v, v, and u v. 17. u = 2, 5, v = 2, u = 2i + 3j, v = i 2j u + v = u + v = -3u + 5v = -3u + 5v = v = v = u v = u v = 19. Find the vector with v = 50 and θ = Find the magnitude and direction of the vector v = i + j. Find (a) u v (dot product) and (b) the angle between u and v to the nearest degree. 21. u = 2,1, v = 3, u = i + 3j, v = 3i + j 23. Determine whether u = 4i and v = i + 3j are orthogonal. 7

8 24. Given u = 3i + 2j, v = i 4j, w = 5i 3j, find u (v + w). 25. Find the work done by the force F = 4i + 20j in moving an object from P(0, 10) to Q(5, 25). 26. A constant force F = 2,8 moves an object along a straight line from point (2, 5) to the point (11, 13). Find the work done if the distance is measured in feet and the force is measured in pounds. Chapter 10 - Conics Graph the ellipse and identify the center, vertices, and foci. 1. x y2 36 = x2 + 16y 2 = 64 Center: Vert: Foci: Center: Vert: Foci: 3. (x 1) (y+2)2 9 = (x + 4) 2 + (y + 3) 2 = 36 Center: Vert: Foci: Center: Vert: Foci: 8

9 Find the standard form of the equation of each ellipse. 5. Foci (0, ±3), vertices (0, ±4) 6. Major axis vertical with length 20; length of minor axis 10; center: (2, -3) 7. Foci (±5, 0), length of major axis Endpoints of major axis: (7, 9) & (7, 3) Endpoints of minor axis: (5, 6) & (9, 6) Convert the equation to standard form by completing the square x 2 + 9y x 36y + 36 = 0 Graph the hyperbola and identify the center, vertices, asymptotes, and foci. 12. y2 16 x2 25 = x2 y 2 = 64 Center: Vertices: Foci: Asymptotes: Center: Vertices: Foci: Asymptotes: 9

10 14. (x + 3) 2 9(y 4) 2 = (y+2)2 9 (x 1)2 25 = 1 Center: Vertices: Foci: Asymptotes: Center: Vertices: Foci: Asymptotes: Find the standard form of the equation of each hyperbola. 16. Foci (0,±4), vertices (0,±1) 17. Vertices (±4, 0), Asymptotes: y = ±2x 18. Endpoints of transverse axis: (0, ±6) 19. Foci (0, ±1), length of transverse axis 1 Asymptotes: y = ±3x Convert the equation to standard form by completing the square. 20. x 2 y 2 2x 2y 1 = 0 Graph the parabola and identify the vertex, directrix, and focus. 21. y 2 = 16x 22. x 2 = 4y Vertex: Dir: Focus: Vertex: Dir: Focus: 10

11 Graph the parabola and identify the vertex, directrix, and focus (x + 4) (y 3) = y 2 12(x + 2) = 0 Vertex: Dir: Focus: Vertex: Dir: Focus: Write an equation in standard form for the parabola satisfying the given conditions. 25. Focus: (8, 0); Directrix: x = Vertex: (2, -3); Focus (2, -5) Find the equation for the parabola whose graph is shown Convert the equation to standard form by completing the square. 29. x 2 + 8x 4y + 8 = 0 11

12 CHAPTER 9 Systems & Matrices Solve the system. 1. y = x2 + 8x y 16 = 2x 2. x y = 4 xy = 12 The matrices A, B, C, D, and E are defined as follows. Carry out the operation if possible. A = [ ] B = [ ] C = [ 1 0] D = [ 3 5 ] E = [ ] A + B 4. 3C D 5. DA 6. B 12

13 7. E 8. B 1 9. E 1 5x + 7y + 4z = Solve the system using inverses of matrices: { 3x y + 3z = 1 6x + 7y + 5z = 1 Solve the system using CRAMER S RULE x + 12y = x y = 5 4x + 7y = 20 5x + 3z = 19 4y + 7z = 17 13

14 Chapter 11 Sequences & Series Find the first five terms of the recursively defined sequence. 1. a n = a n 1 2 ; a 1 = 8 2. a n = a n 1 + a n 2 ; a 1 = 3, a 2 = Find the sum: k 4. Write the sum using sigma notation: k 1 Determine whether the sequence is arithmetic or geometric. Identify the common difference or the common ratio. 5. 3, 3, 3, , 6. 2, 4, 6, 8, 7. Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence -12, -8, -4, 0, 8. The 12 th term of an arithmetic sequence is 32, and the fifth term is 18. Find the 20 th term. 14

15 9. Which term of the arithmetic sequence 1, 4, 7, is 88? 10. Find the partial sum S n of the arithmetic sequence that has a = 1, d = 2, n = 12. A partial sum of an arithmetic sequence is given. Find the sum ( 3 2 ) n 20 n An arithmetic sequence has first term a = 5 and common difference d = 2. How many terms of this sequence must be added to get 2700? 14. Determine the common ratio, the fifth term, and the nth term of the geometric sequence 8, 2, 1 2, 1 8, 15

16 15. The first term of a geometric sequence is 3, and the third term is 4. Find the fifth term Which term of the geometric sequence 2, 6, 18, is 118,098? 17. Find the partial sum of the geometric sequence Find the sum of the infinite geometric series Express as a fraction. 16

17 Chapter 12 Limits 1. For the function g whose graph is given, state the value of the given quantity, if it exists. a) lim x 0 g(x) b) lim x 0 + g(x) c) lim x 0 g(x) d) lim x 2 g(x) e) lim x 2 + g(x) f) lim x 2 g(x) g) g(2) h) lim x 4 g(x) i) g(0) Find the limit algebraically. 2. lim x 3 (x3 + 2)(x 2 5x) lim u 2 u4 + 3u + 6 lim x 4 x 2 + 5x + 4 x 2 + 3x h 4 5. lim x lim x 3 x 3 7. h 0 h lim x 0 (x 3) 2 9 x 8. Evaluate the limits using the function below. x h(x) = { x 2 8 x if x < 0 if 0 < x 2 if x > 2 a) lim x 0 + h(x) b) lim x 0 h(x) c) lim x 1 h(x) d) lim x 2 h(x) e) lim x 2 h(x) 17

18 Find an equation of the tangent line to the curve at the given point. Use: 9. f(x) = 2x x 2 at (1, 1) 10. f(x) = 1 x2 at (-1, 1) f(x) f(a) lim x a x a Find the derivative of the function at the given number. Use 11. f(x) = 2 3x + x 2 at f(x) = x x+1 at 3 lim h 0 f(a + h) f(a) h 18