CHAPTER 8: Polar 1. Convert to polar.

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1 CHAPTER 8: Polar 1. Convert to polar. a. 3,. Convert to rectangular. a. 4, 3 b. 4 4i b. 5cis10 3. Use DeMoivre s Theorem to find a. i 8 4. Graph a. r 4cos3 b. the cube roots of 4 4 3i b. r 3sin 5. Convert the equation 6. Let z1 1cis350 and z 4cis0. Find a. x 3x 4 to polar form a. zz 1 b. r 3cos to rectangular form z1 b. z

2 Honors 7. Find rectangular-coordinate equation for the curve by eliminating the parameter. t x t, y t Honors 8. Find the rectangular-coordinate equation for the parametric equations given. x cos t, y sin t ANSWERS 1a. 4,150 1b cis a., i b. 3a. 56cis0 3b. cis100,cis 0,cis340 4a. 4b. 5a. 5b. r cos 3rcos 4 x y 3x 6a. 48cis10 6b. 3cis y x x 8. x y

3 Chapter 10: Matrices and Systems of Equations Multiply if possible 1. Given A 4 0 and a B 4 0 3, find A 4B b c Solve x y for x and y 4a. Evaluate b. Evaluate by expanding by minors: Solve for y using Cramer s Rule: 4x y z 15 3x y z 1 x y 3z a. Find the inverse b. Solve X c. Solve using matrices: 4 x y 1 3xy

4 7a. Tony flew his ultra-light plane to the next town against a head wind of 10 mph, the trip taking ½ hours. The return trip, under the same wind conditions, took 1 ½ hours. How far is the next town, and what is the speed of the ultra-light? 8. Solve x y 1 14 x y 7b. A collection of dimes and quarters has a total value of five dollars and contains 9 coins. How many of each kind of coin are there in the collection? 9. Graph x 0 xy6 3 y x4 8 to maximize S 4x 3y 10. Solve the system. x y 18 x y Solve the system. 3x y z 1 x 5y 5z 7 4x 3y z 3

Department B takes 5 hours to assemble a bicycle and hours to assemble a tricycle.

5 Chapter 10 (cont) 1. Linear Programming Problem A manufacturer can show a profit on a bicycle of $6 and a profit on a tricycle of $4. Department A requires 3 hours to manufacture the parts for a bicycle and 4 hours to manufacture parts for a tricycle. Department B takes 5 hours to assemble a bicycle and hours to assemble a tricycle. How many bicycles and tricycles should be produced to maximize the profit if the total time available in department A is 450 hours and in department B is 400 hours? ANSWERS b. not possible a c. 31 4a b a. 6b c. 4 7a. 40 mph; 75 mi 7b. 14 quarters; 15 dimes 9. max of 35 at (8, -1) bikes and 75 trikes for Max $ , 10. (3, 3) (-3, -3) (1, 3, )

6 Chapter 9: Vectors 1. Let A3,,1 and 4, 3, B, find. Given u 4i 6 j and v i 5 j find 3u v a. AB b. AB c. A vector equation of the line AB 3. An object that moves with constant velocity so that its position at time t is x, y 3,5 t, 4. a. Find its velocity b. Find its speed c. Find a pair of parametric equations 4a. Find uv if u 3, 1,4 and v,1,4 Find the value of p if vectors u, 3 and v 4 a, 5 are 4b. Parallel d. Find a Cartesian equation 4c. Orthogonal 5. Find the vector v having length v, and direction Find the direction of the vector v 6i 6 3 j

7 7. Find the angle between u 4,,3 and v 6,1,4 8. Find an equation of the plane that passes through the points P, Q, and R. P(8, 1, 1), Q(5,, 0), R(0, 0, 0) ANSWERS 1a. 1,5, 3 1b. 35 1c. r 3,,1 t 1,5, 3. 8i 8 j 3a., 4 3b. 5 5., x3t 3d. xy 11 4a. 9 4b. 3c. y 54t x 5y 11z c. 15 8

8 Chapter 11: Conics Complete the square to graph each conic. Label vertices, foci, asymptotes and directrix. 1. x 6x 8y x 4y 36x 4y x 16y 8x 96y y x 48y Find an equation of the ellipse with vertices (4, 3) and (4, 9) and foci (4, 8) and (4, 4) 6. Find an equation of the parabola with focus (-7, 4) and directrix x 3

HONORS From the definition, write the equation of a parabola with

write the equation of a hyperbola with foci (0, 4) and difference

x3 8 y 1 Parabola: vertex (3,-1) Foci (3, 1); directrix y = -3 4

5 x 16 4 1 3. x1 y3 Ellipse: ctr (-1, 3) Foci 1 3,3 4.

9 HONORS From the definition, write the equation of a parabola with focus (-5, -3) and directrix y 1 HONORS From the definition, write the equation of a hyperbola with foci (0, 4) and difference of the focal radii = 6. ANSWERS 1. x3 8 y 1 Parabola: vertex (3,-1) Foci (3, 1); directrix y = x y3 Hyperbola: ctr (, -3) Foci 13, 3 Asymp: y x x1 y3 Ellipse: ctr (-1, 3) Foci 1 3, y x 4 1 Parabola: vertex (-4, -) Foci , 1 Directrix x x4 y6 6. y 4 0 x H7. x 5 8 y 1 H8. y x 9 7 1

10 Chapter 1: Sequences and Series 1. Find the first seven terms of the sequence a a 3a where a1 4 and a 3 n n1 n. Find the 500 th n term of the sequence 1 3 n 1 4n 1 3. Given the 45 th term of an arithmetic sequence is 04 and the 30 th term is 159, find the 7 th term. 4. Find the sum What term number is 37 in the arithmetic sequence 18, 19.5, 1,? 6. The seats in the new amphitheater are arranged like an open fan. The first row seats 6 people and the second row seats four more, and the next row four more, etc. If there are 30 rows altogether, how many seats does the amphitheater have? 7a. Find a5 for the sequence of positive 3-digit numbers that are multiples of Write the sum without using sigma notation. 10 n n n1 7b. Find the sum of the positive 3-digit numbers that are multiples of Find the sum of Find the sum 14 n1 n

11 11. Find the sum A classic car appreciates in value 3% each year. The original value is $0,000 at the beginning of the first year. Find the value at the end of 5 years. 13. Find the 1 th term in the expansion 0 x y Find the constant term in the expansion of x x Prove by mathematical induction: n3n1 a (3n1) n 16. Prove by mathematical induction: a. n 3 3n n is divisible by 3 n n n 3 3 b = n b. 1 n 1 is divisible by 11 n ANSWERS 1. 4, 3, 18, 45, 144, 43, a a. 73 7b $ x y

12 Chapter 14: Probability 1. A pizza parlor offers two types of crust, 6 types of meat and 8 other toppings. How many different pizzas can be ordered if each pizza must have one meat and one other topping?. How many 6-card hands can be selected from a standard deck that contain a. four hearts b. three kings 3. Two dice are rolled. Let A be the event that the sum is equal to 8. Let B be the event that the numbers are the same on both die. Find 4. Many TV stations have 3-letter call signs beginning with K. How many such call signs are possible if letters can be repeated? P A PB P A B P A B Are event A and B mutually exclusive? 5. A 1-member jury is selected from 9 men and 13 women. Find the probability that the selected jury contains 6. A coin and a die are tossed. Find the probability that the coin turns up tails and the die turns up an even number. a. six men and six women b. at least six men

13 7. Eight people enter a contest. How many ways can 1 st, nd and 3 rd place be awarded. 8. In a class, 75% of the students study for their math final. If 1 students are selected, find the probability that only four studied for the math final. 9. Suppose that you play the following game. A pair of dice is rolled. You get $x if two ones show. Otherwise you pay $1. Find the payout amount that would make this a fair game. 10. Charlie plays a game where he wins $5 if he draws a king form a deck and loses $1 for all other cards. Find the expected value. ANSWERS a b a P A ; PB ; P A B ; P A B ; not mutually exclusive b $

14 Chapter 13 & 15: Limits, Derivatives and Applications 1. Find the limit. Find the derivative by definition: a. x lim x3 x1 x 3 a. f x 3x x b. 3 4x 3x lim x 5 3 x x c. lim x 0 x 3 3 x b. f x x 3. Find f 'bxg a. f x 4 3x ln x 4. Find the equation of the line tangent to 3 y x 3x 4, 4 at 3 x 3 b. f x x cos x 3x c. f x sin x e 3x 5x 6 4x

4 3 5. Graph using calculus: y 3 11 x 3 x x 4x 6 6. A company wishes to manufacture a box with a volume of 36 ft 3 that is open on top and is twice as long as it is wide.

15 Graph using calculus: y 3 11 x 3 x x 4x 6 6. A company wishes to manufacture a box with a volume of 36 ft 3 that is open on top and is twice as long as it is wide. Find the dimensions of the box produced from the minimum amount of material. ANSWERS 1a. 7 1b c. 1 a. f ' x 6x 3 b. f ' x x 5 4 3a y36x x 6 x f ' x 4 3x ln x 15x 8x 4 x 3b f ' x x sin x 3 6x 6x cos x 3x x 3c. f ' x 6 4 x 3 5 cos sin 4 x x x x e x x x x e 3x 5x 6

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

12-13 Spring Semester Exam Review Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Answer the question. 1) In how many ways can you answer the