Precalculus. Cumulative Review Conics, Polar, Parametric, Sequences & Series, Rational Functions. Conics

Transcription

1 Name Precalculus Date Block Cumulative Review Conics, Polar, Parametric, Sequences & Series, Rational Functions Please do all work on a separate sheet of paper Conics Identify each equation If it is a parabola, give its verte, focus, and directri; if it is an ellipse, give its center, vertices, and foci; if it is a hyperbola, give its center, vertices, foci, and asymptotes GRAPH EACH CONIC 1 3 4y 3 16y 19 0 y y 1 4 9y 16 18y 11 4 Sketch the ellipse Find the eccentricity of the ellipse 4 y Sketch the hyperbola Find the equations of the asymptotes 1 y Obtain an equation in general form of each conic described below 6 Hyperbola: center 4, ; verte 1, ; asymptote of y 10 7 Ellipse: center 1,; verte at, ; focus at 8 Parabola: verte at, 3; focus at, , ; goes through the point0, 9 A bridge is built in the shape of a semi-elliptical arch The bridge has a span of 60 feet and a maimum height of 0 feet Find the height, to the nearest tenth, of the arch at distances of 5, 10, and 0 feet from the center

2 Polar Change each of the following points to polar coordinates 10, 11 3, 1 1, 3 Plot each of the following and change each of the points to rectangular coordinates , 14 3, , π - 3, 17-5, Write the polar equation for each of the following y 0 y Write the rectangular equation for each of the following 6 1 r 8 r sin 3 r 4cos Graph each of the following polar equations 4 r cos 5 r 4sin3 Change each of the following to rectangular form cos i sin 7 4cos i sin Change each of the following to comple polar form 8 5i i 31 1 i Evaluate, give your answer in rectangular form cos i sin 4cos i sin cos i sin 4cos i sin

3 Name two other pair of polar coordinates that represents each of the following points , 35 4, 36, 37, Write each epression in the standard a bi form 38 cos i sin 39 3cos i sin Parametrics 40 Graph the parametric equations t 4, y 3t 1; t, Find rectangular equation of each curve Identify the graph of the equation 4t t y 1 t y 5 t Given the parametric equations 3cos t t y 5sin, 0 t 43 Graph the curve 44 Find the rectangular equation Given the parametric equations tan t y 13sec t, 0 t 45 Graph the curve 46 Find the rectangular equation Two semi-trucks are driving loads from Chicago to Denver, a distance of 115 miles The first truck leaves at 8:00 am and averages 50 mph The second truck leaves at 9:00 am Since it has a lighter load, the second truck averages 54 mph 47 Set up two sets of parametric equations to model this situation 48 How long is it until the second truck overtakes the first truck? 49 How far are they from Chicago when the second truck overtakes the first?

4 Gretchen, an outfielder, is 15 feet from home plate after catching a fly ball The runner tags third and heads for home Gretchen releases the ball at an initial velocity of 75 ft/sec at an angle of 5 with the horizontal Assume Gretchen releases the ball 5 feet above the ground and aims it directly in line with the plate 50 Write the parametric equations that represent the path of the ball 51 How far, to the nearest foot, will the ball travel horizontally before hitting the ground? A baseball is hit straight up from a height of 5 feet with an initial velocity of 80 ft/sec 5 Write the parametric equations that represent the path of the baseball 53 How high is the ball after 4 seconds? 54 What is the maimum height of the ball? How many seconds does it take for the baseball to reach its maimum height? A helicopter leaves Central Airport and flies north at 180 mi/h Twenty minutes later a plane leaves the airport and follows the helicopter at 330 mi/hr 55 Write the parametric equations that represent the paths of the helicopter and plane 56 How long does it take the plane to overtake the helicopter?

5 Sequences & Series Write down the first five terms of each sequence 57 n 1 1 a 8n 3 58 a 18 n n 59 a 5, a 4 a 1 n n 60 a 6, a 3 a 1 1 n n 1 Determine whether the given sequence is arithmetic or geometric If the sequence is arithmetic, find the common difference and the sum of the first 10 terms If the sequence is geometric, find the common ratio and the sum of the first 10 terms 61 4,7,10,13,16, 6 6,1,4,48,96, Evaluate each sum k 1 64 k n 1 n Find the indicated term in each sequence 67 8 th term of 1,3,9, th term of 1,, 5, Find the recursive formula for each arithmetic sequence Use an algebraic solution 69 4 th term is 3; 0 th term is th term is 4; 18 th term is 96 Find the recursive formula for each geometric sequence Use an algebraic solution 71 nd term is 36; 9 th 4 4 term is a ; a Find the sum of each infinite sequence k 1 k 1 8 1

6 Tim and Nicole borrowed $50,000 at 575% per annum compounded monthly for 30 years to purchase a home 75 Determine the amount of their monthly mortgage 76 At the end of 30 years, how much interest will they have paid? 77 Suppose Tim and Nicole decide to pay an additional $100 each month on their loan How many months will it take to repay the loan? Erin, age 8, establishes a retirement account in which she deposits $400 per month into an account which pays 4% compounded monthly 78 If she makes her first deposit one month after her 9 th birthday and her last payment on her 65 th birthday, how much money will she have saved for her retirement? 79 At age 40, Erin is doing well enough financially that she can afford to increase her contribution to $500 per month, with the first payment one month after her 40 th birthday How much does she have saved at age 65? You make the decision that it is time to buy a car After spending a considerable amount of time researching, you decide to buy a Hyundai Sonata for $1,000 at an interest rate of 75% for 5 years You also make a down payment of $ Determine the amount of your monthly payment 81 You decide to trade in the car after 4 years How much do you still owe? Jeff has a credit card with $5000 worth of charges His interest rate is 185% 8 If he pays only the minimum payment of $100, how long will it take him to pay off the credit card? 83 How much will he pay in interest? 84 If he wants to pay it off in 3 years, how much should he pay per month?

7 Rational Functions 85 Identify the vertical asymptote(s) of y 3 6? h 86 Identify the horizontal asymptote of 3 87 Identify the horizontal asymptote of m Identify the hole in the graph of f 3 89 Find the equation for the slant asymptote of g 5 90 Write a rational function with the given asymptotes and holes 1, y 3, hole when 1 f Find c such that the graph of f has one vertical 3 c asymptote Justify your answer 91 Let 9 Show the sign analysis you would need to do to sketch the graph of 4 k 5 4 Sketch each of the following functions h k m r 5 4

8 Simplify each of the following Solve each of the following inequalities