Math 370 Exam 5 Review Name Graph the ellipse and locate ... · Exam 5 Review Name ... Graph the ellipse and locate the foci. 1) x 2 64 + y 2 4 = 1 1) Answer:

Math 370Exam 5 Review Name________________________________________________________________________________________________________________________________________Graph the ellipse and locate the foci.

1) x264

+y24

= 1 1)

Answer:foci: (2 15, 0) and (-2 15, 0)

Objective: (9.1) Graph Ellipses Not Centered at the Origin

Graph the ellipse.

2) (x + 2)24

+(y + 1)2

9= 1 2)

Answer:


Convert the equation to the standard form for an ellipse by completing the square on x and y.3) 16x2 + 25y2 - 32x - 150y - 159 = 0 3)

Answer: (x - 1)225

+(y - 3)2

16= 1


Solve the problem.4) The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch.

The width of the roadway is 40 feet and the height of the arch over the center of theroadway is 12 feet. Two trucks plan to use this road. They are both 8 feet wide. Truck 1 hasan overall height of 11 feet and Truck 2 has an overall height of 12 feet. Draw a roughsketch of the situation and determine which of the trucks can pass under the bridge.

4)

Answer: Truck 1 can pass under the bridge, but Truck 2 cannot. At a distance of 4 feet fromthe center of the roadway, the height of the arch is 11.76 feet.

Objective: (9.1) Solve Applied Problems Involving Ellipses

Find the solution set for the system by graphing both of the system's equations in the same rectangular coordinate systemand finding points of intersection.

5) x2 + y2 = 2525x2 + 16y2 = 400

5)

Answer: {(0, 5), (0, -5)}

Objective: (9.1) Additional Concepts

Find the vertices and locate the foci for the hyperbola whose equation is given.6) 49x2 - 100y2 = 4900 6)

Answer: vertices: (-10, 0), (10, 0)foci: (- 149, 0), ( 149, 0)

Objective: (9.2) Locate a Hyperbola's Vertices and Foci

Find the standard form of the equation of the hyperbola satisfying the given conditions.7) Endpoints of transverse axis: (-3, 0), (3, 0); foci: (-8, 0), (8, 0) 7)

Answer: x29

-y255

= 1

Objective: (9.2) Write Equations of Hyperbolas in Standard Form

Convert the equation to the standard form for a hyperbola by completing the square on x and y.8) 9y2 - 16x2 + 18y + 64x - 199 = 0 8)

Answer: (y + 1)216

-(x - 2)2

9= 1

Objective: (9.2) Write Equations of Hyperbolas in Standard Form

Find the location of the center, vertices, and foci for the hyperbola described by the equation.

9) (y + 2)24

-(x - 2)2

9= 1 9)

Answer: Center: (2, -2); Vertices: (2, -4) and (2, 0); Foci: (2, -2 - 13) and (2, -2 + 13)Objective: (9.2) Graph Hyperbolas Not Centered at the Origin

Solve the problem.10) A satellite following the hyperbolic path shown in the picture turns rapidly at (0, 2) and

then moves closer and closer to the line y = 32

x as it gets farther from the tracking station at

the origin. Find the equation that describes the path of the satellite if the center of thehyperbola is at (0, 0).

(0, 2)

y = 32

x

10)

Answer: y24

-x2169

= 1

Objective: (9.2) Solve Applied Problems Involving Hyperbolas

Find the focus and directrix of the parabola with the given equation.11) y2 = 4x 11)

Answer: focus: (1, 0)directrix: x = -1

Objective: (9.3) Graph Parabolas with Vertices at the Origin

Graph the parabola.12) y2 = -16x 12)

Answer:


13) x2 = 18y 13)

Answer:


Find the standard form of the equation of the parabola using the information given.14) Vertex: (6, -4); Focus: (3, -4) 14)

Answer: (y + 4)2 = -12(x - 6)Objective: (9.3) Write Equations of Parabolas in Standard Form

Find the vertex, focus, and directrix of the parabola with the given equation.15) (y - 3)2 = -16(x - 4) 15)

Answer: vertex: (4, 3)focus: (0, 3)directrix: x = 8

Objective: (9.3) Graph Parabolas with Vertices Not at the Origin

Graph the parabola with the given equation.16) (x + 2)2 = 8(y + 2) 16)

Answer:

Objective: (9.3) Graph Parabolas with Vertices Not at the Origin

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

Solve the problem.17) A bridge is built in the shape of a parabolic arch. The bridge arch has a span of 166 feet and a

maximum height of 40 feet. Find the height of the arch at 10 feet from its center.17)

A) 0.1 ft B) 2.3 ft C) 5.2 ft D) 39.4 ftAnswer: DObjective: (9.3) Solve Applied Problems Involving Parabolas

Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curvedescribed by the parametric equations corresponding to the given value of t.

18) x = t3 + 1, y = 9 - t4; t = 2 18)Answer: (9, -7)Objective: (9.5) Use Point Plotting to Graph Plane Curves Described by Parametric Equations

Use point plotting to graph the plane curve described by the given parametric equations.19) x = 2t - 1, y = t2 + 5; -4 t 4 19)

Answer:

Objective: (9.5) Use Point Plotting to Graph Plane Curves Described by Parametric Equations

20) x = 5 sin t, y = 5 cos t; 0 t 2 20)Answer:

Objective: (9.5) Use Point Plotting to Graph Plane Curves Described by Parametric Equations

Eliminate the parameter t. Find a rectangular equation for the plane curve defined by the parametric equations.21) x = 6 cos t, y = 6 sin t; 0 t 2 21)

Answer: x2 + y2 = 36; -6 x 6Objective: (9.5) Eliminate the Parameter

22) x = 2 + sec t, y = 5 + 2 tan t; 0 < t <2

22)

Answer: (x - 2)2 -(y - 5)2

4= 1; x > 3

Objective: (9.5) Eliminate the Parameter

Eliminate the parameter. Write the resulting equation in standard form.23) An ellipse: x = 3 + 2 cos t, y = 5 + 3 sin t 23)

Answer: (x - 3)24

+(y - 5)2

9= 1

Objective: (9.5) Eliminate the Parameter

Find a set of parametric equations for the conic section or the line.24) Hyperbola: Vertices: (3, 0); Vertices: (-3, 0); Foci: (5, 0) and (-5, 0) 24)

Answer: x = 3 sec t, y = 4 tan tObjective: (9.5) Find Parametric Equations for Functions

Solve the problem.25) A baseball pitcher throws a baseball with an initial velocity of 136 feet per second at an

angle of 20° to the horizontal. The ball leaves the pitcher's hand at a height of 4 feet. Findparametric equations that describe the motion of the ball as a function of time. How long isthe ball in the air? When is the ball at its maximum height? What is the maximum height ofthe ball?

25)

Answer: x = 127.8t; y = -16t2 + 46.51t + 4;2.99 sec;1.453 sec;37.8 feet

Objective: (9.5) Understand the Advantages of Parametric Representations

Identify the conic section that the polar equation represents. Describe the location of a directrix from the focus located atthe pole.

26) r = 21 - 2 cos

26)

Answer: hyperbola; The directrix is 1 unit(s) to the left of the pole at x = -1.Objective: (9.6) Define Conics in Terms of a Focus and a Directrix

27) r = 42 + 2 sin

27)

Answer: parabola; The directrix is 2 unit(s) above the pole at y = 2.Objective: (9.6) Define Conics in Terms of a Focus and a Directrix

28) r = 79 - 3 sin

28)

Answer: ellipse; The directrix is 73

unit(s) below the pole at y = -73

.

Objective: (9.6) Define Conics in Terms of a Focus and a Directrix

Graph the polar equation.

29) r = 93 - 3 cos

Identify the directrix and vertex. 29)

Answer: directrix: 3 unit(s) to the left ofthe pole at x = -3

vertex: 32

,

Objective: (9.6) Graph the Polar Equations of Conics

30) r = 124 - cos

Identify the directrix and vertices. 30)

Answer: directrix: 12 unit(s) to the left ofthe pole at x = -12

vertices: 125

, , 4, 0


31) r = 8 1 + 4 cos

Identify the directrix and vertices. 31)

Answer: directrix: 2 unit(s) to the right ofthe pole at x = 2

vertices: -83

, , 85

, 0


Use a graphing utility to graph the equation.

32) r = 3

1 - cos -4

32)

Answer:

Objective: (9.6) Tech: Conic Sections in Polar Coordinates

Write the first four terms of the sequence whose general term is given.33) an = 4n - 2 33)

Answer: 2, 6, 10, 14Objective: (10.1) Find Particular Terms of a Sequence from the General Term

34) an = (-4)n 34)

Answer: -4, 16, -64, 256Objective: (10.1) Find Particular Terms of a Sequence from the General Term

35) an =n + 32n - 1

35)

Answer: 4, 53

, 65

, 1

Objective: (10.1) Find Particular Terms of a Sequence from the General Term

Solve the problem.36) A deposit of $8000 is made in an account that earns 9% interest compounded quarterly.

The balance in the account after n quarters is given by the sequence

an = 8000 1 + 0.094

n n = 1, 2, 3, ...

Find the balance in the account after 6 years.

36)

Answer: $13,646.13Objective: (10.1) Find Particular Terms of a Sequence from the General Term

Write the first four terms of the sequence defined by the recursion formula.37) a1 = -3 and an = an-1 - 3 for n 2 37)

Answer: -3, -6, -9, -12Objective: (10.1) Use Recursion Formulas

Write the first four terms of the sequence whose general term is given.

38) an =n4

(n - 1)!38)

Answer: 1, 16, 812

, 1283

Objective: (10.1) Use Factorial Notation

Evaluate the factorial expression.

39) 9!7! 2!

39)

Answer: 36Objective: (10.1) Use Factorial Notation

40) n(n + 2 )!(n + 3 )!

40)

Answer: nn + 3

Objective: (10.1) Use Factorial Notation

Find the indicated sum.

41)10

i = 7

1i - 4

41)

Answer: 1920

Objective: (10.1) Use Summation Notation

42)5

i = 1

(i + 1)!(i + 2)!

42)

Answer: 153140


Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation.43) 2 + 8 + 18 + . . . + 72 43)

Answer:6

i = 12i2


44) a + 1 + a + 22

+ . . . + a + 55

44)

Answer:5

i = 1

a + ii


45) a + ar + ar2 + . . . + ar11 45)

Answer:12

i = 1ari - 1


Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index ofsummation.

46) (a + 1) + (a + c) + (a + c2) + . . . + (a + cn) 46)

Answer:n

k = 0(a + ck)


Write the first five terms of the arithmetic sequence.47) a1 = -15; d = 3 47)

Answer: -15, -12, -9, -6, -3Objective: (10.2) Write Terms of an Arithmetic Sequence

Use the formula for the general term (the nth term) of an arithmetic sequence to find the indicated term of the sequencewith the given first term, a1, and common difference, d.

48) Find a 31 when a1 = -6 , d = -25

. 48)

Answer: - 18Objective: (10.2) Use the Formula for the General Term of an Arithmetic Sequence

Write a formula for the general term (the nth term) of the arithmetic sequence. Then use the formula for an to find a20,the 20th term of the sequence.

49) a1 = -3, d = 0.5 49)

Answer: an = 0.5n - 3.5; a20 = 6.5Objective: (10.2) Use the Formula for the General Term of an Arithmetic Sequence

Solve the problem.50) To train for a race, Will begins by jogging 13 minutes one day per week. He increases his

jogging time by 4 minutes each week. Write the general term of this arithmetic sequence,and find how many whole weeks it takes for him to reach a jogging time of one hour.

50)

Answer: an = 4n + 9; 13 weeksObjective: (10.2) Use the Formula for the General Term of an Arithmetic Sequence

Find the indicated sum.51) Find the sum of the first 30 terms of the arithmetic sequence: 10, 5, 0, -5, . . . 51)

Answer: -1875Objective: (10.2) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence

52) Find the sum of the odd integers between 30 and 68. 52)Answer: 931Objective: (10.2) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence

Use the formula for the sum of the first n terms of an arithmetic sequence to find the indicated sum.

53)29

i = 1(2i - 1) 53)

Answer: 841Objective: (10.2) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence

Solve the problem.54) A theater has 34 rows with 27 seats in the first row, 30 in the second row, 33 in the third

row, and so forth. How many seats are in the theater?54)

Answer: 2601 seatsObjective: (10.2) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence

Write the first five terms of the geometric sequence.55) a1 = -6; r = -4 55)

Answer: -6, 24, -96, 384, -1536Objective: (10.3) Write Terms of a Geometric Sequence

Use the formula for the general term (the nth term) of a geometric sequence to find the indicated term of the sequencewith the given first term, a1, and common ratio, r.

56) Find a6 when a1 = 9600, r = -12

. 56)

Answer: -300Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence

Write a formula for the general term (the nth term) of the geometric sequence.

57) 3, - 32

, 34

, - 38

, 316

, . . . 57)

Answer: an = 3 -12

n - 1

Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence

The general term of a sequence is given. Determine whether the given sequence is arithmetic, geometric, or neither. If thesequence is arithmetic, find the common difference; if it is geometric, find the common ratio.

58) an = 4n - 2 58)

Answer: arithmetic, d = 4Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence

59) an =32

n59)

Answer: geometric, r = 32

Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence

60) an = 5n2 - 4 60)

Answer: neitherObjective: (10.3) Use the Formula for the General Term of a Geometric Sequence

Solve the problem.61) A hockey player signs a contract with a starting salary of $810,000 per year and an annual

increase of 6.5% beginning in the second year. What will the athlete's salary be, to thenearest dollar, in the eighth year?

61)

Answer: $1,258,729Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence

Use the formula for the sum of the first n terms of a geometric sequence to solve.62) Find the sum of the first 8 terms of the geometric sequence: -8, -16, -32, -64, -128, . . . . 62)

Answer: -2040Objective: (10.3) Use the Formula for the Sum of the First n Terms of a Geometric Sequence

Find the indicated sum. Use the formula for the sum of the first n terms of a geometric sequence.

63)5

i = 1

43

· 2i 63)

Answer: 2483

Objective: (10.3) Use the Formula for the Sum of the First n Terms of a Geometric Sequence

Solve the problem. Round to the nearest dollar if needed.64) To save for retirement, you decide to deposit $2250 into an IRA at the end of each year for

the next 35 years. If the interest rate is 5% per year compounded annually, find the value ofthe IRA after 35 years.

64)

Answer: $203,221Objective: (10.3) Find the Value of an Annuity

Find the sum of the infinite geometric series, if it exists.

65) 4 - 1 +14

-116

+ . . . 65)

Answer: 165

Objective: (10.3) Use the Formula for the Sum of an Infinite Geometric Series

Express the repeating decimal as a fraction in lowest terms.66) 0.58 66)

Answer: 5899

Objective: (10.3) Use the Formula for the Sum of an Infinite Geometric Series

Solve the problem.67) A pendulum bob swings through an arc 80 inches long on its first swing. Each swing

thereafter, it swings only 60% as far as on the previous swing. How far will it swingaltogether before coming to a complete stop? Round to the nearest inch when necessary.

67)

Answer: 200 inchesObjective: (10.3) Use the Formula for the Sum of an Infinite Geometric Series

Use mathematical induction to prove that the statement is true for every positive integer n.

68) 4 + 9 + 14 + . . . + (5n - 1) = n(5n + 3)2

68)

Answer: S1: 4?=

1(5 · 1 + 3)2

4?=

1 · 82

4 = 4

Sk: 4 + 9 + 14 + . . . + (5k - 1) = k(5k + 3)2

Sk+1: 4 + 9 + 14 + . . . + (5k + 4) = (k + 1)(5k + 8)2

We work with Sk. Because we assume that Sk is true, we add the next consecutiveterm, namely5(k+1) - 1, to both sides.

4 + 9 + 14 + . . . + (5k - 1) + (5(k + 1) - 1) = k(5k + 3)2

+ (5(k + 1) - 1)

4 + 9 + 14 + . . . + (5k + 4) = k(5k + 3)2

+ (5k + 4)

4 + 9 + 14 + . . . + (5k + 4) = k(5k + 3)2

+2(5k + 4)

2

4 + 9 + 14 + . . . + (5k + 4) = 5k2 + 13k + 8)2

4 + 9 + 14 + . . . + (5k + 4) = (k + 1)(5k + 8)2

We have shown that if we assume that Sk is true, and we add 5(k+1) - 1 to bothsides of Sk, then Sk+1 is also true. By the principle of mathematical induction, thestatement Sn is true for every positive integer n.

Objective: (10.4) Prove Statements Using Mathematical Induction

Evaluate the given binomial coefficient.

69) 114

69)

Answer: 330Objective: (10.5) Evaluate a Binomial Coefficient

Use the Binomial Theorem to expand the binomial and express the result in simplified form.70) (5x + 2)3 70)

Answer: 125x3 + 150x2 + 60x + 8Objective: (10.5) Expand a Binomial Raised to a Power

71) (x - 2)4 71)

Answer: x4 - 8x3 + 24x2 - 32x + 16Objective: (10.5) Expand a Binomial Raised to a Power

Find the term indicated in the expansion.72) (x - 3y)11; 8th term 72)

Answer: -721,710x4y7Objective: (10.5) Find a Particular Term in a Binomial Expansion

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Math 370 Exam 5 Review Name Graph the ellipse and locate ... · Exam 5 Review Name ... Graph the ellipse and locate the foci. 1) x 2 64 + y 2 4 = 1 1) Answer: