Math 370 Exam 5 Review Name Graph the ellipse and locate the foci. 1) x 6 + y = 1 1) foci: ( 15, 0) and (- 15, 0) Objective: (9.1) Graph Ellipses Not Centered at the Origin Graph the ellipse. ) (x + ) + (y + 1) 9 = 1 ) Objective: (9.1) Graph Ellipses Not Centered at the Origin Convert the equation to the standard form for an ellipse by completing the square on x and y. 3) 16x + 5y - 3x - 150y - 159 = 0 3) (x - 1) 5 + (y - 3) 16 = 1 Objective: (9.1) Graph Ellipses Not Centered at the Origin
Solve the problem. ) The arch beneath a bridge is semi-elliptical, a one-way roadway passes under the arch. The width of the roadway is 0 feet and the height of the arch over the center of the roadway is 1 feet. Two trucks plan to use this road. They are both 8 feet wide. Truck 1 has an overall height of 11 feet and Truck has an overall height of 1 feet. Draw a rough sketch of the situation and determine which of the trucks can pass under the bridge. ) Truck 1 can pass under the bridge, but Truck cannot. At a distance of feet from the 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 system and finding points of intersection. 5) x + y = 5 5x + 16y = 00 5) {(0, 5), (0, -5)} Objective: (9.1) Additional Concepts Find the vertices and locate the foci for the hyperbola whose equation is given. 6) 9x - 100y = 900 6) vertices: (-10, 0), (10, 0) foci: (- 19, 0), ( 19, 0) Objective: (9.) 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) x 9 - y 55 = 1 Objective: (9.) 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) 9y - 16x + 18y + 6x - 199 = 0 8) (y + 1) 16 - (x - ) 9 = 1 Objective: (9.) 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 + ) - (x - ) 9 = 1 9) Center: (, -); Vertices: (, -) and (, 0); Foci: (, - - 13) and (, - + 13) Objective: (9.) 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, ) and then moves closer and closer to the line y = 3 x as it gets farther from the tracking station at 10) the origin. Find the equation that describes the path of the satellite if the center of the hyperbola is at (0, 0). (0, ) y = 3 x y - x 16 9 = 1 Objective: (9.) Solve Applied Problems Involving Hyperbolas Find the focus and directrix of the parabola with the given equation. 11) y = x 11) focus: (1, 0) directrix: x = -1 Objective: (9.3) Graph Parabolas with Vertices at the Origin
Graph the parabola. 1) y = -16x 1) Objective: (9.3) Graph Parabolas with Vertices at the Origin 13) x = 18y 13) Objective: (9.3) Graph Parabolas with Vertices at the Origin Find the standard form of the equation of the parabola using the information given. 1) Vertex: (6, -); Focus: (3, -) 1) (y + ) = -1(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) = -16(x - ) 15) vertex: (, 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 + ) = 8(y + ) 16) 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 0 feet. Find the height of the arch at 10 feet from its center. A) 0.1 ft B).3 ft C) 5. ft D) 39. ft D Objective: (9.3) Solve Applied Problems Involving Parabolas 17) Parametric equations and a value for the parameter t are given. Find the coordinates of the point on the plane curve described by the parametric equations corresponding to the given value of t. 18) x = t 3 + 1, y = 9 - t ; t = 18) (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 = t - 1, y = t + 5; - t 19) Objective: (9.5) Use Point Plotting to Graph Plane Curves Described by Parametric Equations
0) x = 5 sin t, y = 5 cos t; 0 t 0) 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. 1) x = 6 cos t, y = 6 sin t; 0 t 1) x + y = 36; -6 x 6 Objective: (9.5) Eliminate the Parameter ) x = + sec t, y = 5 + tan t; 0 < t < ) (x - ) - (y - 5) = 1; x > 3 Objective: (9.5) Eliminate the Parameter Eliminate the parameter. Write the resulting equation in standard form. 3) An ellipse: x = 3 + cos t, y = 5 + 3 sin t 3) (x - 3) + (y - 5) 9 = 1 Objective: (9.5) Eliminate the Parameter Find a set of parametric equations for the conic section or the line. ) Hyperbola: Vertices: (3, 0); Vertices: (-3, 0); Foci: (5, 0) and (-5, 0) ) x = 3 sec t, y = tan t Objective: (9.5) Find Parametric Equations for Functions Solve the problem. 5) A baseball pitcher throws a baseball with an initial velocity of 136 feet per second at an angle of 0 to the horizontal. The ball leaves the pitcher's hand at a height of feet. Find parametric equations that describe the motion of the ball as a function of time. How long is the ball in the air? When is the ball at its maximum height? What is the maximum height of the ball? 5) x = 17.8t; y = -16t + 6.51t + ;.99 sec; 1.53 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 at the pole. 6) r = 6) 1 - cos 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 7) r = + sin parabola; The directrix is unit(s) above the pole at y =. Objective: (9.6) Define Conics in Terms of a Focus and a Directrix 8) r = 7 9-3 sin 7) 8) ellipse; The directrix is 7 3 unit(s) below the pole at y = - 7 3. Objective: (9.6) Define Conics in Terms of a Focus and a Directrix Graph the polar equation. 9 9) r = 3-3 cos Identify the directrix and vertex. 9) directrix: 3 unit(s) to the left of the pole at x = -3 vertex: 3, Objective: (9.6) Graph the Polar Equations of Conics
1 30) r = - cos Identify the directrix and vertices. 30) directrix: 1 unit(s) to the left of the pole at x = -1 vertices: 1,,, 0 5 Objective: (9.6) Graph the Polar Equations of Conics 31) r = 8 1 + cos Identify the directrix and vertices. 31) directrix: unit(s) to the right of the pole at x = vertices: - 8 3,, 8 5, 0 Objective: (9.6) Graph the Polar Equations of Conics
Use a graphing utility to graph the equation. 3 3) r = 3) 1 - cos - Objective: (9.6) Tech: Conic Sections in Polar Coordinates Write the first four terms of the sequence whose general term is given. 33) an = n - 33), 6, 10, 1 Objective: (10.1) Find Particular Terms of a Sequence from the General Term 3) an = (-) n 3) -, 16, -6, 56 Objective: (10.1) Find Particular Terms of a Sequence from the General Term 35) an = n + 3 n - 1 35), 5 3, 6 5, 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 + 0.09 n n = 1,, 3,... Find the balance in the account after 6 years. $13,66.13 Objective: (10.1) Find Particular Terms of a Sequence from the General Term 36) Write the first four terms of the sequence defined by the recursion formula. 37) a1 = -3 and an = an-1-3 for n 37) -3, -6, -9, -1 Objective: (10.1) Use Recursion Formulas
Write the first four terms of the sequence whose general term is given. n 38) an = (n - 1)! 1, 16, 81, 18 3 Objective: (10.1) Use Factorial Notation 38) Evaluate the factorial expression. 9! 39) 7!! 0) 36 Objective: (10.1) Use Factorial Notation n(n + )! (n + 3 )! n n + 3 Objective: (10.1) Use Factorial Notation 39) 0) Find the indicated sum. 10 1 1) i - i = 7 19 0 Objective: (10.1) Use Summation Notation 1) ) 5 i = 1 (i + 1)! (i + )! 153 10 Objective: (10.1) Use Summation Notation ) Express the sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 3) + 8 + 18 +... + 7 3) 6 i i = 1 Objective: (10.1) Use Summation Notation ) a + 1 + a + +... + a + 5 5 5 a + i i i = 1 Objective: (10.1) Use Summation Notation )
5) a + ar + ar +... + ar 11 5) 1 i = 1 ar i - 1 Objective: (10.1) Use Summation Notation Express the sum using summation notation. Use a lower limit of summation not necessarily 1 and k for the index of summation. 6) (a + 1) + (a + c) + (a + c ) +... + (a + c n ) 6) n k = 0 (a + c k ) Objective: (10.1) Use Summation Notation Write the first five terms of the arithmetic sequence. 7) a1 = -15; d = 3 7) -15, -1, -9, -6, -3 Objective: (10.) 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 sequence with the given first term, a1, and common difference, d. 8) Find a 31 when a1 = -6, d = - 5. 8) - 18 Objective: (10.) 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 a0, the 0th term of the sequence. 9) a1 = -3, d = 0.5 9) an = 0.5n - 3.5; a0 = 6.5 Objective: (10.) 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 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. an = n + 9; 13 weeks Objective: (10.) Use the Formula for the General Term of an Arithmetic Sequence 50) Find the indicated sum. 51) Find the sum of the first 30 terms of the arithmetic sequence: 10, 5, 0, -5,... 51) -1875 Objective: (10.) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence 5) Find the sum of the odd integers between 30 and 68. 5) 931 Objective: (10.) 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. 9 53) (i - 1) 53) i = 1 81 Objective: (10.) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence Solve the problem. 5) A theater has 3 rows with 7 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? 601 seats Objective: (10.) Use the Formula for the Sum of the First n Terms of an Arithmetic Sequence 5) Write the first five terms of the geometric sequence. 55) a1 = -6; r = - 55) -6,, -96, 38, -1536 Objective: (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 sequence with the given first term, a1, and common ratio, r. 56) Find a6 when a1 = 9600, r = - 1. 56) -300 Objective: (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, - 3, 3, - 3 8, 3 16,... 57) an = 3-1 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 the sequence is arithmetic, find the common difference; if it is geometric, find the common ratio. 58) an = n - 58) arithmetic, d = Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence 59) an = 3 n 59) geometric, r = 3 Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence 60) an = 5n - 60) neither Objective: (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 the nearest dollar, in the eighth year? $1,58,79 Objective: (10.3) Use the Formula for the General Term of a Geometric Sequence 61) Use the formula for the sum of the first n terms of a geometric sequence to solve. 6) Find the sum of the first 8 terms of the geometric sequence: -8, -16, -3, -6, -18,.... 6) -00 Objective: (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 3 i 63) i = 1 8 3 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. 6) To save for retirement, you decide to deposit $50 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 of the IRA after 35 years. $03,1 Objective: (10.3) Find the Value of an Annuity 6) Find the sum of the infinite geometric series, if it exists. 65) - 1 + 1-1 16 +... 65) 16 5 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) 58 99 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 swing altogether before coming to a complete stop? Round to the nearest inch when necessary. 00 inches Objective: (10.3) Use the Formula for the Sum of an Infinite Geometric Series 67)
Use mathematical induction to prove that the statement is true for every positive integer n. n(5n + 3) 68) + 9 + 1 +... + (5n - 1) = 68) S1:? = 1(5 1 + 3)? = 1 8 = Sk: + 9 + 1 +... + (5k - 1) = Sk+1: + 9 + 1 +... + (5k + ) = k(5k + 3) (k + 1)(5k + 8) We work with Sk. Because we assume that Sk is true, we add the next consecutive term, namely 5(k+1) - 1, to both sides. + 9 + 1 +... + (5k - 1) + (5(k + 1) - 1) = + 9 + 1 +... + (5k + ) = + 9 + 1 +... + (5k + ) = k(5k + 3) k(5k + 3) + 9 + 1 +... + (5k + ) = 5k + 13k + 8) + 9 + 1 +... + (5k + ) = + (5k + ) + (k + 1)(5k + 8) k(5k + 3) (5k + ) + (5(k + 1) - 1) We have shown that if we assume that Sk is true, and we add 5(k+1) - 1 to both sides of Sk, then Sk+1 is also true. By the principle of mathematical induction, the statement Sn is true for every positive integer n. Objective: (10.) Prove Statements Using Mathematical Induction Evaluate the given binomial coefficient. 11 69) 330 Objective: (10.5) Evaluate a Binomial Coefficient 69) Use the Binomial Theorem to expand the binomial and express the result in simplified form. 70) (5x + )3 70) 15x3 + 150x + 60x + 8 Objective: (10.5) Expand a Binomial Raised to a Power 71) (x - ) 71) x - 8x3 + x - 3x + 16 Objective: (10.5) Expand a Binomial Raised to a Power
Find the term indicated in the expansion. 7) (x - 3y)11; 8th term 7) -71,710xy7 Objective: (10.5) Find a Particular Term in a Binomial Expansion