Name: Date: Practice Midterm Exam Sections 1.2, 1.3, , ,

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1 Name: Date: Practice Midterm Exam Sections 1., 1.3,.1-.7, , a108 Please develop your one page formula sheet as you try these problems. If you need to look something up, write it down on your formula sheet. Complete longer problems on a separate sheet of paper. Chapter 1., 1.3, Evaluate the function at the specified value of the independent variable and simplify. y, y 1 f ( y) = y + 3 y, 1 < y 1 3 y + 3 y, y > 1 1 f 3 A) 4 7 B) 1 3 C) 1 3 D) 10 9 E) Find the domain of the function g( w) = 36 w. A) 6 w 6 B) w 6 or w 6 C) w 0 D) w 6 E) all real numbers Page 1

2 3. Use the graph of the function to find the domain and range of f. A) B) C) D) E) Page

3 4. Use a graphing utility to graph the function and visually determine the intervals over which the function is increasing, decreasing, or constant. A) B) C) D) E) 5. Describe the right-hand and the left-hand behavior of the graph of 5 3 m( x) = ( x + 4 x 5x + 1). 1 A) Because the degree is odd and the leading coefficient is positive, the graph falls to the left and falls to the right. B) Because the degree is odd and the leading coefficient is negative, the graph rises to the left and falls to the right. C) Because the degree is odd and the leading coefficient is negative, the graph falls to the left and rises to the right. D) Because the degree is odd and the leading coefficient is positive, the graph rises to the left and rises to the right. E) Because the degree is even and the leading coefficient is negative, the graph rises to the left and falls to the right. Page 3

4 6. Find all real zeros of the polynomial mutiplicity of each. f ( x) x + 9 x + 14x 4 3 = and determine the A) x = 0, multiplicity ; x = 7, multiplicity 1; x =, multiplicity 1 B) x = 7, multiplicity ; x =, multiplicity C) x = 0, multiplicity ; x = 7, multiplicity 1; x =, multiplicity 1 D) x = 7, multiplicity ; x =, multiplicity E) x = 0, multiplicity 1; x = 7, multiplicity 1; x = 7, multiplicity 1; x =, multiplicity 1 7. Find a polynomial function with following characteristics. Degree: 4 Zero:, multiplicity: Zero: 4, multiplicity: Falls to the left, Falls to the right Absolute value of the leading coefficient is one A) 4 3 y = x + 6 x + 5 x 96 x + 8 B) 4 3 y = x + 6 x + 3 x + 16 C) 4 3 y = x + 8 x 3 x + 0 x 16 D) 4 3 y = x + 1 x 5 x + 96 x 64 E) 4 3 y = x + 1 x + 96 x 3 8. Use long division to divide. x 4 x 3 x + 4 x 1 ( ) ( ) A) x 4 x + 4 B) x + 4 x 4 C) 68 x + 13 x 4 x x + 4 x 1 D) 5x 1 x + 4 x 4 + x + 4 x 1 E) 4 x 4 x + 4 x 4 x + 4 Page 4

5 9. Use synthetic division to divide ( x 3 x + 7 x ) ( x + 4) A) 3 x x 8 B) 3 x + 13 x + 3 C) 3 x 7 x + 4 D) 3 x 5 x E) 3 x + x Simplify ( 3 7 i ) ( 3 + 7i ) A) 0 B) 84i C) i D) i E) 6 8i. and write the answer in standard form i Simplify 5 + 4i and write the answer in standard form. A) i B) i C) i D) i E) i 1. 1 Combine i 4 i and write the answer in standard form. A) i B) i C) i D) i E) i Page 5

6 13. Write 3 f ( x) = x x + 9 x 9 as a product of linear factors. A) ( )( ) x = x 1 x + 3 B) x = ( x 1) ( x 3i) C) ( )( ) x = x 1 x 3 D) x = ( x 1)( x + 3i)( x 3i) E) x = ( x + 1)( x 1)( x + 3) 14. Find all zeros of the function f ( x) ( x 1 )( x + 3 i)( x 3i) A) x = 1, 3 i, 3i B) x = 1, 3i C) x = 1, 3, 3 D) x = 1, 3 i, 3i E) x = 1 =. 15. Determine the equations of the vertical and horizontal asymptotes of the graph of the 5x function f ( x) =. x 9 A) horizontal: y = 5 ; vertical: x = 3 and x = 3 B) horizontal: y = 3 ; vertical: x = 5 C) horizontal: x = 5 ; vertical: y = 3 and y = 3 D) horizontal: y = 5 ; vertical: x = 3 E) horizontal: x = 3 and x = 3; vertical: y = x + 4 x + 3 Determine the domain of the function f ( x) =. x + 1 A) Domain: all real numbers except x = 1 and 3 B) Domain: all real numbers except x = 1 C) Domain: all real numbers except x = 1 and 3 D) Domain: all real numbers except x = 1 and 3 E) Domain: all real numbers Page 6

7 17. x + 0 Given f ( x) = x 5 9 x + 0. Determine the domain of f ( x ) and find any vertical asymptotes. A) domain: all real numbers except x = 4 and x = 5 vertical asymptote: none B) domain: all real numbers except x = 5 vertical asymptote: x = 4 and x = 5 C) domain: all real numbers except x = 0 vertical asymptote: x = 5 D) domain: all real numbers except x = 4 and x = 5 vertical asymptote: x = 4 E) domain: all real numbers except x = 4 and x = 5 vertical asymptote: x = 5 Chapter Given C = 14, a = 16.9, and c = 1.3, use the Law of Sines to solve the triangle (if possible) for the value of b. If two solutions exist, find both. Round answer to two decimal places. A) b = B) b = and 16.0 C) b = D) b = and E) not possible 19. Given A = 10, b = 1, and a = 10, use the Law of Sines to solve the triangle (if possible) for the value of c. If two solutions exist, find both. Round answer to two decimal places. A) c = 19.5 B) c =.04 and 1.60 C) c = 0.38 D) c = 1.06 and E) not possible 0. Determine the area of a triangle having the following measurements. Round your answer to two decimal places. C = 77 39', a = 13, and b = Given a = 8, b = 4, and c = 5, use the Law of Cosines to solve the triangle for the value of A. Round answer to two decimal places. Page 7

8 . In the figure below, a = 7, b = 10, and θ = 46. Use this information to solve the parallelogram for c. The diagonals of the parallelogram are represented by c and d. Round answer to two decimal places. a β c θ d b figure not drawn to scale 3. A triangular parcel of land has sides of lengths 850, 760, and 450 feet. Approximate the area of the land. Round answer to nearest foot. 4. In the figure below, ( a, b ) = (,8) and ( c, d ) = ( 8,5 ). Find (, ) x y so that u = v. Page 8

9 5. Find the magnitude of vector v. 6. Find the component form of vector v. 7. Find the component form of vector v with initial point ( 6,1) and terminal point ( 4,3). 8. Find the magnitude of vector v with initial point ( 5, 5) and terminal point ( 1,6). Page 9

10 9. Using the figure below, sketch a graph of the given vector. [The graphs in the answer choices are drawn to the same scale as the graph below.] 30. Let u = 10,3 and v = 9,13. Find u v. 31. Given u = 5 i + j and v = i + 3j, determine 6 u v. 3. Find the magnitude and direction angle of v = 5 i 5j. Round direction angle to nearest degree. 33. Find the component form of v if v = 6 and the angle it makes with the x-axis is 10. A) 3 3,3 B) 6,6 3 C) 6 3, 6 D) 3,3 3 E) 3,3 Page 10

11 34. An airplane is flying at a bearing of 143 with an airspeed of 330 kilometers per hour. Because of the wind, its ground speed and direction are, respectively, 310 kilometers per hour and 138. Find the direction and speed of the wind. Round your answer to two decimals. A) 14.91, B) , C) 37.01, D) 14.91, E) , kilometers per hour 5.3 kilometers per hour kilometers per hour 5.3 kilometers per hour 5.30 kilometers per hour 35. Given u = 5, 7 and v = 4,1, find u v. 36. Given vectors u = 3, 1 and v = 5,3 ( 4u 3 ) v u, determine the quantity indicated below. 37. Use the dot product to find the magnitude of u if u = 5, Find the angle between the vectors u and v if u = 3, and v = 1, to two decimal places.. Round answer 39. Determine whether u and v are orthogonal, parallel, or neither. 4 3 u =,, v = 16, Determine whether u and v are orthogonal, parallel, or neither. u = 4, 3, v = 6,0 41. Determine whether u are v and orthogonal, parallel, or neither. u = 3,5, v = 5,15 Page 11

12 4. Find the absolute value of the complex number 1 + 4i. A) 5 B) 3 5 C) 17 D) 4 17 E) Find the trigonometric form of the complex number below. A) 7π 7π cos + i sin 6 6 B) 7π 7π 4 cos + i sin 6 6 C) 4π 4π cos + i sin 3 3 D) 7π 7π 4 cos + i sin 3 3 E) 4π 4π 4 cos + i sin i 44. Perform the operation below and leave the result in trigonometric form. ( + i ) ( + i ) 4 cos64 sin cos11 sin Find the trigonometric form of the complex number shown below. 8i 46. Find the trigonometric form of the complex number shown below. 5 Page 1

13 47. Find the trigonometric form of the complex number -3-3i. A) B) C) D) E) 48. 5π 5π Find the standard form of the complex number 5 cos + i sin 6 6. A) i B) i C) i D) i E) i 49. Perform the operation shown below and leave the result in trigonometric form cos π isin π 4 cos π isin π Use DeMoivre's Theorem to find the indicated power of the following complex number. π π cos + i sin Page 13

14 51. Find all solutions to the following equation. x = 0 Chapter 8 (Not on the Midterm Exam save for the Final Exam) 5. n 3 If ( ) a n n = 1, find the first five terms of the sequence. 14 n Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 7, 1, 17,, Write an expression for the apparent nth term of the sequence. (Assume that n begins with 1.) 5, 5, 5, 5, 5, Write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of n. (Assume that n begins with 1.) a = 4, a = a 1 k+ 1 k 56. Write the first five terms of the sequence. (Assume that n begins with 0.) a n = n 5 1! ( n + ) 57. Simplify the following factorial expression. 6n +! ( ) ( 6 n)! 58. Find a formula for a n for the arithmetic sequence. a = 14, a = Write the nth term of the arithmetic sequence as a function of n. a =, a = a k+ 1 k Page 14

15 60. Find the partial sum. 10 n= 1 ( 5 n + 4) 61. Write the nth term of the geometric sequence as a function of n. a = 5, a = a 1 k+ 1 k 6. Find the indicated nth term of the geometric sequence. 7th term: a5 =, a11 = 81 59, Find the sum of the finite geometric sequence. Round to the nearest thousandth. 4 i= 0 ( ) i 64. Use sigma notation to write the sum Find the sum of the infinite series. i i= Find the sum of the following expression. Round your answer to three decimal places. 5 50(0.85) n n= Find the sum of the following infinite geometric series Use mathematical induction to prove the formula for every positive integer n. Show all your work. n ( 5 n + ) = ( 5 n + 9) Page 15

16 Calculate the binomial coefficient: Use the Binomial Theorem to expand and simplify the expression. ( w + ) Find the number of distinguishable permutations of the group of letters. 7. Solve for n given 30 n 1P4 = n + 1P You are given the probability that an event will not happen. Find the probability that the event will happen. 1 P( E ) = Use the Binomial Theorem to expand and simplify the expression. ( 5 x 4y ) Use the Binomial Theorem to expand the complex number. Simplify your result. ( 3 + 4i ) 4 Page 16