Math 153 Final Exam Extra Review Problems This is not intended to be a comprehensive review of every type of problem you are responsible for solving, but instead is meant to give you some extra problems to practice in addition to what’s on your old quizzes, hw, worksheets, class notes, etc. 1) For each, I) find the Difference Quotient (Make sure to simplify and that your h cancels!) II) For each difference quotient that you found, find, if it exists, lim → III) For the functions in part d) and e) only, find the value of the difference quotient if x = 3 and h = .5 2) a) If you have 600m of fencing and need to enclose the 5-sided field pictured, what dimensions maximize the total area and what’s the total maximum area? 3) If you have 10,000 feet of fencing and need to enclose the 3 -sided field pictured, what dimensions maximize the total area and what’s the total maximum area? (This is referring to the AROC you found in part a) c) Find the average rate of change over the interval [4,7] (This is referring to the AROC you found in part a) c) Find the average rate of change over the interval [4,7] d) Find the average rate of change over the interval [-1,3] 7) The revenue generated from selling x items is given by R(x) = 1000x − x . a) What’s the revenue if 160 items are sold? b) What quantity x maximizes the revenue? c) What’s the maximum revenue? d) How many items should be sold in order for there to be no revenue? 8) A cannonball fired out to sea from shore follows a parabolic trajectory given by ℎ( ) = − + 10 where h is in feet and t is in seconds. Determine the maximum height the ball reaches, the time at which the height is maximized, and the time at which the ball hits the water. (also graph b,c, and d) 11) Put each in the form + . Identify the real and imaginary parts of your solution. 12) For each, put the equation in standard/transformation form by completing the square. Find the vertex, all intercepts, the axis of symmetry, and the intervals of increase and decrease. Sketch a graph. a) ( ) = −2 + 8 + 2 b) ( ) = 5 − 20 + 4 c) ℎ( ) = − + − 1 14) Graph each. Label at least two points, any intercepts or asymptotes. Also, state the domain and range: a) ( ) = − d) ( ) = − +1 +3 b) ( ) = √−3 − 6 e) ( ) = 2ln( − 6) c) ( ) = log − 15) d) ( ) = − e) ( ) = − ( ), d) ( ) = √ − + , ( )= − ( )= + e) ( ) = − 21) Solve each. Write your answer in interval and set-builder notation. Sketch the solution set on a number line. See 1.7 for help. a) < 3 (make sure you do NOT clear the fraction!!) b) ≥ (make sure you do NOT clear the fraction!!) c) 10 d) 4 +7 − 12 ≤ 0 (Hint: factor, make a sign chart) − 16 > 0 22) Solve. Write your solutions as x,y-coordinates: a) 3 + 2 = 26 5 +7 =3 23) Find, if it exists, lim b) 2 = 24 − +4=0 ( ) c) − =2 √ = → 24) For the circle 3 + 3 + 2 − 24 = −44, put the equation in standard form by completing the square. State the center and radius. Find all intercepts. 29) Solve each. Write your solution in interval and set notation and sketch on a number-line: a) |4 − 5| + 3 < 6 d) + 1 − 2 < −4 b) + 1 − 2 ≥ −1 c) + 1 − 2 ≥ −4 33) 34) 35) If sec = and is in QIV, find the exact values of sin , cos , tan ,cot ,csc 36) 37) Write cos in terms of csc , t in Quadrant IV 38) A circle has a radius of 10 inches. At a central angle of 30˚, Find the area of the circular sector formed and find the arc length subtended. 39) One end of a ladder is resting the top of a wall. The base of the ladder is 8 feet from the wall. If the angle of elevation of the ladder is 60˚, find the length of the ladder. 40) Find the exact value of each of the following if it exists: a.sin (1) b. cos f.tan(tan (7)) g. (− √ ) c.sin (− ) d.cos ( cos (− )) 41. Solve the triangle: a. ∠ = 90°, c. = 2, = 4, = 5 (−4) e. cos h. tan (− √ = 6, ∠ = 30° ( cos(− )) ) b. ∠ = 30° = 3, = 7, 42. Graph each. Label all key points. Show a full period. State the amplitude, period, phase shift, domain, range, if applicable. a. =3 (2 − ) 43. Find the exact value of =5 b. (4 + ) c. = ( + ) cot cos 44. From the top of a 200 foot cliff, the angle of depression to a point is 30°. How far is the point from the base of the cliff? 45. A satellite passes directly over two observation stations,340 miles apart. At one instant, when the satellite is between the stations. Its angle of elevation is measured to be 60° at one station and 75° at the other. How far apart are the stations? How high is the satellite? 46. From the desert in Merzouga, Morocco, I measure the angle of elevation to an airplane to be 75°. My friend is 670 km away from me at the beach in Agadir, Morocco, and he measures the angle of elevation to the same plane (at the same instant) to be 60°. How far is the plane from the desert? How high is the plane? 47. Prove each: a. b. c. e. d. 1 sin x sinx 1 4 tan x sec x 1 sinx 1 sinx f. sin2 x tan x 1 cos2 x g cos x sin x 6 3 48. A triangular lot has sides of length 4, 6, and 8 meters. Find the largest angle. Leave your answer in exact form. 49. 49. Find the exact value of each: sin , sin 15°, cos 22.5° 50. Give the domain and range of each (use appropriate notation): = sec , = sin , = tan = cos , = csc , 51. From where he is perched in a tree, at a height of 20 ft., a hunter measures the angle of depression to an ibex below to be 52°. How far is the ibex from the base of the tree? 52. From where I am standing (at point C) on one side of an arroyo, I choose points A and B, both on the other side, which are 30 ft. apart. I measure BAC 56 and ABC 23 . Approximate the distance from A to C. Leave your answer in exact form. ( − ) = tan 53. Prove each a. c. 2 csc(2 ) tan = sec b. cos d. 54.. Suppose I know that cos(2 ) = 55. Suppose I know that cos( ) = + = sin( − ) = −2 sin = cos(2 ) . 2 cos (6 ) − 1 = cos(12 ) x in Quadrant II. Find sin( ), cos( ) and tan( ). x in Quadrant I. Find sin(2 ), cos(2 ) and tan(2 ). 56 Rewrite the expressions in terms of only the first power of cosine . cos (6 ) sin (6 ) . cos 57. Write the given expression in terms of x and y only. a. cot(csc−1 x) b. (cos + cot ) c. (2 csc ) 58. Find all solutions. Then, find all solutions in [0,2 ). . 2sin θ + sin . 3tan =1 −1=0 b. −2sin θ − 7cos −2= 0 e. tan (5θ) = tan(5 ) c. sin 2 t 7 cos t 4 3 cos2 t f. 3 sin 2 − 6 sin 59. The height of a wave (in inches, after t seconds) is given by ( ) = 5 initially? What happens to the height as t increases? Explain your reasoning. =0 (4 ) What’s the height (4 ) What’s the height 60. The height of a wave (in inches, after t seconds) is given by ( ) = 5 initially? What happens to the height as t increases? Explain your reasoning. 61. Sketch the curve represented by the parametric equationsx = t − 5, y = √t − 2. Find a rectangular coordinate equation for the curve by eliminating the parameter. 62. Sketch the curve represented by the parametric equationsx = 6 sin , y = 6 cos . Find a rectangular coordinate equation for the curve by eliminating the parameter. 63. Sketch the curve represented by the parametric equationsx = cos , y = 1 + cos .Find a rectangular coordinate equation for the curve by eliminating the parameter. 64. Find parametric equations for a line passing through (-2,-8) and (-5,1). Find a rectangular coordinate equation for the curve by eliminating the parameter. 65. Find parametric equations for the circle + = 4. 66. Convert each point in rectangular coordinates to polar coordinates with > and ≤ < . (x, y) = (-3, 3), (x, y) = (4√3, -4), (x, y) = (0, -10) 67. Convert the polar coordinates to rectangular coordinates (x, y) ( , ) = −2, ( , ) = (5,0), , ( , ) = −1, 68. Convert each equation to rectangular form, simplify, and graph: 69. Evaluate/Simplify. Write your answer in the form a + bi: (− ) =− , = 7, cos = −1 + (3 − ), , 70. For each complex number, graph, find the modulus and the conjugate: z = −8 + 6i, = −3 71. Write each in polar form with 0 ≤ <2 , 72. For z1 = 1 − √3, z2 = −1 − , find , > 0: , , , = 6 − 6√3 , = 9, . Write all answers with = −5 + 5 > and ≤ 73. For each vector, find the magnitude and direction. Graph. Then, find and graph + , − Practice writing these answers both in component form and in terms of the unit vectors , . = −4 + 2 , = −3 − 3 , = −2 74. Express each vector v both in component form and in terms of i and j: < . + v has length 5 and direction , v has length 1 and direction 330° 75. Find each limit, if it exists lim → lim → lim → lim → → lim −7 lim → lim → → + lim → √1 + 5ℎ + 1 ℎ 76. For each, find and simplify the difference quotient. Then, find the derivative, ′( ) using the limit definition. ( ) = −2 + 3 − 5, ( )= , ( ) = √ + 1, ( )= .
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