Dual-Enrollment Final Exam Preparation

Transcription

1 Dual-Enrollment Final Exam Preparation Dates: May 7 th and 8 th : Part 1 (75 minutes) questions covering 1 st Semester Material May 9 th and 10 th Part 2 (75 minutes) Questions covering 2 nd Semester Material Exam: Will be a mix of multiple choice and free response (Mostly Free Response) Will be worth a 1/3 of your 2 nd semester grade Review: To help prepare you have A List of Formulas you need to know (The Bolded ones will be given to you) 2 nd Semester Cumulative Reviews A List of Topics Possible Covered (Not all will be) A set of 125 Practice Questions Opportunity: If you submit neat and organized work for the 125 Practice Questions (All Completed with Work Shown) by the start of the Final May 7 th. You will get the opportunity to OMIT 1 question from each Part of the Final (Your Pick)

2 Formulas You Must Know (To be Successful) log(a x ) = xlog(a) log(xy) = logx + logy log ( x y ) = logx logy log a b = logb loga = lnb lna A = Pe rt N t = N 0 e rt sin θ = y r sin θ = y cos θ = x r cos θ = x tan θ = y x sin θ tan θ = cos θ csc θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ cos θ cot θ = sin θ sin 2 θ + cos 2 θ = 1 tan 2 θ + 1 = sec 2 θ 1 + cot 2 θ = csc 2 θ cos θ = sin(90 θ) cot θ = tan(90 θ) sin(2x) = 2 sin x cos x cos(x + y) = cos x cos y sin x sin y cos(x y) = cos x cos y + sin x sin y cos(2x) = cos 2 x sin 2 x = 1 2sin 2 x = 2cos 2 x 1 csc θ = sec(90 θ) sin A a sin B = b = sin C c c 2 = a 2 + b 2 2ab cos C b 2 = a 2 + c 2 2ac cos B sin(x + y) = sin x cos y + cos x sin y Area = 1 2 ab sin C a 2 = b 2 + c 2 2bc cos A sin(x y) = sin x cos y cos x sin y Area = s(s a)(s b)(s c), where s = a+b+c 2

3 Part 1: Possible Topics Covered Chapter 1 o Identify a conic from Ax 2 + By 2 + Cx + Dy + E o Put a conic section in (h, k) form o Solve Inequalities and put in Interval Notation Chapter 2 o Sketch Polynomials and Rational Functions o Solve Polynomial Equations, Polynomial Inequalities, and Rational Equations o Apply the Rational Root Theorem (Find all possible rational roots) o Divide Polynomials either Synthetic or Long o Apply Factor Theorem to find solutions Chapter 3 o Find Domain o Perform/Solve Compositions o Find Discontinuities (Holes, Vertical Asymptotes, and Jumps) o Find Inverses o Determine if a function is Even, Odd, or Neither o Sketch and Evaluate Piece-wise Functions o Express Absolute Value as Piece-wise Function o Find the Difference Quotient Chapter 4 o Expanding Logarithms o Simplifying Logarithms o Apply the Change of Base Formula (and Simplify) o Solve Logarithmic and Exponential Equations o Find/Apply Exponential Growth and Decay (Including Doubling and Half-Life)

4 Part 2- Possible Topics Chapter 5 o Given a trig function and quadrant Find another trig function o Evaluate off Unit Circle o Find Reference Angles o o o o Write as a Trig Function less than 90 or 45 Convert between Radians and Degrees Find Arc Length Find Sector Area Chapter 6 o Graph any of the six trig functions o Find Domain of Trig Function o Find Range of Trig Function o Write a sine/cosine Equation (includes modeling word problems) o Evaluate Arc Functions Chapter 7 o Evaluate using Fundamental Identities o Simplify and Verify using Fundamental Identities o Evaluate using Sum, Difference and Double Angle Identities o Simplify and Verify using Sum, Difference and Double Angle Identities o Solve Basic Trig Equations on [0,2π] o Solve Trig Equations using Identities on [0,2π] o Solve a Trig Equations for all solutions Chapter 8 o Find Angles and Sides of Triangle using Law of Sines and Law of Cosines o Find Area of Triangle using Trig o Find Area of a Triangle using Heron s Formula o Find Area of a Polygon

5 Practice Questions Chapter 1 1. Identify the type of conic section 16y 2 25x 2 50x 425 = 0 2. Identify the type of conic section x + 2y y + 44 = 0 3. Identify the type of conic section x 2 + 9y 2 + 2x 72y = 0 4. Put in (h, k) form 9x 2 + 4y x 24y + 36 = 0 5. Put in (h,k) form 9x 2 16y 2 54x + 64y 127 = 0 6. Put in (h,k) form y 2 + 4x = y 7. Put in (h,k) form 3x 2 + 3y 2 6x + 3y = 4 8. Solve the following inequality (express in interval notation) 6x 2 2x 20 < 0 Chapter 2 9. Sketch the polynomial f(x) = x 2 (x + 2)(x 1)(x + 1) Sketch P(x) = 2x2 +5x x 2 +x 11. Solve: x + 7 x 8 = Solve: 2x 2 (4x 1) = x(1 4x) 13. Solve: 2x = x 14. Solve: 3x 3 7x 2 2x + 8 = Solve: = 0 x 2 x Solve: x 5 6x 3 = 5x 17. Divide: 2x 3 3x 2 + x + 1 by x Divide: 2x 3 + 3x 2 2x 3 by 2x Find the other solutions of P(x) = x 3 7x x 27 given that 2 ± i 5 are solutions 20. Find k, such that x 3 is a factor of 3x 3 9x 2 + kx 12 Chapter Find the domain f(x) = x2 3x+6 x 2 3x 10

6 22. Find the domain f(x) = 2x x Even, Odd, or Neither f(x) = (x 2 + x) Even, Odd, or Neither f(x) = x x+x Describe any discontinuities in the following equation 26. Describe any discontinuities in f(x) = x2 4 x 2 +3x Let f(x) = x + 2, g(x) = x 2 2x. Find f[f(4)] 28. Let f(x) = x + 2, g(x) = x 2 2x. Find (g f)(x) 29. Find the inverse of f(x) = x 4 x 2 3x 2 +2x 12x 2 +5x 2 3kx + 4 x < Find k, that would make f(x) continuous f(x) = { 2x 8 x 4 x + 3 x Find f(2) if f(x) = { 3 0 x < 2 2x 1 x Express f(x) = 2x as a piecewise function 33. Let f(x) = x 2 + 4x 5. Find the difference quotient of f(x) 34. Let f(x) = x 2 2. Find the difference quotient of f(x) x + 3 x Sketch f(x) = { 3 0 x < 2 2x 1 x 2 x x < x < Sketch f(x) = { x 2 0 x 2 x x > 2 Chapter Evaluate 3 2 log log Evaluate log 25 8 log Evaluate e lne3ln2 40. Evaluate log log Expand ln(x 3 y) Expand log ( x2 y z )

7 43. Simplify to a single log: log x + 2 log y 3 log z 44. Simplify to a single log: ln ln 36 ln Rewrite using the change of base formula log 9 7, then get decimal apporximation 46. Solve: 3 2x+1 = Solve: log 2 (x + 2) + log 2 5 = Solve: 8 + 2e x = Solve: e 2x 4e x + 3 = Solve: 2 4x+1 = 3 x 51. Solve: ln 1 x = A population of bacteria doubles after 10 hours. What is the growth rate of the bacteria? 53. A house worth 200,000 in 1980 is now worth 325,000 in What is the relative growth rate? Chapter A sector has a radius of 15 cm and a central angle of 60. What is the sector s arc length? 55. A sector has an arc length of 6.4 cm and an area of cm 2. What is the central angle, θ? 56. A sector has a central angle of 120 and an arc length of 9cm. What is the sector s area? 57. A Point, P, moves on a circle, with radius 6 cm, at a speed of π radians per second. How far does P 6 move after 8 seconds? 58. Convert 7π 6 radians to Degrees 59. Convert 320 to Radians 60. Given P on a terminal ray at (5, 7). Find all 6 cosθ 61. Given: sec θ = 4 and sin θ > 0. Find tan θ. 62. Given cot θ = 2 and cos θ > 0. Find all sec θ

8 63. Evaluate each trig function a. tan 5π 3 b. cos 2π 3 c. csc 5π 4 d. cot 3π 2 e. sin 3π 2 f. sec 5π 5 g. cos π Evaluate sin 2 3π 4 + cos2 4π Evaluate 3 tan 7π 4 2 cot 5π Find all θfor 0 < θ 2π a. sec θ = 2 3 b. tan θ = 1 c. cos θ = 2 2 d. csc θ = undefined 67. Rewrite as a trig function less than 90 degrees a. cos 320 b. sec 250 c. cot Rewrite as a trig function less than 45 degrees Chapter 6 a. csc 115 b. tan Sketch y = 2 sec(x + π) 70. Sketch y = csc (3x π 2 ) Sketch y = 2 tan(2x) Sketch y = 1 2 cos 1 2 (x π 5 ) 3

9 73. Write a sinusoidal equation that has a maximum at( π 6, 3) and a minimum at (3π 2, 1) 74. Find Domain and Range: f(x) = 3 csc 4 (x π 3 ) Find Domain and Range: f(x) = 2 tan (3x 3π 4 ) Find Domain and Range: f(x) = 3 sin(2x) Bilbo became the first hobbit to orbit the planet when one of Gandalf s spells went horribly wrong. Bilbo s distance from the equator varied sinusoidally with time. After 45 minutes, Bilbo was the farthest away at 3200 miles. Half a cycle later he was the closest at 1600 miles. It takes Bilbo 80 minutes to orbit the planes. Write an equation that models how far Bilbo is away from the equator. 78. A tsunami is a fast-moving ocean wave caused by an underwater earthquake. The water first goes down, from its normal level, then rises an equal distance above its normal level, then returns to its normal level. Generally the period of a tsunami is around 15 minutes. Suppose a tsunami with a height of 10 meters hits Honolulu, Hawaii which has a normal water depth of 9 meters. Write an equation that models the water height. 79. Write an equation for the graph shown 80. Evaluate a. cos -1 ( 3 2 ) b. arctan 3 c. sin -1 (cos π 2 ) d. cos (tan 1 3) e. sin(sin ) f. cos (arcsin ( 1 2 )) g. sin (arctan ( 3))

10 Chapter Simplify csc 2 x(1 cos 2 x) 82. Simplify cota seca sina 83. Simplify (cscθ cotθ)(secθ + 1) 84. Simplify sina tana + sin (90 A) 85. Simplify cos ( π 3 + θ) + cos (π 3 θ) 86. Simplify cos(x + y) cos(x) + sin(x + y) sin (x) 87. Simplify cos 2 (4A) sin 2 (4A) 88. Simplify (1 + tan 2 x)(1 + cos 2x) 89. Evaluate cot(20 ) cos(20 ) sin(20 ) 90. Let cos(x) = 3 and tan(x) < 0. Find sin (2x) Evaluate 2sin(67.5 ) cos (67.5 ) 92. Let sin(α) = 4, sin(β) = 1. Let π < α < β < π. Find sin (α β) Evaluate sin(285 ) 94. Evaluate cos ( π 4 + θ) if cos(θ) = 1 2 with θ in Q4 95. Prove sinθ sinθ+cosθ = tanθ 1+tanθ 96. Prove 2 csc(2x) tan(x) = sec 2 (x) 97. Prove sin(3x) = 3 sin(x) 4sin 3 (x) 98. Prove = 1 sinx 1+sinx 2sec2 x sec(θ)+tan (θ) 99. Prove = (secθ + tanθ)2 sec(θ) tan (θ) 100. Solve 2 sin(2x) + 3 = sin(2x) = Solve 2sin 2 x 3 sin x + 1 = Solve cos 2 x 3 cos x 4 = Solve tan ( x 3 ) = 1

11 105. Solve (tanθ 1)(secθ 1) = Solve sin 2 (x) cos 2 (x) = 1 + cos (x) 107. Solve tan (μ) = 2sin (μ) 108. Solve 3(1 cos(x)) = sin 2 x Chapter Given: a = 12, <A = 80, <B = 40. Find side length b A satellite dish can track the speed of a plane by recording the distance to the plane at two points in time and the angle which the dish rotates. A satellite measured the distance to a plane at 35 miles. After 15 minutes, the dish measured the distance to the plane at 58 miles. If the dish rotated 132 degrees, how far did the plane travel? 112. Find the altitude of an isosceles triangle with base 4.24 feet. The vertex angle of the triangle measures 85 degrees Given: a = 3, b = 4, C = 40. Find the area of triangle ABC 114. A triangular parcel of land has sides measuring 25 yards, 31 yards, and 50 yards. What is the area of the land? 115. Find the area of a regular octagon inscribed in a circle with radius 6 cm.

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