Pre-Calculus 40 Final Outline/Review:

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1 Pre-Calculus 40 Final Outline/Review: Non-Calculator Section: 13 multiple choice and 8 open ended. Calculator Section: 7 multiple choice and 13 open ended. First Semester Topics: o Logarithmics and Exponentials (5 questions): Evaluate, solve and investment o Polynomials (2 questions): Solve using Fundamental Theorem of Algebra and Regressions. o Rationals (2): Simplify, Identify domain. o Inverse and composition (2): Operations, composition and Proof. Second Semester Topics: Trigonometry: o Right Triangle Trig (2 questions) o Area (2): formulas given o Arc Length, Linear/Angular Speed (2): formulas given o Evaluate Unit Circle Values (6 questions): including - given one trig, evaluate others, using sum and difference and double angle formulas (given) o Inverses (2 questions): single and composition o Sinusoidal Waves (3): Amplitude, Period, Phase shift, domain and range o Graph sine or cosine by hand (1) o Solve Triangles (2 questions): using law of Sines and Law of Cosines (given) Parametric, Polar and Vectors: o Convert between Parametric and Rectangular (1) o Using parametric (1): Calculate height, distance given velocity and angle of elevation o Convert between Polar and Rectangular (3 questions): Points and equations o Graph Polar(1): calculator section o Determine Vector (2 questions): given points, magnitude and direction o Force (1) o Determine Magnitude and properties of Vectors (2): sum, difference, scalar multiplication.

2 First Semester Topics: 1. Determine if the following equations are polynomials. If they are polynomials, determine the degree, leading coefficient and constant term. a. f(x) = 3x b. g(x) = 4x 5 3x c. h(x) = 5x 3 2x Determine whether the x value is a solution to the equation 0 = x 4 + 6x 3 + 9x x + 20 a. x = 2 b. x = -1 c. x = 0 3. Determine the remainder when 3x 2 + 2x 1 is divided by (x + 2). 4. Determine all real solutions of 2x 3 x 2 7x + 6 = 0. NONCALCULATOR! Show Factored Form. 5. Determine all real solutions of x 3 + 3x 2 4x 12 = 0. NONCALCULATOR! Show Factored Form. 6. Simplify the Rational Expressions: a. 3p2 9p 30 6p 2 +6p 12 b. x2 +8x+15 2x 2 18

3 7. Determine the domain, holes, vertical asymptotes, horizontal or oblique asymptotes: a. f(x) = x2 4 b. h(x) = x+4 x 2 2x 8 x 2 +x Add, Subtract or multiply the Rational Expressions: a. x+1 + x+2 b. 4(x 2) x 3 x c. ( x2 3x+2 x 2 1 ) (3x x 4 ) 9. Evaluate (non-calculator): a. log 3 81 b. log c. ln e Solve the equation (non-calculator): a. 32 x+1 = 8 5x b. ln(3x 1) + 2 ln(3) = ln(x x) c. log 3 27 = 2x Paula invests $250 at 3.75% compounded monthly. Determine the amount in her account after 15 years. 12. Frederico invests $100 at 3.75% compounded continuously. How long will it take for him to double his investment? 13. Angelo needs $12,000 to make his dream trip to Greece. If he plans the trip for 10 years from now and plans on investing money at 4.65% continuously, how much does he need to invest now?

4 Second Semester Topics: 14. Write each degree measure in radians as a multiple of π and each radian measure in degrees. a. 136 b. 45 c. 3π 4 d. 5π Find the length of the intercepted arc with the given central angle measure in a circle with the given radius. Round to the nearest tenth. a. 8π 3, r = 1.5cm b. 105, r = 10 ft 16. Find the area of a sector of a circle with diameter 18 feet formed by a 75 o angle. 17. A car with a tire with a radius of 16 inches is rotating at 450 revolutions per minute. a. Find the angular speed, in radians per minute. Round to the nearest tenth. b. Find the linear speed of the tire s rim, in miles per hour. 18. The point (-1, 5) lies on the terminal side of an angle in standard position. Find the values of the six trigonometric functions of. 19. Find the exact value of each trigonometric function, if defined. If not defined, write undefined. a. tan( 45 ) b. cos ( 3π 2 ) c. csc (5π 6 ) d. sin(900 ) e. sec (π 2 ) 20. Find the exact values of the five remaining trigonometric functions of. a. sec θ = 11, tan θ < 0 b. cosθ = 1, csc θ > 0 3 3

5 21. For each function below: Identify the domain, range, amplitude, period, frequency, phase shift, vertical shift, and x-interval. Neatly graph each function on a separate sheet of paper. Fully label the scale on each axis and include an x-y table. a. y = 2.5 sin x b. y = cos ( π 2 x) + 1 c. y = 1 3 sin(4x) 1 d. y = cos (x + π 3 ) 2 a. c. b. d. 22. Find the exact value of each of the following WITHOUT using a calculator! a. tan 1 ( 3 ) = b. 3 sin 1 ( 3 2 ) = c. arccos ( ) = d. 2 2 tan 1 (tan ( 2π )) = 3 e. sin 1 (sin( π)) = f. cos (arctan ( 5 2 )) = g. sin (cos 1 ( 3 5 )) = 23. A pilot needs to begin his descent when his plane is 7.5 km above ground and 200 km straight to the airport. At what angle should his decent be so that he can fly in a straight line from the point of initial decent to the ground? How much ground will he pass from the point of initial descent until he touches down at the airport?

6 24, Solve the following triangles. Round to the nearest hundredth. a. a = 11cm, b = 6 cm, A = 22ᵒ b. a = 13 m, b = 12 m, c = 8m c. a = 9 cm, b = 10 cm, C = 42ᵒ d. a = 5 cm, A = 36ᵒ, B = 42ᵒ e. A = 63, a = 18in, b = 25in f. A = 20 o, a = 4mm, b = 6mm 25. Determine the area of each triangle to the nearest tenth. a. A = 95, b = 12m, c = 18 m b. a = 44, b = 47, c = Find the value of each expression using the given information. a. If tan θ = 7 and sin θ > 0, find cos θ. b. If cos θ = 3 and 180 < θ < 270, find cot θ If sin α = 5 and cot α < 0, determine: 9 1. tan α b. sin( α) 28. If cos α = 5 and sin α > 0, determine: 5 a. sec α b. csc α

7 29. Find the exact value of each trigonometric function. a. cos 15 b. sin 19π c. tan 255 d. cos25 cos35 sin25 sin sin α = 5 and cos β = 1, where cos α < 0 and π β 0. Find: sin (α β) sin α = and cos β =, where π α π and 0 β π. Find: sin (α + β) sin α = 3 5 and tan β = 5 12, where 3π 2 α 2π and sin β < 0. Find: cos (α β) 33. In ABC below, find the following values. a. sin A = b. tan C = c. cos A =

8 34. A blimp was flying above Fairfield the other day at an altitude of 425 meters. Emily was in the blimp and she saw the high school. She calculated the angle of depression from the blimp to the high school was about 48. If she dropped a rock out of the blimp and the rock fell straight to the ground, how far away from the high school would the rock land? Round your answer to the nearest meter. 35. Find the rectangular coordinates of: a. (4, 120 ) b. (-2, 3π/4) c. (3, -π/3) 36. Find one set of polar coordinates for the following rectangular coordinates if r > 0: a. (3, 6) b. (-2, 7) c. (-1, -7) 37. Name the polar coordinates of points A and F graphed below if: a. r>0 and 0 θ 360 b. r<0 and 0 θ 360

9 38. Write the polar equations in rectangular form: a. r = -6sinθ b. r = 2cscθ 39. Write the Polar equations in Rectangular form and Graph By HAND a. r = 3 secθ b. θ = π/6 c. r = 5cos θ 40. Write the rectangular equations in polar form, then graph: a. x 2 + y 2 = 16 b. (x 2) 2 + y 2 = Name the Graph of the polar equation and Sketch its graph using a Graphing Calculator. a. r = 2 2sinθ b. r = 3 + 2cosθ

10 42. Graph the number 3 + 4i in the complex plane and find its absolute value. 43. Find 5r 2s if r = 3, 9 and s = 3,6 44. Determine the direction angle and magnitude of the following vectors. Parts a and b (non-calculator) and part c (calculator) a. 2, 2 b. 0, 7 c. 2, Let AB be the vector with initial point A(10, 4) and terminal point B( 1, 3). Write AB as a linear combination of the vectors i and j. 46. Find the component form of AB with initial point A( 12, 7) and terminal point B(8, 2). 47. Find the component form of AB given v = 12 and direction angle θ = 5π. NONCALCULATOR. 3

11 48. A plane takes off at 220 miles per hour at an angle of 51 with the ground. Find the magnitude of the horizontal and vertical components of its velocity. Round to the nearest tenth.(calc) 49. Charles leaves his apartment and walks 55 north of west for 1000 feet and then walks 300 feet due north to go bowling. How far and at what quadrant bearing is Charles from his apartment? (CALC) Parametric Formulas: x = tv 0 cos θ y = tv 0 sin θ 1 2 gt2 + h Write the following parametric equations in rectangular form: a. x = 3t 1, y = 2t b. x = 4 cos θ y = 2 sin θ 51. Suppose Mr. Coyne hit a golf ball with an initial velocity of 150 feet per second at an angle of 30 o to the horizontal. Round all answers to the nearest hundredth. (CALC) a. Write a set of parametric equations that describe the position of the ball as a function of time. b. How long is the golf ball in the air? c. When is the ball at its maximum height? d. What is the maximum height of the golf ball? e. His goal was to hit the golf ball at least 600 feet. Did he reach his goal? How far away did the golf ball land?