Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Review for Final Exam Note: all students will take final exam in two parts, over two days. Logarithmic and Exponential Functions 1. Determine a formula for the exponential function given in the table below. X 70 71 72 73 74 F(x) 32 8 2 0.5 0.125 1 π(π₯ ) = 32 ( )π₯β70 4 2. Write and solve an exponential equation. Round answer to the next whole number. The population of a town in Alaska was 9500 in 1985. Since then, the population has decreased an average of 1.3% per year. In what year do you predict that the population will drop below 3000 people? In the year 2073 3. Use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms: a. log π₯ log π₯ β 2 log π¦ π¦2 1 1 ln 5 + ln π¦ 3 3 3 b. ln β5π¦ 4. Use properties of logarithms to write the expression as a sum or difference of logarithms or multiples of logarithms. Then, list the transformations that change f(x) = log x into g(x). a. log 1000 3 βπ₯ 1 3 β 3 log π₯ 1 translate up 3 units; reflect over x-axis; vertical shrink factor of 3 b. log (π₯β1)2 10 2 log(π₯ β 2) β 1 translate down 1 unit and right 2 units; vertical stretch factor of 2 5. A bacteria population is increasing 12.2% each day. How long will it take the population to quadruple? Write an equation and solve it. Round to the nearest whole day. β 12 or 13 days 6. Solve the equation. Round to the thousandths place. a. 2(3π₯β2) = 23 x β 2.175 b. 3(2)π₯ = 51 x β 4.087 Review for Final Exam 7. Georgia needs $5000 for a down payment on a car in 6 years. Her bank promises an interest rate of 1.15% yearly. How much should she invest now to guarantee that she will have enough money for the car in 6 years? Principal is β $ 4668.47 Trigonometry Basics 5π 9 8. Convert 100° into radians. Give an exact, simplified answer: πππππππ 9. Find all six trigonometric functions of A in βABC: 5 Csc A = 13 5 5 cm Cos A = 12 Sec A = 13 12 C Cot A = 12 5 B Sin A = 13 13 A 12 cm 5 Tan A = 12 B 10. Solve the right βABC if a = 25 and b = 7.3. 16.28 ° 11. Solve the right βABC if B = 48° and a = 7 B 26.044 units 48 ° 25 units 10.46 units A 7 units C 7.77 units 73.72 ° A 7.3 units C Review for Final Exam 12. The point (4,9) is on the terminal side of angle ΞΈ. Evaluate the six trigonometric functions for ΞΈ: Sin ΞΈ = Tan ΞΈ = 9 Csc ΞΈ = β97 9 Cot ΞΈ = 4 β97 9 Cos ΞΈ = 4 β97 Sec ΞΈ = β97 4 4 9 13. Find one positive angle and one negative angle that are co-terminal with an angle of 19π β13π 8 8 3π 8 2π radians: 14. Calculate the length of an arc intercepted by a central angle of radians in a circle with a radius 3 4π of 2 centimeters. ππππ‘ππππ‘πππ 3 Sinusoidal Functions and Problem-solving 15. Determine whether f(x) is a sinusoid. If yes, determine the values of a, b, and h so that f(x) β a sin (b (x β h)) + 0. Round to two decimal places if necessary. a. f(x) = sin x β 3 cos x = 3.16 sin 1 ( x β 1.25) b. f(x) = Ο sin3 x β 4Ο sin 2x not sinusoidal because coefficients are 3 and 2 Review for Final Exam c. f(x) = 3 cos 2x β 2 sin 2x = -3.61 sin 2 ( x β 0.491) 16. In the equation sin x = 0.5, solve for x in the interval π 2 β€ π₯ β€ π. x= 5π 6 17. The average daily air temperature (β) for Fairbanks, Alaska, from 1975 to 2004, can be 2π modeled by the equation: π(π₯) = 37.3 sin [ (π₯ β 114)] + 26, where x is the time in days. 365 January 1 is represented by x = 1. On what days do you expect the average temperature to be 32β? Day 123 and Day 287, which by the way is May 3rd and October 14th in the year 2014 Review for Final Exam 18. On a particular Labor Day, the high tide in southern California occurs at 7:12 AM. At that time, you measure the water at the end of the Santa Monica Pier to be 11 feet deep. At 1:24 PM it is low tide and you measure the water to be only 7 feet deep. Assume the depth of the water is modeled by a sinusoidal function of time. a. At what time on that Labor Day does the first low tide occur? 1 AM b. What was the approximate depth of the water at 9:00 PM? 10.5 feet c. What is the first time on that Labor Day that the water is 9 feet deep? 4:06 AM d. 2 19. Specify the period and amplitude of the function f(x) = 4 sin 3 (π₯): Amplitude is 4; period is 3Ο 20. State the amplitude, period, phase shift, and vertical displacement of the function 2 π₯β3 f(x) = 3 cos ( 4 ) + 1 as compared to the parent function f(x) = cos x: Amplitude is 2 , 3 period is 8Ο, phase shift is 3, and vertical displacement is 1 21. Write an equation of the sine function with amplitude vertical translation down 5 units: 1 f(x) = 2 sin π(π₯ + 3) β 5 ½ , period 2, phase shift left 3 units, and Review for Final Exam 22. Solve βABC, if a = 3.2 inches, b = 7.6 inches, and c = 6.4 inches: C 7.6 inches 74.7 ° 3.2 inches A 80.8 ° 24.6 ° B 6.4 inches 23. Solve βDEF if d = 41 meters, D = 22.9°, and F = 33°: F 33 ° 87.2 meters 41 meters D 124.1 ° 22.9 ° 27.0 meters E 24. Two triangles can be formed using the measurements C = 68°, a = 19 feet, and c = 18 feet. Solve both triangles: Review for Final Exam B B 10.15 ° 33.85 ° 180 - Angle A = new angleA 18.0 feet 19.0 feet 18 feet 19 feet 101.85 ° 78.15° 3.42 feet C 10.8 feet C A 68 ° A 68 ° 25. What is the measure of the largest angle in a triangle with sides 117 cm, 102 cm, and 160 cm? 160 cm 117 cm 93.62 ° 102 cm Trigonometric Identities π 26. Simplify the expression: cot ( 2 β π₯) csc (-x) = β sec x Review for Final Exam 27. Simplify the expression csc ΞΈ cot2 ΞΈ + csc ΞΈ 28. Simplify the expression = π ππ2 π½+ πππ‘ 2 π½+ πππ 2 π½ ππ π 2 π½ csc3 ΞΈ = 1 29. Determine all the solutions to the equation 4 cos2 x β 4 cos x + 1 = 0 in the interval [0, 2Ο): X= π 3 πππ 5π 3 radians 30. Write the expression 5 β 6 sin2 ΞΈ β 5 cos ΞΈ in factored form as an algebraic expression of a single trigonometric function: 2 6 cos x β 5 cos x β 1 31. Determine all the solutions to the equation tan Ο sin2 Ο = tan Ο in the interval [0, 2Ο): X = 0Ο , π 2 ,π πππ 3π radians 2