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Final Review Math 1508
Problem 1. The fuel efficiency (in miles per gallon) of an SUV depends on its weight according to the formula E = 0_._ 0000016 x 2 − 0_._ 016 x + 54 (1800 ≤ x ≤ 5400), where x is the weight of an SUV in pounds. According to the model, what is the weight of the least fuel-efficient SUV? Problem 2. Two airplanes leave Kennedy airport in New York at 11 am. The air traffic controller reports that they are traveling away from each other at an angle of 100°. The first airplane travels 405 mph and the second airplane travels at 409 mph. How far apart are they at 11:30 am? Problem 3. Manipulating only with one side of the identity, prove that a) !
sin ( 2 x ) =
2 tan x 1 + tan^2 x b)
tan
2
x " sin
2
x = (sin
2
x )(tan
2
x )
c) ! sin t + sin 3 t cos t + cos 3 t = tan 2 t d) ! csc 2 " = csc " 2 cos " Problem 4. Given is the function !
g ( x ) =
3 x^2 + 3 x " 6 4 x^2 " 8
a) Find all zeros, if any. b) Find the equations of vertical asymptotes, if any, c) Find the equations of horizontal asymptotes, if any. d) Find the y - intercept. e) Using the information you obtained in parts a)-d), sketch the graph of. Problem 5. Solve the system of equations: a. ! "
x y z x y z x z b. ! 2 x " y = 1 2 x " x 2 = y
Problem 6. Let !
f ( x ) =
2 " 3 x 3 x + 5 a) Find the inverse function, ! f " 1 ( x ). b) Find the domain of ! f. c) Find the domain of ! f " 1 . d) Find the range of ! f. e) Find the range of ! f "^1.
Problem 7. Find all the solution for the following equations: a. ! x 4 " 2 x 3
- x 2
- 2 x " 2 = 0 b. ! ln( x + 1 ) " ln( 2 x " 1 ) = 1 c. ! 2 e^2 x^ = 4 (1.2) x Problem 8. Soon after taking an aspirin, a patient has absorbed 300 mg of the drug. If the amount of aspirin in the bloodstream decays exponentially, with half being removed every two hours, find, to the nearest 0.1 hour, the time it will take for the amount of aspirin in the bloodstream to decrease to 100 mg. Problem 9. Find all solutions for the equations: a. ! 2 cos^2 x = 1 " cos x b. ! cos x = sin 2 x Problem 10. A population of animals oscillates between a low of 1300 on February 1 and a high of 2200 on August 1. Let t represents the time in months with t=0 representing January 1. a) Find a formula for the population, P, in terms of the time, t, in months. b) Find and interpret the amplitude, period, and midline. c) Calculate the population by March 1. Problem 11. An observer is near a river and wants to calculate the distance across the river. He measures the angle between his observations of two points on the shore, one on his side and one on the other side, to be 28º. The distance between him and the point on his side of the river can be measured and is 300 feet. The angle formed by him, the point on his side of the river, and the point directly on the opposite side of the river is 128º. What is the distance across the river? Problem 12. If! " ,! 2
, csc 7
cos = =# is in the 4th^ quadrant, and! is in the 3rd
quadrant, find the exact value of sin( " + !).
Problem 13. An airplane flies at an altitude of 6 miles toward a point directly over an observer. Let ! " be the angle of elevation to the plane and let x be the horizontal distance between the airplane and the observer. Find ! " as a function of x and find the angle of elevation ! " when the horizontal distance between the airplane and the observer is ! x = 7 miles.
