Math 11, Precalculus Mathematics Sample for the Final Exam. Phone use is not allowed on this exam. You may use a standard two sided sheet of note paper and a calculator. The actual final exam consists of multiple choice questions. That allows for 4 minutes 4 seconds per question. (1) If the graph of f(x) is depicted to the right then graph f ( 1 x 1) + 1. a. b. c. d. 1
() Which of the following is the graph of 1 (x 1) + 1? a. b. ( ) 0, ( 1, 1) ( 1, 1) ( ) 0, 1 c. d. ( 1, 1) ( ) 0, ( ) 0, 1 ( 1, 1) () For the graph below, find the intervals of increase (, ) (1, 1) (4, ) ( 4, ) (, 1) a. ( 4, ) (, 1) (1, 4) b. ( 4, ) (, 4) c. (, ) ( 1, ) d. (, ) ( 1, 1) (1, )
(4) In which quadrant does the angle 017π 7 lie? a. Quadrant 1 b. Quadrant c. Quadrant d. Quadrant 4 () In which quadrant does the angle π 7 lie? a. Quadrant 1 b. Quadrant c. Quadrant d. Quadrant 4 (6) In which quadrant does the angle 4 radians lie? a. Quadrant 1 b. Quadrant c. Quadrant d. Quadrant 4 (7) In which quadrant does the angle 017 degrees lie? a. Quadrant 1 b. Quadrant c. Quadrant d. Quadrant 4 (8) What is the domain of g(x) = ln(x(x ))? a. (0, ) b. (, 0) (, ) c. [0, ] d. (, 0] [, ) (9) What is the domain of g(x) = (x(x ))? a. (0, ) b. (, 0) (, ) c. [0, ] d. (, 0] [, ) (10) What is the domain of g(x) = ln(1 4 x)? a. [, 4] b. (, 4] c. [, 4) d. (, 4) (11) Find the domain of ln(x(4 x) + 10) a. [ 14, + 14 ] b. ( 14, + 14 ) c. (, 14] [ + 14, ) d. (, 14) ( + 14, ) (1) Find the maximum value of x(4 x) + 10 a. b. 14 c. 4 d. 10 e. ln() f. ln(14) g. ln(4) h. ln(10) (1) Find input which produces the maximum value of x(4 x) + 10 a. b. 14 c. 4 d. 10 e. ln() f. ln(14) g. ln(4) h. ln(10) (14) Find the maximum value of ln(x(4 x) + 10) a. b. 14 c. 4 d. 10 e. ln() f. ln(14) g. ln(4) h. ln(10) (1) Find input which produces the maximum value of ln(x(4 x) + 10) a. b. 14 c. 4 d. 10 e. ln() f. ln(14) g. ln(4) h. ln(10)
4 For the next three functions, find the horizontal and vertical asymptotes, the x- intercepts and the y-intercepts (16) x 1 x + 1 Horizontal Asymptotes: a. y = 1 b. y = c. y = 1 d. y = 0 Vertical Asymptotes a. x = b. x = 1/ c. x = d. x = 1/ x-intercepts a. x = b. x = 1/ c. x = d. x = 1/ y-intercepts a. y = 1 b. y = c. y = 1 d. y = 0 (17) (18) x 1 (x 1)(x + 1) Horizontal Asymptotes: a. y = 1 b. y = c. y = 1 d. y = 0 Vertical Asymptotes a. x = 1 and x = 1 b. x = 1/ and x = 1/ c. x = and x = d. None of the above x-intercepts a. x = 1 and x = 1 b. x = 1/ and x = 1/ c. x = and x = d. None of the above y-intercepts a. y = 1 b. y = c. y = 1 d. y = 0 x 1 (x 1)(x + 1) Horizontal Asymptotes: a. y = 1 b. y = c. y = 1 d. y = 0 Vertical Asymptotes a. x = 1 and x = 1 b. x = 1/ c. x = 1 d. None of the above x-intercepts a. x = 1 and x = 1 b. x = 1/ c. x = 1 d. None of the above y-intercepts a. y = 1 b. y = c. y = 1 d. y = 0
(19) Which of the following is identically equal to log(a b)? a. log(a) log(b) b. log(a) + log(b) c. log(a) b d. a log(b) e. log(a) b f. log(a) log(b) g. log(a) log(b) h. log(a + b) i. log(b) a j. log(a b) (0) Which of the following is identically equal to log(a b ) a. log(a) log(b) b. log(a) + log(b) c. log(a) b d. a log(b) e. log(a) b f. log(a) log(b) g. log(a) log(b) h. log(a + b) i. log(b) a j. log(a b) (1) Which of the following is identically equal to log( b a)? a. log(a) log(b) b. log(a) + log(b) c. log(a) b d. a log(b) e. log(a) b f. log(a) log(b) g. log(a) log(b) h. log(a + b) i. log(b) a j. log(a b) ( a ) () Which of the following is identically equal to log b a? a. log(b) b log(a) b. log(a) a log(b) c. a log(b) log(a) d. b log(a) log(b) () Express log() in terms of log(4) and log(10) a. log(10) 1 log(4) b. log(10) log(4) c. log(10) log(4) d. log(10) + log(4) e. log(4) log(10) (4) Express log(40) in terms of log(4) and log(10) a. log(10) 1 log(4) b. log(10) log(4) c. log(10) log(4) d. log(10) + log(4) e. log(4) log(10) () Express log(0) in terms of log(4) and log(10) a. log(10) + 1 log(4) b. log(10) + log(4) c. log(10) + log(4) d. log(10) log(4) e. log(4) log(10) (6) Someone has slipped grams of Nobelium-8 into your Coffee. Nobelium-8 has a half life of one hour. You can safely drink the coffee once the concentration of Nobelium-8 is below. grams. How long do you have before you can drink your coffee? a. log (1/6) hours b. log (6) hours c. log (1/) hours b. log () hours
6 (7) A sample of mold is growing exponentially inside your coffee cup. At t = 0 there is (1/) grams of mold. After three days there is one gram of mold in your coffee cup. How much mold will there be in 0 days? a. 10 grams b. 9 grams c. 0 grams b. 9 grams (8) What is the inverse function to 17 e ln(x)+1? (Remember to Simplify) x x a. 17 b. 17e c. e x x 17 d. 17 1 (9) What is the inverse function to 17 x+1? a. (log 17 (x)) b. (log 17 (x 1)) c. log 17 (x 1) d. log 17 (x) 1 (0) Solve log (x + ) + log (x ) = log (x) + 1 a. x = 1 b. x = 4 and x = 1 c. x = 4 d. There are no solutions. (0.1) How many solutions are there to log (x + 1) + log (x 1) = log (x) + a. No solutions b. One solution c. Two solutions d. Three Solutions (1) Solve log (x) log (x 10) =? a. x = 0 b. x = 80 c. x = 4 7 4 d. x = 4 e. There are no solutions. () How many solutions to cos (x) sin(x) + 1 = 0 lie in the interval (0, π)? a. None b. One c. Two d. Three e. Four () How many solutions to tan (x) tan(x) 1 = 0 lie in the interval (0, π)? a. None b. One c. Two d. Three e. Four (4) How many solutions to cos (x) cos(x) + 1 = 0 lie in the interval (0, π)? a. None b. One c. Two d. Three e. Four () The ghost of Oscar Wilde floats 0 meters ahead of you. The line between you and Oscar is 0 above horizontal. How far above the ground is Oscar Wilde s ghost? (in meters) The figure to the right is helpful Oscar Wilde you 0 0 meters a. 0 tan(0 ) b. 0 tan(0 ) c. 0 cos(0 ) d. 0 cos(0 ) d. 0 sin(0 ) e. 0 sin(0 )
(6) Start at the coordinate (1, 0) and rotate counter clockwise an angle of 017π/ radians. What is the y-coordinate of your end position? a. 0 b. 1 c. d. e. 1 f. 1 g. h. i. 1 (7) Start at the coordinate (1, 0) and rotate counter clockwise an angle of 017π/ radians. What is the x-coordinate of your end position? a. 0 b. 1 c. d. e. 1 f. 1 g. h. i. 1 (8) Start at the coordinate (1, 0) and rotate counter clockwise an angle of 017π/ radians. What is the slope of the ray from the origin to this point? a. 0 b. undefined c. 1 d. e. f. 1 g. h. (9) Start at the coordinate (1, 0) and rotate counter clockwise an angle of 017π/ radians. What is the length of the arc you have traced? a. 017π b. π c. π d. 4π (40) You wish to construct a circular track which has the property that for every radian you travel around the track you walk 0 feet. What is the radius of this track? a. 10π feet b. 0π feet c. 10 feet d. 0 feet (41) What is the period of sin(x 10)? a. π b. π c. π d. π e. π (4) What is the period of tan(πx 10)? a. π b. π c. 1 d. e. π (4) What is the period of cos(x/π)? a. π b. π c. d. π e. π (44) What is cos ( sin 1 ( 1 ))? a. 1 b. 0 c. 6 d. 6 e. 4 6 (4) What is cos ( cot 1 ( ))? a. 1 b. 0 c. 9 9 d. 4 9 9 e. 9 9 (46) What is tan ( cot 1 ( ))? a. 1 b. 0 c. d. e. 4 (47) What is cos ( cos 1 ( 4 ))? a. 4 b. c. 4 d. 4 7
8 (48) Two chickens cross the road at the same time. Their paths cross making an angle of 0. If the first chicken is traveling at a rate of 1 foot every second and the other is traveling feet every seconds then after one minute what is the distance between them? The solutions to this problem are all incorrect. See solutions for correct answer. a. 90 feet b. 10 feet c. 0 7 feet d. 10 0 6 feet (49) If cos(θ) = 1/4 and π + π/ < θ < 6π then what is tan(θ)? a. 1 b. 1 1 c. 1 16 d. 1 4 e. 1 f. 1 1 g. 1 16 h. 1 4 (0) If cos(a) =.1 then cos(a) =... a..98 b..99 c..8 d..9 e.. f..98 g..99 h..8 i..9 j.. (1) If sin(a) =.1 = 1 10 and π < a < π then sin(a) =... a. 11 100 b. 11 0 c. 10 0 d. 10 100 d..9 e.. g. 11 100 f. 11 0 h. 10 0 i. 10 100 j..9 e.. () If tan(a) = then tan(a) =... a. 4 b. c. 4 d. e. 4 f. g. 4 h. () What is sin 1 (sin(017π/))? a. π b. 7π c. π d. π e. 7π f. π (4) What is cos 1 (cos(017π/))? a. π b. 7π c. π d. π e. 7π f. π () What is tan 1 (tan(017π/))? a. π b. 7π c. π d. π e. 7π f. π
9 (6) The triangle to the right is not a right triangle and is not drawn to scale. What is cos(α)? α 9 10 11 a. b. d. e. 4 4 c. 1 f. 1 (7) The triangle to the right is not a right triangle and is not drawn to scale. What is sin(α)? a. 11 b. 11 c. 11 d. 11 0 e. 11 0 f. 11 0 π 6 α 10 11 (8) The triangle to the right is not a right triangle and is not drawn to scale. What is its area? π 6 9 10 a. 4 b. 4 c. 90 d. 4 e. 4 f. 90