Math 112, Precalculus Mathematics Sample for the Final Exam.

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1 Math 11, Precalculus Mathematics Sample for the Final Exam. Solutions. There is no promise of infallibility. If you get a different solution, do not be discouraged, but do contact me. (1) If the graph of f(x) is depicted to the right then graph f(x 1). () If the graph of f(x) is depicted to the right then graph f( x). () For the two graphs above label the intervals of increase and decrease. (1) The interval of increase is ( 4, 1). The interval of decrease is (1, 4) () The interval of increase is (, 1). The interval of decrease is ( 1, ) (4) In which quadrant does the angle 014π lie? () In which quadrant does the angle π lie? Quadrant 1 (6) In which quadrant does the angle 4 radians lie? 1

2 () In which quadrant does the angle 014 degrees lie? (8) What is the domain of g(x) = ln(x(x + ))? (, ) (0, ) (9) What is the domain of g(x) = (x(x + ))? (, ] [0, ) (10) What is the domain of g(x) = ln(1 4 x)? (, 4] (11) Find the domain of ln(x(4 x) + 10) ( 14, + 14) (1) Find the radius of the circle given by x + 4x + y 6y = 1 (1) Find the maximum value of x(4 x) (14) Find input which produces the maximum value of x(4 x) + 10 (1) Find the maximum value of ln(x(4 x) + 10) ln(6) (16) Find input which produces the maximum value of ln(x(4 x) + 10) (1) Where are the horizontal and vertical asymptotes? Where are the x-intercepts? Where are the y-intercepts Horizontal Vertical x-int y-int 1 1/ 1/ 1 x + 1 x 1 ±1 ± 1/ 1 0 ±1 1/ 1 none 1/ ±1 1

3 (18) Simplify (a) log(a b) log(a) + log(b) (b) log(a b ) b log(a) (c) log( b a) log(a)/b ( ) 1 (d) log b a a a log(b) log(a) (19) Express log() in terms of log(4) and log(10) log(10) log(4) (0) Express log(40) in terms of log(4) and log(10) log(10) + log(4) (1) Express log(0) in terms of log(4) and log(10) log(10) + log(4) () 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? 0 tan(0 ) () Someone has slipped grams of Nobelium-8 into your Coffee. Nobelium-8 has a half life of one hour. If you can safely drink the coffee once the concentration of Nobelium-8 is below. grams, then how long do you have before you can drink your coffee? log (6) (4) A sample of mold is growing exponentially inside your coffee cup. At t = 0 there is. 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? 9 grams () What is the inverse function to 1 ln(x)+1? e log (x/1) 1 (6) What is the inverse function to 1 x+1? log 1(x) 1 () Solve log (x + 6) log (6x ) = log (x) x = Notice the lack of a 6 minus solution. (8) How many solutions are there to the equation log (x) + log (x 10) = 9? One

4 4 (9) How many solutions to cos (x) sin(x) + 1 = 0 lie in the interval (0, π)? One (0) How many solutions to tan (x) tan(x) + 1 = 0 lie in the interval (0, π)? None. (1) Start at the coordinate (1, 0) and rotate counter clockwise an angle of 014π/ radians. What is the y-coordinate of your end position? () Start at the coordinate (1, 0) and rotate counter clockwise an angle of 014π/ radians. What is the x-coordinate of your end position? 1 () Start at the coordinate (1, 0) and rotate counter clockwise an angle of 014π/ radians. What is the slope of the ray from the origin to this point? (4) Start at the coordinate (1, 0) and rotate counter clockwise an angle of 014π/ radians. What is the length of the arc you have traced? 014π/ () What is the definition of an even function? What is the definition of an odd function? Sketch a function which is even and not odd. Sketch a function which is odd and not even. Sketch a function which is both even and odd. Sketch a function which is neither even nor odd. Answers will vary (6) 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? What is its circumference? 0 feet () What is the period of sin(0)? π (8) What is the period of tan(πx 10)? 1 (9) What is the period of cos(x/π)? π (40) prove that tan 4 (x) cot 4 (x) = 4 cos(x) (False identity. Bad question) 1 cos(4x) (41) What is cos ( sin 1 ( 1 ))? 6 (4) What is cos ( cot 1 ( ))? 9 (4) What is tan ( cos 1 ( )) 4? (44) What is cos ( cos 1 ( )) 4 4? (4) 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? feet. (46) If cos(θ) = 1/4 and π + π/ < θ < 6π then what is tan(θ)? 1

5 (4) If cos(a) =.1 then cos(a) = ± 11 (48) If sin(a) =.1 then sin(a) = (49) If tan(a) =.1 then tan(a) = (0) What is sin 1 (sin(014π/))? π (1) What is cos 1 (cos(014π/))? π () What is tan 1 (tan(014π/))? π () The triangle to the right is not a right triangle and is not drawn to scale. What is cos(α)? α (4) The triangle to the right is not a right triangle and is not drawn to scale. What is sin(α)? 100 sin(0 ) 0 α 100

Math 112, Precalculus Mathematics Sample for the Final Exam.

Math 112, Precalculus Mathematics Sample for the Final Exam. 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