MATH 175: Final Exam Review for Pre-calculus

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1 MATH 75: Final Eam Review for Pre-calculus In order to prepare for the final eam, you need to be able to work problems involving the following topics:. Can you find and simplify the composition of two functions and determine the domain of the new function? Pro.. Find f g and determine the domain of the composite function. Put your answer in interval notation. a. f() = ; g() = f() = ; g() =. Can you find the difference quotient of a given function? f ( h) f ( ) Pro. Find the difference quotient of f; that is find h simplify., h 0, for each function. Be sure to a. f ( ) f ( ) (Hint: Rationalize the numerator). Can you graph rational functions by hand after algebraically or numerically finding their asymptotes or holes, if any and then verify your results using a graphing utility? Pro.. Sketch a complete labeled graph for the following function after listing any (and all) vertical asymptotes, holes, horizontal asymptotes, intercepts, and domain. f( ) ( 5) ( ) ( 6 5)( 5). Can you determine whether a function has an inverse function and, if possible, find it? Pro.. Give a reason why you epect the function f ( ) to have an inverse f Sketch the graph f( ) and f aes.) ( ). ( ) on the same set of aes, clearly labeling each. (use ZDecimal for the Pro. Why doesn t the function f ( ) ( 7) have an inverse function? Give a reason (which could be in the form of a labeled sketch with comments). Pro. Algebraically find the inverse function of range of f and f. f. Algebraically determine the domain and Revised February 06

2 5. Can you define and graph the basic eponential and logarithmic functions and their transformations by hand? Pro5. Sketch a labeled graph of the function f( ) e if ln( ) if 6. Can you use the properties of logarithms to epand or condense an epression? Pro 6. Write each epression as a single logarithm. a. 5log log log log log Prob 6. Write each epression as the sum and/or difference of logarithms. Epress powers as factors. a. log 5, ln, 7. Can you solve eponential and logarithmic equations algebraically and/or graphically using the laws of logarithms where appropriate? Pro7. Solve the following equations for. Show your steps and give eact answers. a log ( ) c. log 6 log d Can you find the eact values of the trigonometric functions using the unit circle, point in the plane, and the right triangle definitions of the trigonometric functions? Pro8.. Sketch a unit circle with the given values marked on it Then find the eact functional value requested. a. Find sin() if = 5/ Find csc() if = -5/6 Pro8.. If sin(and cos : (a.) What Quadrant is the terminal side of in? () Give the eact values of cos( and of tan( cos( tan Revised February 06

3 Pro 8. Find the eact value of each of the following under the given conditions. sin, where cos where 5 a. sin cos c. sin d. tan 9. Can you use the basic trigonometric identities to prove other identities and to find eact values of the other trigonometric functions given the eact value of one function? Pro9. Complete the following identities with an equivalent epression. a. tan(-) = cos(a)sin(b) sin(a)cos(b)= c. + tan () = d. cos(a) = e. cos(a)sin(a) = f. cos A ( ) g. cos (5 )+ sin (5)= Pro9.. Use the appropriate sum or difference formula to find the eact value of: (Show your steps.) a. 5 cos = sin( )cos(8 ) sin(8 )cos( ) Pro9. Prove the identity, show all work. tan( ) sec( ) sec( ) tan( ) 0. Can you graph basic trigonometric functions and transformations of the sine and cosine function by hand? Pro0. The simple harmonic motion of a spring that is initially displaced from its rest position is given by the equation y 5cos where is in seconds and y is in centimeters. a. Sketch a labeled graph of this equation for time = 0 to = 8. What is the maimum displacement? c. Mark with an asterisk and give the coordinates for each time that the spring is cm below the rest position. Pro 0. Graph two periods of the function f sin phase shift.. State the amplitude, period and Revised February 06

4 . Can you solve trigonometric equations both algebraically and graphically? Pro. Solve each equation on the interval 0 a. sin sin c. sin cos sin d. cos sin 0 e. cos sin 0. Can you solve right and oblique triangles? Pro. Which of the following statements about the angle is true? a. sin() / cos() 5 / c. tan() /5 d. sin() /5 e. sin() / 5 Pro. Find the length of side h in the triangle, where angle A measures 0 o and the distance from C to A is 5. B h 0 C 5 A. Can you evaluate inverse sine, inverse cosine, and inverse tangent function values eactly when possible and, if not possible, approimate using a graphing utility? Pro. a. What is the domain and range for the function f() cos ()? Give eact values in radians of cos ( /). Pro. a. What is the domain and range for the function f() tan ()? Give eact values in radians of tan ( ). Pro. a. What is the domain and range for the function f() sin ()? Give eact values in radians of sin (sin(8 / )). Pro. a. Give eact values in radians of cos (sin(5 /). Give the eact value of tan(cos ( / )). Revised February 06

5 Pro.5 Find all real numbers that satisfy the equation : tan(). Can you solve applications involving eponential, logarithmic, and trigonometric functions algebraically or by using a graphing utility? Pro. The half-life of calclinium is 00 days. If you start with 00 milligrams, write down the epression for the amount of calclinium remaining after t days. Use your result to find the time when % of the calclinium you started with remained. Pro. a. Define all 5 variables in the formula: nt r A P. n Use this formula and algebra to determine how long it would take an investment of $500 to grow to $5000 at an annual interest rate of 7.5% and weekly compounding. An answer without the required steps will not receive full credit. Pro. The annual rate of growth of the world s population in 005 was k =.5%. The world s population in 005 was 6,5,058,790. a. Let t = 0 represent 005 and use the uninhibited growth model to predict the world s population in 05. Using this model determine how long it would take for the population to double. Pro.. A pilot in a plane at an altitude of,000 feet observes that the angle of depression to a nearby airport is 6 o. How many miles is the airport from a point on the ground directly below the plane? A labeled sketch of the situation is required. Revised February 06 5

6 Answers to the Final Eam Review questions for Math 75 Pro.. a. f g = ; Domain:, 0 0,, Pro.. a. g f = ; Domain: [, ) ( )( h ) h Pro.. f( ) ( 5) ( ) ( 6 5)( 5) ( 5) ( ) ( )( 5)( 5) Hole at (-5, /), vertical asymptote: = -, horizontal asymptote: y=, -intercept, 0 Domain:, 5 or, 5 5,,, y-intercept 0, 6 Pro.. Use your graphing calculator to graph f ( ) and y, then use DRAWINVERSE from the DRAW menu to see the graph of f y f ( ) ( ) 0 Revised February 06 0 y f ( ) 6

7 Pro. ( ) ( 7) is not because 7 ( y ) / y f. Pro. f. The domain of f is,,. The range of f is,, The domain of f is,,. The range of f is,, Pro5. f( ) y e if e if ln( ) if (, e ) y ln( ) if Pro6. (a) log (b) log log log log Pro6. (a) (b) ln ln ln Pro7. (a) ln(5) /(ln(7/ 5) (b) (c) = 7 (d) Revised February 06 7

8 Pro8.. (a) sin(5 / ) / (b) csc( 5 / 6) Pro8.. (a) Quadrant II (b) cos( ) 7 /, tan( ) 7 / Pro8.. (a) sin (b) cos (c) sin 56 (d) tan Pro9. (a.). - tan() () sin(b - A) = sin(a B) (c.) sec () (d.) cos Asin A cos A sin A (e.) sin(a) (f.) ( cos( A)) / (g.) Pro9. (a.) ( ) / () sin(60 ) / Pro9. cos( ) ( sin( )) cos( ) cos( )sin( ) sin( ) tan( ) sec( ) sec( ) tan( ) ( sin( )) ( sin( )) cos ( ) cos( ) cos( ) Pro0. a.graph y 5cos ma displacement is 5 cm and c. Revised February 06 8

9 Pro0. y Amplitude: Period: Phase shift: Pro. 7 a., 6 6 5, c., d. 5,, 6 6 e. 7,. Solve right and oblique triangles. Pro. (d.) is true, i.e. sin() /5 Pro. h 5/ cos(0 ).65 Pro. a. The domain is [,] and the range is [0, ] cos ( / ) /. Revised February 06 9

10 Pro. a. The domain is (, ) and the range is ( /, / ) tan ( ) /. Pro. a. The domain is [,] and the range is [ /, / ] sin (sin(8 / )) /. Pro. a. cos (sin(5 /) 5 / 6. tan(cos ( / )). Pro.5 tan () n.905 n with n =,,,.. Pro. The amount after t days is until only % is left. A( t) 00 e ( t ln() /00) 00e t and it will take about 60 days Pro. ( nt * ) r a. A P. A is the amount after t years, P is the Principal (original amount invested), r is the n interest rate written as a decimal, n is the number of compounding periods per year, and t is the number of years..075 Solve Pro. a. The population model is (5* T ) for t. Then ln(0 / ) t ~6 yrs. (5ln(.075/ 5)) P( t) 6, 5,058,790 ln() The doubling time predicted for this model is td 60. years t e. P(0) 7,7,7,50 Pro.. The plane is 000 feet 5,06.7feet or approimately 8.5 miles from the airport. tan(6 ) Plane Horizontal 6 (angle of depression),000 ft 6 Ground? Airport Revised February 06 0

MATH 175: Final Exam Review for Pre-calculus

MATH 175: Final Exam Review for Pre-calculus MATH 75: Final Eam Review for Pre-calculus In order to prepare for the final eam, you need too be able to work problems involving the following topics:. Can you graph rational functions by hand after algebraically