MATH 175: Final Exam Review for Pre-calculus

Transcription

1 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 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, and horizontal asymptotes. 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 inverse f ( ). Sketch the graph f( ) and labeling each. (use ZDecimal for the aes.) f 3 3 f ( ) to have an ( ) on the same set of aes, clearly Pro. Why doesn t the function f ( ) 3( 7) have an inverse function? Give a reason (which could be in the form of a labeled sketch with comments). 3. Can you define and graph the basic eponential and logarithmic functions and their transformations by hand? Pro3. Sketch a labeled graph of the function f( ) e if ln( ) if. Can you solve eponential and logarithmic equations algebraically and/or graphically using the laws of logarithms where appropriate? Pro. Solve the following equations for. Show your steps and give eact answers. a log (3 ) 5. 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? Pro5.. 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

2 Pro5.. If sin( and cos : (a.) What Quadrant is the terminal side of in? () Give the eact values of cos( and of tan( cos( tan 6. 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? Pro6. 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 )= Pro6.. Use the appropriate sum or difference formula to find the eact value of: (Show your steps.) 5 a. cos = sin( )cos(8 ) sin(8 )cos( ) Pro6.3 Prove the identity, show all work. tan( ) sec( ) sec( ) tan( ) 7. Can you graph basic trigonometric functions and transformations of the sine and cosine function by hand? Pro7. 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.

3 8. Can you solve trigonometric equations both algebraically and graphically? Pro8. Find all real numbers that satisfy: sin () 3sin() 9. Can you solve right and oblique triangles? Pro9. Which of the following statements about the angle is true? a. sin( ) 3/ cos( ) 5 / 3 c. tan( ) 3/5 d. sin( ) /5 e. sin( ) /3 5 Pro9. 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 0. Can you evaluate inverse sine, inverse cosine, and inverse tangent function values eactly when possible and, if not possible, approimate using a graphing utility? Pro0. a. What is the domain and range for the function f() cos ()? Give eact values in radians of cos ( /). Pro0. a. What is the domain and range for the function f() tan ()? Give eact values in radians of tan ( 3). Pro0.3 a. What is the domain and range for the function f() sin ()? Give eact values in radians of sin (sin(8 / 3)). Pro0. a. Give eact values in radians of cos (sin(5 /3). Give the eact value of tan(cos ( / 3)). Pro0.5 Find all real numbers that satisfy the equation : tan() 3

4 . 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 33% of the calclinium you started with remained. Pro. a. Define all 5 variables in the formula: A P r n ( nt). 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.3 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.. Can you convert rectangular coordinates to polar coordinates and vice versa Can you convert rectangular equations to polar equations and vice versa?.. Convert the polar coordinates ( 3, 3 / ) to rectangular coordinates... Convert the rectangular coordinates (-6., -3) to polar coordinates with [0, )..3. Write the equations of the following polar curves in rectangular coordinates. a. r = 3 = 3.5 c. r sin.. Write the equations of the following rectangular coordinate curves in polar coordinates. a. y 6 ( 3) y 9 c. y

5 Answers to the Final Eam Review questions for Math 75. Pro.. f( ) ( 5) ( ) ( 6 5)( 5) ( 5) ( ) ( )( 5)( 5) Hole at (-5, 3/), vertical asymptote: = -, horizontal asymptote: y= 3 6 Pro.. Use your graphing calculator to graph 3 3 f ( ) and y, then use DRAWINVERSE from the DRAW menu to see the graph of y f ( ) f ( ) 0 0 y f ( ) Pro. y f ( ) 3( 7) is not because 7 ( y ) / 3.

6 Pro3. f( ) e if ln( ) if y e if (, e ) y ln( ) if Pro. (a.) 3ln(5) /(ln(7/ 5) () (c.) no solution Pro5.. (a.) sin(5 / ) / () csc( 5 / 6) Pro5.. (a.) Quadrant II () cos( ) 7 /, tan( ) 3 7 / 7 Pro6. (a.). - tan( ) () sin(b - A) = sin(a B) (c.) sec ( ) (d.) cos A sin A cos A sin A (e.) sin(a) (f.) ( cos( A )) / (g.) Pro6. (a.) ( 3 ) / () sin(60 ) 3 / Pro6.3 cos( ) ( sin( )) cos( ) cos( )sin( ) sin( ) sec( ) tan( ) ( sin( )) ( sin( )) cos ( ) cos( ) cos( ) tan( ) sec( )

7 Pro7. a.graph y 5cos ma displacement is 5 cm c..59. and Pro8. sin ( ) 3sin( ) (sin( ) )(sin( ) ) 0. 7 So k or k where k 0,,,3, Solve right and oblique triangles. Pro9. (d.) is true, i.e. sin( ) /5 Pro9. h 5/ cos(0 ) Pro0. a. The domain is [,] and the range is [0, ] cos ( / ) /. Pro0. a. The domain is (, ) and the range is ( /, / ) tan ( 3) / 3. Pro0.3 a. The domain is [,] and the range is [ /, / ] sin (sin(8 / 3)) / 3. Pro0. a. cos (sin(5 / 3) 5 / 6. tan(cos ( / 3)). Pro0.5 tan (3) n.905 n with n =,, 3,.. Pro. The amount after t days is A( t) 00 e ( t ln() /00) t 00e and it will take about 60 days until only 33% is left.

8 ( nt * ) r Pro. a. A P n amount invested), r is the interest rate written as a decimal, n is the number of compounding periods per year, and t is the number of years. Solve A is the amount after t years, P is the Principal (original (5* T ) Pro.3 a. The population model is P (0) 7, 37, 7,50 for t. Then The doubling time predicted for this model is t ln(0 / 3) (5ln(.075/ 5)) 0.05t P( t) 6, 5,058,790 e. ln() t D 60.3 years ~6 yrs. Pro.. The plane is the airport. 000 feet 5,06.7feet or approimately 8.5 miles from tan(6 ) Plane Horizontal 6 (angle of depression),000 ft Ground? Airport 6.. ( 6 /, 6 / ).. ( 7., tan (3/ 6.) ).3. a. y 9 (tan 3.5) y c. ( y ).. a. r r 6cos c.