Final Exam Review Advanced Functions

Transcription

1 Final Eam Review Advanced Functions. Determine, if possible, the equation of the polynomial function, given that the following set of points lie on the graph a) (, -), (,), (,5), (,8), (5,), (6,) b) (,-), (,0), (,0), (,98), (5,6), (6,96) c) (,-9), (,-0), (,-), (,0), (5,), (6,6). Completely factor the following: a) b) c) 8 5 d) a) A particular radioactive isotope is known to decay at a rate proportional to the amount present. If its half-life is 500 years, how long will it take for a g sample to decay to the point where its mass is.9g? b) Given that the half-life of radium is 690 years, how much will remain of g of radium after years? c) If radioactive carbon has a half-life of 550 years, what will remain of g after 000 years? d) If 00 bacteria in a colony reproduce eponentially and in hours there are 00 bacteria, how many bacteria are in the colony after hours?. a) An earthquake of magnitude. on the Richter scale struck northern Peru on May, 90. On December 6, 90, an earthquake of magnitude 8.6 on the Richter scale struck Gansu, China. Compare the intensities of the two earthquakes. b) While going home yesterday, Jeff was waiting for the subway. Just as the train entered the station, a child started screaming right beside Jeff. Compare the intensities of the two sounds, if the noise level from the subway was 90 db and the noise from the screaming child was 00 db. 5. If f ( ) and g ( ), find: a) fg() b) gf() c) fg() d) gf() 6. Evaluate the following logarithms. a) log b) log 8 c) log. Write as a single logarithm: log log y log 8. Graph y, y log & y log ( ). State the domain, range and any asymptotes for each. 9. Solve a) log log log b) log ( ) log c) log ( ) log ( ) d) 0 95

2 0. Consider the graph of the function f ( ) 5 8. a) Find the slope of the secant that joins the points =- and =. b) Find the slope of the tangent line at the point when = (solve algebraically).. Find the slope of the tangent to the curve of y 5 when =9 by choosing two appropriate points on either side of the equation.. For what value of k is a factor of f ( ) k?. Solve for : a) b) 0. Prove the given identities: cos csc a) tan csc sec b) tan c) sec y tan y tan y cot cot y sin cos csc d) e) cos f) cos sin cot sin tan cot cot cos g) csc sin 5. Write the following as a single epression using addition, subtraction and double angle formulas: a) sin cos( ) cos sin( ) b) sin cos 6. Solve the following equations for 0. a) cos 0 b) 5( sin ) sin c) cos sin 0. Review the graphs of the tangent, cotangent, secant and cosecant functions. 8. State the transformations, domain & range and sketch one cycle of the graph for y cos( ) Given cos where, find: 5 a) sin b) cos 0. Find the eact value of the following: a) sin b) tan 5 c) sec 5 6. Graph the following functions. Ensure to label your graph, aes and intercepts clearly. a) y ( )(5 )( ) b) y ( ) ( )( ) c) y. a) Find the family of cubic functions whose -intercepts are, - and 5. b) Find the particular member of the above family whose graph passes through the point (, 0).

3 . Solve the following inequalities and graph your solutions on a number line: a) ( ) b) 6 0. Define an even function and an odd function. Describe their properties. 5. Eplain whether f ( ) is an even function, an odd function or neither. 6. Graph f() = and f ( ) ( ) on the same aes. State the domain and range of the transformed function.. Graph the following functions, f(), and their reciprocals,. State the domain f ( ) and range of each. a) f() = 5 b) f() = 8. Find the following instantaneous rates of change algebraically (t in seconds, s(t) in meters). a) s ( t).9t 5000 when t = b) s( t) t t when t = 9. An empty 0 liter water tank is placed underneath a closed tap. At t = 0 (in minutes) the tap is slowly opened, until, at t = it is all the way open. At t = there is liters of water in the tank, and water is flowing into the tank at a constant rate of L / minute. The tap is left fully open until t = 0, when the tap is gradually closed so that at t =, the water tank is full and the tap is closed. The tank is left alone for minutes, until Andy, in a fit of rage, kicks a hole in the tank. Water starts flowing out of the tank slowly, but the hole keeps getting bigger and bigger. By t =, the tank is empty. a) Draw a graph to illustrate this situation. b) Find the average rate of change in the first minutes. c) Find the average rate of change over the entire minutes. Eplain. d) Find the average rate of change over between t=5 and t= minutes. e) Mark where the instantaneous rate of change is L/min with an X on the graph. 0. An orange is spherical. Suppose it grows so that it s volume increases at an average rate of cm /day. a) Epress the radius of the orange as a function of time t in days. b) Determine the radius of the orange si weeks after it begins growing.

4 Answers. a) y b) y 6 c) y. a) ( )( )( ) b) ( )( 8)( ) c) ( )( )( ) d) ( )( )( ). a) years b) g c) g d) 900. a) China was.9 times more intense b) 0 times more intense 0 5. a) b) c) d) ( ) 6. a) b) ¼ c) 0 y. log 8. y ER y 0, yer y 0 y log 0, ER yer 0 y log ( ), ER yer 9. a) b) c) d). 0. a) b). ¼.. a) =, -/, 5 b) = -,. LS=RS 5. a) sin( + ) b) cos a) =,. b) =.8, 5.55 c) = 5,, Amplitude:, Period:, Phase Angle Shift: right, Vertical Shift: 0, ER y, 9. a) /5 b) /5 yer

5 0. a) b) c). Check your solutions on your graphing calculator.. a) f ( ) k( )( )( 5) 5 b) f ( ) ( )( )( 5). a) <5/ b). Even: f ( ) f ( ) Odd: f ( ) f ( ) etc. 5. Neither 6. ER y, yer. a) For f(): b) For f(): ER yer ER and for y 6.5, yer : f ( ) and for 5, y 0, : f ( ) ER yer,, ER y 0, yer 8. a) -9. m/s b) 5 m/s 9. a) b).5 L/min c) 0 L/min d).l/min e) minute or minutes 0. a) t r b). cm