Rancho Bernardo High School/Math Department Honors Pre-Calculus Exit Exam

Transcription

1 Rancho Bernardo High School/Math Department Honors Pre-Calculus Eit Eam You are about to take an eam that will test our knowledge of the RBHS Honors Pre-Calculus curriculum. You must demonstrate genuine understanding of ke concepts and master of problem solving skills in order to be prepared for AP Calculus AB/BC. Please read all directions carefull. Errors made due to not following directions will result in loss of points. Unless otherwise stated in the directions, our answers should be stated as eact values. This means no decimal approimations. You are permitted to use a Scientific Calculator onl with no graphing capabilit. This test should be completed in approimatel three hours. You will score our own test and tall up our points. Based on our score ou can determine our preparedness level for AP Calculus AB/BC. There are a total of 00 points possible on the test. Score 0-00 not recommended to proceed into AP Calculus AB/BC 0-00 recommended to proceed into AP Calculus AB/BC

2 RBHS Honors Pre-Calculus Eit Practice Eam Instructions: Show all relevant work. Your work should be neat, organized and eas to follow. Simplif all answers to lowest terms. Give eact answers. Onl a Scientific Calculator is permitted!!. State the domain of each of the following: ( pts. each) 5 a.) f( ) b.) g( ) log9 c.) h( ) ln 6 d.) i( ) Sin e.) j ( ) f.) g ( ) 6. Determine which of the following functions are odd, even, or neither. ( pts. each) a.) f( ) 5 6 b.) g( ) 9 c.) h ( ) 5 d.) i( ) Tan e.) ( ) sin j f.) g ( ) 6. Given: 5 P( ) 5 and Q ( ) 7 a.) List all possible rational roots of P ( ) 0 ( pts) b.) Write P ( ) in completel factored form: (6 pts) c.) Find values of for which P( ) Q( ) (6 pts) d.) Find values of for which P( ) Q( ) ( pts)

3 . Write the polnomial function for the given graph. Find the eact equation containing the point (, 7). Leave in factored form. (5 pts) 5. Find the value of the constant k so that P( ) k 8 has i as one of its roots. (5 pts) #6-7: Provide a reasonable sketch of each function. State the domain, range, and zeros of the function (give eact values). f 6 6. f 7. ( pts) ( pts) Domain ( pts) Domain ( pts) Range ( pts) Range ( pts) Zeros ( pts) Zeros ( pts)

4 8. Water is leaking out of a conical tank at a rate of ft /min. If the height is feet and the base radius is feet, a.) Find the total volume of the tank. ( pts) b.) Epress the volume of the tank as a function of its height, h. ( pts) c.) How long will it take to completel empt the tank? ( pts) #9 : Solve the following equations over the Real numbers. Give eact answers. (5 pts. each) 9. e e log log

5 5. The following graph shows one full period of f numbers., a periodic function. The function is defined for all real a.) State the period of f. ( pt) b.) State the amplitude of f. ( pts) c.) Find f 00. ( pts) d.) Find f 8. ( pts) #6 9: Sketch the graph of each transformation of the function shown above to the left. ( pts. each) 6. f 7. f 8. f 9. f 0. Write the function g( ) in piecewise form. (6 pts.) 5

6 #: Solve each for 0.. a.) b.) sincos sin cot csc 0 (6 pts. each). For what values of, for 0, is cos sin? (6 pts.) # 6: Given, where sin and cot, find the EXACT value of each: (5 pts each). sin. tan 5. cos 6. sin #7 8: Solve for all in radians. Give eact answers. (5 pts each) csc 7. 6tan

7 #9 0: Simplif. Give eact answers. ( pts each) sin cos 0. sin Tan 5. Find the angle of elevation for the line 7 in degrees to the nearest tenth. ( pts). State the domain and range of each of the following: ( pt each) sin Domain: a.) Range: b.) tan Domain: Range: c.) csc Domain: Range: d.) Cos Domain: Range:. Given CAT, mt = 0, c = 6 cm, and t = cm. Solve the triangle. Find angles and side lengths to the nearest tenth. (7 pts) 7

8 . Three ships are assigned to rescue an astronaut as his spaceship plunges into the ocean. The ships are at the vertices of a triangle with side lengths 5, 7, and 0 kilometers. Find the measure of the largest angle of this triangle to the nearest tenth of a degree. (5 pts) 5. For each of the following, graph over the indicated domain. List and label all pertinent information. (6 pts each) a.) cot for. b.) cos for. P lies in the first quadrant of the parabola a.) Epress the area of the shaded triangular region as a function of the coordinate of P. ( pts) 6. A point, 6 as shown in the diagram. no data Function Plot 6 8 P(, ) b.) What is the domain of the function? ( pts) = 8

9 7. Given the series: (6) a.) Write the series in summation notation using k 0 as a lower limit. ( pts) b.) Write the series in summation notation using k as a lower limit. ( pts) 8. Given the infinite geometric series: ( ) ( ) ( ) f( ) a.) Find the interval of convergence of the series. ( pts) b.) Solve for all values of if the sum of the series converges to. ( pts) c.) Find f () if it eists. ( pts) d.) Find f ( ) if it eists. ( pts) 9. A rectangular flat piece of cardboard is going to be used to make a rectangular open top bo b cutting out squares of equal size, each of which has side of length, from each corner. If the dimensions of the cardboard are inches b 8 inches, set up the equation for the volume of the bo in terms of length that is cut out. State the domain for volume of the bo. (5 pts) 9

10 0. Find f ( ) for each of the following. State the domain of a.) f 5 b.) f f ( ). ( pts each) c.) f d.) f ln. Answer the following given the function 5 f ( ). ( pts each) a.) If f ( ), find the value of. b.) If f( ) 8, find the value of.. Sketch the graph of f ( ) given the graph of f( ). ( pts each) a.) b.) 0

11 f Find each of the following from the chart shown above given that both the function f and its inverse eist for all. ( pt each) a.) f b.) Find if 5 f. c.) Find if f d.) f f 6 e.) 5 f f f.) f ( ) 00 Velocit of a Rocket (ft/sec) Time (sec). A model rocket, launched from the ground level, burns fuel for a few seconds, accelerating the rocket upward. After the rocket engine burns out, the rocket coasts upward for a while and then begins to fall. A parachute pops out shortl after the rocket starts down in order to slow the rocket. Use the graph above to answer the questions a-f below. ( pts each) a.) At what time(s), in seconds, is the rocket s velocit 50 ft/sec? b.) At what time, in seconds, did the rocket s engine burn out? c.) At what time, in seconds, did the rocket reach its highest point? d.) At what time, in seconds, did the rocket s parachute pop out? e.) What does the area under the curve represent in contet to the rocket s flight? Recall d = rt. f.) Approimate the maimum height of the rocket?

12 Answers:. a.) b.) (0,9) c.) 0 d.) All Real Numbers e.) f.) (,). a.) odd b.) even c.) neither e.) even d.) odd f.) odd. a.) 0,, b.) ( )( ) c.), d.) (,). ( ) ( ) 5. k = ; roots = i, -i, 6. Domain: (, ) 6. Range: (, ) Zeros:, 0 7. Domain: 6,6 7. Range:, Zeros: 8 a.) V 6 cubic feet 8 b.) V h c.) 8 minutes 9. ln 0..,... 0,

13 5. a.) 9 c.) - b.) 5 d.) 6. reflects across -ais with double the period 7. shifts down then reflects across -ais 8. reflect across the -ais then the -ais (smmetr about the origin) 9. everthing to the left of the -ais is ignored and everthing to the right of the -ais remains the same and then that is reflected across the -ais (even) 0., 0 g( ), 0,. a.) b.) 5 7 0,,,,,. 5 5, k, k 8. 6 k, k 5 6 k, k o a.) Domain: All Real Numbers b.) Domain: k, k 6 Range: All Real Numbers Range: 7 c.) Domain:, k k d.) Domain: Range:, Range: 0

14 . ma9. o, mc 0.9 o, a o 5. a.) b.) 6. a.) (6 ) A b.) 0 7. a.) ( ) (7n ) n n b.) k 0 ( n 7) 6 k ( ) (7n 9) ( n ) n n 8. a.) 5 b.) 7, c.) d.) does not eist 8 9.) v ( )(8 ) domain 0 0. a.) ( ) 5 b.) c.) d.) e ( ),. a.) b.). a.) b.). a.) b.) c.) -8 d.) 5 e.) f.). a.), 6. sec b.) sec c.) 8 sec d.) sec e.) distance traveled b the rocket f.) about 650 feet