Honors Calculus Midterm Review Packet

Transcription

1 Name Date Period Honors Calculus Midterm Review Packet TOPICS THAT WILL APPEAR ON THE EXAM Capter Capter Capter (Sections. to.6) STRUCTURE OF THE EXAM Part No Calculators Miture o multiple-coice, matcing, sort response Part Scientiic Calculators Open ended questions (6 pts) (5 pts) Te maimum possible score is points. Tere is no ormula seet. HOW TO PREPARE FOR THE MIDTERM EXAM Answer as many problems as possible rom tis review packet and ceck your answers. Only use a calculator i it is absolutely necessary. I you do not own a graping calculator, use desmos.com to practice. Review old tests and quizzes rom te st and nd marking periods. Ask questions on our review day. Stay ater scool or additional review. Get enoug rest te nigt beore. Wake up early and eat a ealty breakast. RULES AND EXPECTATIONS ON EXAM DAY Bring at least two sarpened pencils wit BIG erasers. You may only use a calculator tat as been RESET BY ME. Bring tings tat will elp you study or your oter eams in case you inis early. I you inis early, you must remain quiet until all eams ave been collected! NO ONE MAY LEAVE te room unless it is an absolute emergency. You are required to remain in te eam room te entire time. No one gets to leave early. Do not ask or passes to go places. You will ave a ten minute break between your eams to use te restroom. Midterm Eam Scedule Date Session Session Tuesday, January 9t Wednesday, January t Tursday, January st Friday, February st /5 5/6 6/7 8/9 7/8 9/

2 PRACTICE PROBLEMS: CHAPTER Scientiic Calculator Practice Question ) Use te Intermediate Value Teorem to determine i ( ) [, ] as zeros in te interval. Do not attempt to locate te zeros. Indicate i te teorem gives no inormation. No Calculator Practice Questions Use te table o values to determine te it. sin )..... ()? ) () ? ) State te interval(s) on wic te unction ( ) 5) Is te unction ( ) continuous at is continuous.? Wy or wy not? 6) Use te deinition o continuity to sow tat ( ) 9 6 < > is continuous at. 7) Find te constant a tat makes ( ), continuous on te entire real number line. a 5, > 8) Given: ( ) Find: 9) I ( ) c c g ( ) ( ) g c ( ) ( ) 8 ( ) ( ), ind. c ( ) 5 Study Tip: Break te study guide problems up into sections, and do a section eac day. I you ave any questions, come to etra elp and get tem answered!

3 ) I te unction as a vertical asymptote, state its equation. I te unction as a removable discontinuity, state te coordinates o te ole. ( ) ) Use te given grap to determine te -values 5 at wic te unction ( ) [ ] is not continuous. Use te given graps to ind eac it. ) [ sec ] ( ) sec ) ( ) sin ( ) sin ( ) Find eac it. ) ( ) sin 5) 6 6 6) 7 7) cos 5 8) sin 9) ) ( 5 ) ) ( ), were ( ),, <. ) Draw te grap o ( ) and ind ( )

4 Use te grap to answer eac questions # -. ) ( ) ) ( ) 5) ( ) 6) ( ) 7) () 8) ( ) 9) () ) ( ) ) Wat type o discontinuity eists at? ) Wat condition o te deinition o continuity is not satisied at? PRACTICE PROBLEMS: CHAPTER No Calculator Practice Questions. ) Use te it deinition o a derivative to sow tat te derivative o ( ) is ( ) 6 ) Determine te cot ( ) cot. (Just say te answer. Do not attempt to work it out.) Find te derivative o eac unction. ) g( ) sec 7 ( ) ) ( ) 5 5 s t 5) ( ) 6) ( ) sin t t y 7) Find te second derivative: ( ) 5 5 8) Given tat ( ) 8, ind ( 8). 9) Find te ourt derivative, ( ) ( ), o te unction ( ) sin. ) Given: ( ) ( ) and ( ), ( ), g( ) 5, and g ( ) g( ) Find: ( )

5 ) Find te slope o te line tangent to te grap o y at te point (, ). ) Find te equation o te normal line to te curve y 9 at. ) Consider te curve given by y y 6. dy d y y y a) Use implicit dierentiation to sow tat. b) Find all points on te curve wose -coordinate is. c) Write an equation or te tangent line at eac o tose points. sin sin ) Find te coordinates o all points on te grap o ( ) tangent lines in te interval [, ]. 5) Te graps o,, and are sown. Wic grap is wic? tat ave orizontal 6) Te grap o a piecewise-deined unction is sown below. Te grap as a vertical tangent line at and orizontal tangent lines at and. Wat are all values o, < <, at wic is continuous but not dierentiable? Study Tip: Studying or -5 minutes at a time (wit minute breaks in between) is te most eective way to retain inormation.

6 7) A particle moves along a line so tat at time t, were t, its position is given by te unction t s t cost. Wat is te velocity o te particle wen its acceleration is zero? ( ) 8) Te unction is deined on te closed interval [, 8]. Te grap o its derivative is sown. Te point (, 5) is on te grap o y ( ). Find te equation o te line tangent to te grap, 5. o y ( ) at te point ( ) Scientiic Calculator Practice Questions 9) A potato is launced upward rom te top o a 6-oot building wit an initial velocity o 8 eet per s t t t. second. Te position unction is given by ( ) 6 8 6? Include units o measure. b) At wat moment in time will te potato reac its maimum eigt? Include units o measure. c) At wat moment in time will te potato it te ground? Include units o measure. d) Wat is te velocity o te potato upon impact wit te ground? Include units o measure. a) Wat is te average rate o cange on te interval [, ] ) Two runners, A and B, run on a straigt racetrack or t seconds. Te grap, wic consists o two line segments, sows te velocity, in meters per second, o Runner A. Te velocity, t v t t. in meters per second, o Runner B is given by te unction ( ) a) Find te velocity o Runner A and te velocity o Runner B at time t seconds. Indicate units o measure. (, ) (, ) b) Find te acceleration o Runner A and te acceleration o Runner B at time t seconds. Indicate units o measure. Velocity o Runner A (meters/sec) time (seconds)

7 ) A 5 oot ladder is leaning against te side o a ouse. Te base o te ladder is pulled away at a rate o eet per second. How ast is te top o te ladder moving down te side o te ouse wen te base o te ladder is 5 eet rom te ouse? Include units o measure. ) A ot air balloon rising straigt up rom a level ield is traced by a range inder eet rom te lit o point. At te moment te range inder s angle is, te angle is increasing at a rate o.5 radians / min. How ast is te balloon rising at tat point? Include units o measure. ) Salt is poured rom a conveyer belt at a rate o t / min, orming a conical pile wit a circular base. Te eigt and diameter o base are always equal. How ast is te eigt o te pile increasing te moment wen te pile is eet ig? Include units o measure. Formula: V r No Calculator Practice Questions PRACTICE PROBLEMS: CHAPTER ) Use te given grap to determine te absolute ) Te grap o is given along wit its and relative etrema. inlection points. State te open intervals were te unction is concave up or concave down. ) Wy can t Rolle s Teorem be applied to te unction ( ) sec on te closed interval [ ] (Select all tat apply.) ( ) is not continuous on [ ], ( ) is not dierentiable on ( ) ( ) ( ).,,.? ( ) sec

8 ) State te domain, determine were it is continuous, identiy any absolute maimum values, identiy any absolute minimum values, identiy any relative maimum values, and identiy any relative minimum values. Domain: Continuous on: Absolute maimum value: Absolute minimum value: Relative maimum value: Relative minimum value: 5) Suppose is continuous on [, 5] and dierentiable on (, 5) and tat ( ) 6 ( 5 ) 8 (, 5) suc tat ( c)., and. According to te Mean Value Teorem (MVT), tere is some value c in te open interval 6) Find te critical numbers or te unction ( ) ( 8 ) 7) State te open intervals on wic ( ) ( ) 8) Find te relative etrema o te unction ( ) is increasing or decreasing. 6. ( ) 9) Given: ( ) sin on te closed interval [, ] a) State te open intervals were te unction is concave up or concave down. b) State te coordinates o te points o inlection.. Find eac it. ) 7 5 ) 5 ) Determine te equations o any asymptotes. ) ( ) ) ( ) 5) ( ) 6

9 6) A unction is continuous or all real numbers. Te graps o and are given below. ( ) y ( ) y (, ) a) Over wat interval(s) is increasing? b) Over wat interval(s) is decreasing? c) At wat value(s) o does ave a relative minimum? d) At wat value(s) o does ave a relative maimum? e) Were is concave up? ) Were is concave down? g) List any inlection points. Scientiic Calculator Practice Questions 7) Let ( ) on te closed interval [, ]. Find all numbers c tat satisy te MVT. 8) Find te absolute etrema o te unction ( ) on te closed interval [, ] 9) Given: ( ) a) State te open intervals were te unction is concave up or concave down. b) State te coordinates o te points o inlection. c) Use te Second Derivative Test to identiy any relative etrema. ) Given: ( ) a) State te domain. b) Find te intercepts. c) Determine te symmetry. d) State te equations o any asymptotes. Describe te end beavior. e) Determine were te unction is increasing or decreasing. ) Determine te relative etrema. g) Determine te concavity and points o inlection. ) Take all o tis wonderul inormation and draw me a gorgeous unction..

10 CHAPTER ANSWERS ) Since ( ) and ( ). 6, by te Intermediate Value Teorem, tere eists at least one, suc tat ( c). number c in te interval [ ] ) ) ) Continuous on te interval [, 9) ( 9, ). 5) No, because ( ) 6) Step : ( ) 6 ( )( ) Step : ( ) ( ) 9 ( ) 6 ( ) ( ) 6 Since te let and rigt-and its bot equal 6, ( ) 6. Step : Since ( ) ( ) 7) a 6, te unction is continuous at. is undeined. 8) ( ) ( ) g( ) g c 8 ( ) ( ) c c c ( ) ( 5) 5 9) ( ) ( ) ) Vertical asymptotes: and ) n were n is any integer ecept, Hole: (, ) ) ) ) 5) 8 6) 7 7) 8) 9) ) ) ) ( ) ) ) 5) DNE 6) 7) 8) 9) Undeined ) ) Removable ) ( ) ( )

11 CHAPTER ANSWERS ) ( ) ( ) ( ) ( ) ( ) 6 6 [ 6 ] 6 ( ) 6 csc ) g ( ) sec 7 ( ) tan( ) ) ) ( ) 5) ( ) 6) s ( t) 5 t cos t sin t t y 8 8) 9) 7) ( ) ( ) ( ) 6 sin ) 5 ) 5 ) y a) dy y y b) (, ) (, ) d y ) ( ) (, ), and c) At (, ) : y At (, ) : y and 5) : line, : parabola, : cubic unction 6) and 7) 9a) 8 t/sec 9b). 5 seconds ater being launced 9c) 5 seconds ater being launced 9d) t/sec 8) y a) Runner A: 6.67 m/s b) Runner A:. m/s ). 5 t/sec Runner B: 6.86 m/s Runner B:.7 m/s ) 8 t/min ). 8 t/min 6 5

12 CHAPTER ANSWERS ) ( ) : Relative Min ) Concave Up: (, ) and (, ) ( ) : Absolute Ma & Relative Ma Concave Down: (, ) and (, ) ( ) : Relative Min ( ) : Relative Ma ) ( ) is not continuous on [ ]. ( 5) : Absolute Min ( ) is not dierentiable on ( ). ) Domain: [, ] 5) ( c) Continuous on: [, ] ecept at Absolute Ma: ( ) 5 and ( ) 5 Absolute Min: None 6) and Relative Ma: ( ) 7) Increasing: (, ) and (, ) Relative Min: None Decreasing: (, ) and (, ) 8) Relative Min: ( ), No Relative Ma at because ( ), is undeined at 9a) Concave Up: (, ) and (, ) Concave Down: (, ) and (, ) 9b) Inlection Points: (, ), (, ), and (, ) ) 7 ) ) ) VA: HA: ) VA: None 5) VA: y HA: y and y OA: y 6a) (, ) and (, ) 7) c 6b) (, ) 8) Absolute Ma: ( ) 8 Absolute Min: ( ) 6c) Relative Min at 6d) Relative Ma at 6e) (, ) 9a) Concave Up: (, ) Concave Down: (, ) 6) (, ) and (, ) 9b) Inlection Point: (, ) 6g) (, () ) 9c) Relative Ma: ( ) Relative Min: ( ) 7 a) (, ) ) Relative Ma: ( ) Relative Min: ( ) b) -int:, ± y-int: g) Concave Up: (, ) Concave Down: (, ) c) symmetric wit origin Inlection Point: (, ) d) As, ( ) ) As, ( ) e) INC: (, ) and (, ) DEC: (, ) and (, )