AP Calculus Summer Packet

Transcription

1 AP Calculus Summer Packet Istructor: Mr. Adrew Nichols questios to Help Sessio: Moday, July, :0 5:0 Due Date: Tuesday, August Prerequisites Quiz: Thursday, August Istructios: Work o this packet over the summer. This packet is due o the first day of class ad will be graded for completio ad accuracy. This is your first graded assigmet of the fall semester i AP Calculus. You must SHOW YOUR WORK to receive credit. You must complete this packet ad pass the Prerequisites Quiz to keep your positio i AP Calculus. If you fail to complete this packet or fail the Prerequisite Quiz, I will recommed that you drop AP Calculus. The aswers to all of these problems are attached. CHECK YOUR ANSWERS. Strategies for Success: ) Start workig o this packet NOW while you have the support of your PreCalculus teacher. ) Review the etire packet first ad idetify problems that you do t recogize or have struggled with i the past. ) Idetify a study group that you ca work with over the summer either i perso or olie. 4) Ask a more eperieced studet for help. 5) Use iteret resources like Kha Academy, CoolMath, ad YouTube. 6) Mr. Nichols if you get stuck ad eed help. 7) Do t procrastiate. This packet will take loger tha you epect to complete ad puttig it off oly makes it more stressful. Submissio: Write the aswers to problems 7 i this packet. Work all other problems o your ow paper. Orgaize your work ad aswers so that they ca be easily graded. Whe you had i your work, remove problems 7 from the packet ad staple them to your work ad aswers for the remaiig problems. We are what we repeatedly do. Ecellece, therefore, is ot a act, it is a habit. Who are you? What do you wat to be? Look at the habits of your heart, your mid, ad your behavior. You! Me! We are what we repeatedly do. Where our attetio goes, we go! - Ukow Success is a lousy teacher. It seduces smart people ito thikig they ca t lose. Bill Gates

2 SKILLS NEEDED FOR CALCULUS A workig kowledge of these topics is importat for success i AP Calculus. Items marked with ad asterisk (*) are vital. The practice problems at the ed of this packet cover these topics. I. Algebra A. *Epoets (operatios with iteger, fractioal, ad egative epoets) B. *Factorig (GCF, triomials, differece of squares ad cubes, sum of cubes, groupig) C. Ratioalizig (umerator ad deomiator) D. *Simplifyig ratioal epressios E. *Solvig algebraic equatios ad iequalities (liear, quadratic, higher order usig sythetic divisio, ratioal, radical, ad absolute value equatios) F. Simultaeous equatios II. Graphig ad Fuctios A. *Lies (itercepts, slopes, write equatios usig poit-slope ad slope itercept, parallel, perpedicular, distace ad midpoit formulas) B. Coic Sectios (circle, parabola, ellipse, ad hyperbola) C. *Fuctios (defiitio, otatio, domai, rage, iverse, compositio) D. *Basic shapes ad trasformatios of the followig fuctios (absolute value, ratioal, root, higher order curves, log, l, epoetial, trigoometric. piece-wise, iverse fuctios) E. Tests for symmetry: odd, eve III. Geometry A. Pythagorea Theorem B. Area Formulas (Circle, polygos, surface area of solids) C. Volume formulas D. Similar Triagles IV. Logarithmic ad Epoetial Fuctios A. *Simplify Epressios (Use laws of logarithms ad epoets) B. *Solve epoetial ad logarithmic equatios (iclude l as well as log) C. *Sketch graphs D. *Iverses * V. Trigoometry A. *Uit Circle (defiitio of fuctios, agles i radias ad degrees) B. Use of Pythagorea Idetities ad formulas to simplify epressios C. ad prove idetities D. *Solve trigoometric equatios E. *Iverse Trigoometric fuctios F. Right triagle trigoometry G. *Graphs VI. Limits A. Cocept of a limit B. Fid limits as approaches a umber ad as approaches

3 Toolkit of Fuctios I Calculus it is epected that studets kow the basic shape of may fuctios ad be able to graph their trasformatios without the assistace of a calculator. () For each fuctio below, accurately sketch its graph ad give its domai ad rage. Costat: y c Liear: y Quadratic: y Cubic: y Square Root: y Cube Root: y Reciprocal: y y y y y Absolute Value: y

4 Greatest Iteger:y = Semicircle: y Sie: y si Cosie: y cos Taget: y ta Epoetial: y a, a Epoetial: y a, 0a Natural Epoetial y e Logarithmic y log b, b Logarithmic y log b, 0b Natural Log y l

5 Polyomial Fuctios A fuctio P is called a polyomial if P( ) a a... a a a - 0 Where is a oegative iteger ad the umbers a0, a, a are costats. Eve degree Leadig coefficiet sig Odd degree Leadig coefficiet sig Positive Negative Positive Negative Number of roots equals the degree of the polyomial. Number of itercepts is less tha or equal to the degree. Number of beds is less tha or equal to (degree ). Roots with eve multiplicity bouce off of the -ais. Roots with odd multiplicity cross the -ais. Ratioal Fuctios Locatig Asymptotes () Use the followig ratioal fuctios to ivestigate asymptotes. Before you begi, (a) Factor the umerator ad deomiator of each fuctio, (b) Determie the domai for each, ad (c) Graph the fuctio o your calculator i the stadard viewig widow ad sketch the graph o the coordiate ais. f( ) 9 g ( ) 6 D: D: h ( ) k ( ) 4 D:

6 **I the otes below meas approaches. So is read approaches. The statemet y meas that y grows without boud or y goes to positive ifiity. Vertical Asymptotes Each domai restrictio is the locatio of either (a) a poit discotiuity (i.e. hole i the graph) or (b) a vertical asymptote (a vertical lie that, as the fuctio approaches the lie, y ). () Classify each domai restrictio as either a vertical asymptote or poit discotiuity by eamiig the graph of the fuctio: f( ) Poit discotiuities at Vertical asymptotes at h ( ) Poit discotiuities at Vertical asymptotes at g ( ) Poit discotiuities at Vertical asymptotes at k ( ) Poit discotiuities at Vertical asymptotes at (4) How could you tell the differece betwee a poit discotiuity ad a vertical asymptote with out lookig at the graph ad just eamiig the equatio of the fuctio? Write your aswer i the space provided below. Horizotal Asymptotes The ed behavior of a fuctio y f determies its horizotal asymptotes (a horizotal lie that the fuctio approaches as ). (a) If a fuctio has ifiite ed behavior, the it has o horizotal asymptotes, but (b) If y c as, the y c is a horizotal asymptote for y f ( ). (5) Determie the ed behavior of each fuctio ad write the equatios of ay horizotal asymptotes. I the fuctio f : As, y As, y Horizotal asymptotes: y I the fuctio h : As, y As, y Horizotal asymptotes: y I the fuctio g : As, y As, y Horizotal asymptotes: y I the fuctio k : As, y As, y Horizotal asymptotes: y

7 Summary of Horizotal Asymptotes for Ratioal Fuctios p ( ) Let f( ) be a ratioal fuctio, i.e., p ( ) ad ( ) q ( ) Let degree of p( ) ad m degree of q( ). Let a leadig coefficiet of p( ) ad b leadig coefficiet of q( ). q are polyomials. (a) If m, the f( ) has the horizotal asymptote y 0. a (b) If m, the f( ) has the horizotal asymptote y. b (c) If m, the f( ) has o horizotal asymptote. 8 (6) Make a detailed sketch of the graph of f( ) without the use of your graphig calculator. Attach your sketch to back of this packet. Trigoometric Formulas Arc Legth of a circle: L = r or L = Area of a sector of a circle: Area = Solvig parts of a triagle: d r 60 r or Area = d 60 r Law of Sies: a b c si A si B si C Law of Cosies: a b c bc cos A b a c ac cos B c a b ab cos C Area of a Triagle: Area = bc sia or Area = ac sib or Area = ab sic Hero's formula: Area = s( s a)( s b)( s c), where s a b c

8 Trigoometric Idetities You are epected to memorize the Reciprocal, Quotiet, ad Pythagorea Idetities. Reciprocal Idetities: csc A = si A sec A cosa cot A = taa Quotiet Idetities: Pythagorea Idetities: si A ta A cot A = cos A cosa sia si A + cos A = ta A + = sec A + cot A = csc A You do ot eed to memomize ay of the followig idetities, but you do eed to kow how to use them. Sum ad Differece Idetities: si(a + B) = sia cosb+cosa sib si(a B) = sia cosb - cosa sib cos(a + B) = cosa cosb sia sib cos(a B) = cosa cosb + sia sib ta (A + B) = ta A ta B ta A ta B ta (A B) = ta A ta B ta A ta B Double Agle Idetities: si(a) = sia cosa ta(a) = ta A ta A cos(a) = cos A - si A cos(a) = cos A cos(a) = si A Half Agle Idetities: A cos A si A cos A cos A cos A ta cos A Geometric Area Formulas: You are epected to memorize these formulas. Area of a trapezoid: A hb b Area of a triagle: A bh Area if a circle: A r Circumferece of a circle: C r or C d

9 Uit Circle You are epected to memorize the First Quadrat of the Uit Circle ad be able to eted that kowledge to the other quadrats as eeded. The sitee poits marked o the Uit Circle below correspod to quadratal agles, special agles, ad their multiples. (7) Label each poit with (a) the correspodig positive agle measure betwee 0 ad 60, (b) positive radia measure betwee 0 ad π, ad (c) its (,y)-coordiates. (8) For a agle θ i stadard positio whose termial side itersects the Uit Circle at (,y), si θ = cos θ = ta θ = csc θ = sec θ = cot θ =

10 THE REMAINING PROBLEMS IN THIS PACKET SHOULD BE WORKED ON YOUR OWN PAPER AND ATTACHED TO THE PACKET WHEN HANDED IN. (9) Recall the properties of epoets: (i) b = b (ii) If >, b bb b b, times. (iii) (iv) 0 b a a m m a m m a a (v), where b 0 m b b m m a a (vi), where b 0 m b b Simplify ad write without usig egative epoets: (a) y (b) 5y 4 y (0) Epress usig a sigle radical ad positive iteger powers: (vii) If b 0, b b (viii) a m a m m, where a 0 a a (i) If b0 ad, b () If b y b. ad m ad are itegers m m with >, b b b 8 yz (c) 4 yz m () Epress usig positive ratioal epoets ad without radicals: 5 y z 0 4 () Epad each epressio ad combie like terms: (a) a b (b) a b () Factor completely (a) 9 y y Hit: use groupig (b) (c) (c) y Hit: factor as a differece of squares first, the as the sum ad differece of cubes (d) 5 4 Hit: first factor out the GCF: (e) Hit: first factor out the GCF:.. (4) Ratioalize the deomiator: (5) Ratioalize the umerator: (6) Simplify: 4 (7) Use Ratioal Root Theorem ad sythetic divisio to help factor the followig polyomial fuctios completely. State all of the factors ad roots. p 4 6 (b) q (a)

11 (8) Use the Ratioal Root Theorem to eplai why caot be a root of 5 where c ad d are itegers. (9) Eplai why 4 use the sigs of f 4 c d 5, g 7 5 must have a root i the iterval0,. Eamie the graph ad g 0 ad g to justify your aswer. (0) Solve the followig equatios ad iequalities algebraically. Check your solutios by graphig. (a) 4 4 (d) (f) 0 6 (b) (c) (e) 9 0 (g) 4 () Solve each system of equatios algebraically ad the check your aswer by graphig. y 0 4 y (a) (b) y y () Write the equatio for the lie described by the give iformatio. (a) Passes through the poit, ad has slope. (b) Passes through the poit 4, ad is perpedicular to y 4. (c) Passes through, ad is parallel to y. 5 () A small-appliace maufacturer fid that it costs $9,000 to produce,000 toaster oves per week ad $,000 to produce,500 toasters per week. (a) Epress the cost as a liear fuctio of the umber of toaster oves produced ad sketch its graph. (b) What is the slope of the graph ad what does it mea i the cotet of this problem? (c) What is the y-itercept of the graph ad what does it mea i the cotet of this problem? (4) Fid the domai ad rage of each fuctio. Recall that domai restrictios may result from ay of the followig (i) Deomiator must be 0 (ii) Argumet (iput) for a log must be > 0 (iii) Radicad (iput) for a eve root must be > 0 For rage restrictios, use algebraic reasoig ad, if all else fails, your graphig calculator. (a) y ylog (b) (c) y 4 (d) y (e) y 5 (f) y

12 (5) Sketch the graph of the piecewise fuctio f give below o the iterval, rage of f o this domai? if 0 f if 0 if (6) Use the fuctios f, g, ad h defied below to compute each iverse or compositio. f g 4 h l w (a) (b) g h (c) w (d) f g( ) for 4. What is the 4 (e) hg f () (7) A stoe is dropped ito a lake, creatig a circular ripple that travels outward at a speed of 60 cm/secod. (a) Epress the radius r of this circle as a fuctio of time t (i secods). (b) If A is the area of the circle as a fuctio of the radius, fid A r ad eplai its meaig. (8) Does y 9 have a iverse fuctio? Justify your aswer. (9) Let f, g, ad h 4 f g (a) (b) f g h the fid (0) Let s 4 ad t the fid the domai ad rage of () Epress the fuctio F F f g h. s t. as the compositio of three fuctios f, g, & h so that () Use your Toolkit of Fuctios ad your kowledge of trasformatios to sketch the graphs of the followig fuctios without usig your calculator (ecept to check your aswer). (a) y (e) ysi (b) y 5 (g) f 0, 5 5 (f) y l (c) y (d) y () Idetify each fuctio as odd, eve, or either ad justify your aswer algebraically. Use these algebraic tests: f f for all. (i) f is ad eve fuctio if ad oly (ii) f is ad odd fuctio if ad oly f f for all (a) f 4 (b) f 6 f (c) f si (d) f (e) f

13 (4) Fill i each blak with either eve or odd. (a) The product or quotiet of eve fuctios is. (b) The product or quotiet of odd fuctios is. (c) The product or quotiet of a eve ad a odd fuctio is. (5) The graph of a equatio is symmetric with respect to the (i) y-ais if the equatio is uchaged whe replacig with. (ii) -ais if the equatio is uchaged whe replacig y with y. (iii) origi if the equatio is uchaged whe replacig with AND y with y. Test each fuctio for symmetry: (a) 4 y (b) y si (c) y cos (d) y y (e) (6) Eve fuctios are symmetric with respect to the. Odd fuctios are symmetric with respect to the. (7) Recall the properties of logarithms: (i) logb 0 (ii) log b (iii) log b b (iv) log m log m log b b b m (v) log log m log p (vi) log m plog m b b b b (vii) If log m log, the m b b b Simplify each logarithmic epressio below usig the properties of logarithms. (a) log (b) log log8 log log9 7 (c) log (d) log w w (e) l e (f) l (g) l e (8) Solve each equatio for : log log 4 (b) (a) 6 6 log log00 log (c) 5 (9) Uder ideal coditios a certai bacteria populatio is kow to double i size every three hours. Suppose that there are iitially 00 bacteria. (a) What is the size of the bacteria populatio after 5 hours? (b) What is the size of the bacteria populatio after t hours? (c) How log will it take the bacteria populatio to grow to 50,000?

14 (40) The populatio of a certai species of tree frogs i a limited eviromet ad iitial populatio of 00, ca be predicted usig Pt e t (a) Use your calculator to graph this fuctio i a appropriate viewig widow. Sketch the graph. (b) Based o the graph, what seems to be the maimum umber of frogs this eviromet ca accommodate? (c) Estimate how log it takes for the populatio to reach 900 frogs. (d) Fid the iverse of this fuctio ad eplai its meaig. (e) Use the iverse fuctio to fid the time required for the populatio to reach 900. Compare with the result of part (c). (4) Simplify each epressio usig trigoometric idetities: ta csc (a) (b) cos (c) csc ta si sec ta (4) Verify the idetity si ta. (4) Solve each equatio for 0 : (a) cos cos (b) si (c) cos si 0 (44) Evaluate each epressio. Note that si arcsi (a) si (b) arcsi (c) arcta (45) Sketch the graph of each fuctio ad state its domai ad rage: (d) si cos (a) Arcsie: y si, (b) Arccosie: y cos (b) Arctaget: y ta (46) Solve for i each diagram. (a) (b) (47) The rollercoaster car show i diagram below takes.5 secods to go up the degree iclie segmet AH ad oly.8 secods to go dow the curved drop from H to C. The car covers horizotal distaces of 80 feet o the iclie ad 60 feet o the drop. A H 80 ft B 60 ft C (a) What is the maimum height of the rollercoaster above poit B? (b) Fid the distaces AH ad HC. (c) How fast (i ft/sec) does the car go up the iclie? (d) What is the average speed of the car as it goes dow the drop? (e) Is HC a over- or uder- approimatio of the true distace traveled by the car from H to C?

15 (48) Idetify the amplitude, period, horizotal shift, ad vertical shift of each fuctio compared to its paret. Sketch the graph of each fuctio o the iterval,. (a) y si (b) ycos (49) Familiarize yourself with the followig CALC ( d TRACE) commads o your TI-84 graphig calculator: (i) Root (iii) Maimum (ii) Miimum (iv) Itersect Give 4 f 0 : (a) Fid a appropriate calculator viewig widow that shows all of the importat features of the fuctio. Report Xmi, Xma, Ymi, ad Yma. Sketch the resultig graph o your paper. (b) Fid the -coordiates all of the roots of the fuctio accurate to three decimal places. (c) Fid the (,y)-coordiates of all local maimums accurate to three decimal places. (d) Fid the (,y)-coordiates of all local miimums accurate to three decimal places. 0 f 0.5 accurate to three decimal places. (e) Fid the value of f, f, f, ad (50) Graph the followig fuctios ad fid their poits of itersectio usig the Itersect commad o your calculator (do ot use TRACE). Sketch the graphs o your paper ad state your aswer accurate to three decimal places. (a) y 5 7 y 0. 0 (b) si y e y (c) y l cos y

16 AP Calculus Summer Packet Aswers You ca ot copy these aswers ad receive credit for the summer packet. You must submit complete solutios, ot just aswers, to receive credit.. All graphs must be eatly sketched with correct domai ad rage listed. Check your aswers o your graphig calculator.. All graphs must be eatly sketched with correct with correct factorizatios ad domais give. f has o poit discotiuities ad has vertical asymptotes at.. g has a poit discotiuities at ad has a vertical asymptote at. h has poit discotiuities at ad has o vertical asymptotes. k has a poit discotiuities at 0 ad has a vertical asymptote at. 4. Hit: compare the factors of the umerator ad deomiator. f, as, y 0, so y 0 is a horizotal asymptote. 5. I I I g, as, y, so y is a horizotal asymptote. h ad k, as, y ad as, y, so either has ay horizotal asymptotes. 6. Neatly sketch the graph. Be sure to label your aes ad idicate the positios of all -itercepts, vertical, ad horizotal asymptotes. Check your graph o your calculator. 7. Label the Uit Circle as istructed. You must be able to reproduce at least the first quadrat from memory. 8. si y, cos, ta y, csc y, sec, cot y. You should memorize these formulas. 9. a) 8 y b) 5 y 4 c) 4 y z y 5 y z. a) a ab b b) a ab b c) y y 4. a) y 9 y b) c) d) e)

17 a) p b) q ca ot be a ratioal root of f sice is ot a factor of Sice g 0 0, g 0, ad g is a cotiuous fuctio, 0 0. a) 5 or b) 5 c), 0, or 5 d) or 0 e) 0 or f) g) g for some i ; i iterval otatio:, 5, ; i iterval otatio:, 0, ; i iterval otatio:,0, ; i iterval otatio: 5 8, 8. a), or,4 b), or,0. a) y or y b) y 4 or y 7 c) y 5 or y 5 5 0,.. a) y6 000 ; sketch the graph eatly beig sure to label ad scale the aes. b) The slope, $6 per toaster ove, is the cost of each additioal toaster produced. c) The y-itercept, $000, is the base cost of operatig the factory eve if o toasters are produced. 4. Domai Rage,,,0 0, a) b),, c),, d), 0, e), 0, f),,, 0, 5. Neatly sketch the graph of each piece of the fuctio o its restricted domai. Be sure to scale your aes. Domai:,

18 6. a) g 4 b) h e c) w 4 d) e) for 4 f g 6 h g f l a) r t 60 b) t A r t 600 t which is the area of the circle as a fuctio of time. 8. No, its graph does ot pass the Horizotal Lie Test. OR No, because, y both whe ad whe (for eample). f g b) f g h 8 9. a) 0. D :,, R : 0,. F f g h where f, g, ad h. Sketch each graph eatly. Be sure to scale your aes carefully. Check your aswers o your graphig calculator.. a) odd?) odd d) either f) eve b) eve c) odd e) odd 4. a) eve b) eve c) odd 5. Each fuctio is oly symmetric to the: a) y-ais b) origi c) y-ais d) -ais e) y-ais 6. y-ais; origi 7. a) b)?) c) d) 45 e) f) 0 g) 8. a) or 6 b) 0 l5 c).465 l 9. a) 00 bacteria Pt 00 t b) l 500 c) t hours l 40. a) Sketch a copy of the graph o your paper. Be sure to label ad scale the aes carefully. b) Approimately 000 frogs c) A little less tha 4.5 years. 00, 000 d) tp l P or simplified a bit: 900P t P l 00, P t years which is a little less tha 4.5 years as predicted i (c). e) 4. a) b) si c) 4. Hit: use the Pythagorea idetities to show that the left-had side simplifies to. 4. a) b) 6,,7 6,or 4 c) 44. a) b) 4 c) 4 d)

19 45. Check your graphs o your graphig calculator. Be sure to label your aes. Domai Rage y si,, y y cos, 0, ta,, 46. a) Note: is the legth of the leg opposite the 50 agle b) Note: the path of the roller coaster is alog the lie segmet ad the alog a curved path from H to C. This was iadvertetly omitted from the diagram. a) feet b) feet c) 8. feet per secod d) feet per secod e) Uder approimatio 48. (a) (b) Amplitude Period 4 Horz. Shift 0 Vert. Shift a) Aswers will vary, but should use a widow that shows all of the major features of the fuctio: two local miimums, oe local maimum, four -itercepts, ad both positive ad egative ed behaviors. Sketch ad label your graph carefully o your paper. b) -itercepts at.5, 0,, ad 5 c).067,0.0 d) 0.890, 8.48 ad.948, e) f 8, f 0, f 0 0, f Iclude eat, labeled sketches of each system with your aswers. 5.77, , ,0.60 a),, ad b).88,.67 c).4,0.050 ad 0.5,.98