Chapter 3 Review: Applications of Derivatives · Curve Sketching 10.Optimization HW Sets Topics...
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1 Name: ____________________ Period: ______ Date: ___________ AP Calc BC Mr. Mellina Chapter 3 Review: Applications of Derivatives Topics: 1. Extreme Value Theorem 2. Rolle’s Theorem 3. Mean Value Theorem 4. Intervals on Which a Function is Increasing or Decreasing 5. 1 st Derivative Test for Relative Extrema & Linearization 6. Motion Along a Line 7. Points of Inflection 8. 2 nd Derivative Test for Relative Extrema 9. Curve Sketching 10. Optimization HW Sets Topics 1-3: Chapter 3 Review Set A Topics 4-6: Chapter 3 Review Set B Topics 7-9: Chapter 3 Review Set C Topic 10: Chapter 3 Review Set D
Transcript of Chapter 3 Review: Applications of Derivatives · Curve Sketching 10.Optimization HW Sets Topics...
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Name: ____________________ Period: ______ Date: ___________ AP Calc BC
Mr. Mellina
Chapter 3 Review: Applications of Derivatives
Topics:
1. Extreme Value Theorem 2. Rolle’s Theorem 3. Mean Value Theorem 4. Intervals on Which a Function is Increasing or Decreasing 5. 1st Derivative Test for Relative Extrema & Linearization 6. Motion Along a Line 7. Points of Inflection 8. 2nd Derivative Test for Relative Extrema 9. Curve Sketching 10. Optimization
HWSets
Topics1-3:Chapter3ReviewSetATopics4-6:Chapter3ReviewSetBTopics7-9:Chapter3ReviewSetCTopic10:Chapter3ReviewSetD
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Topic1:ExtremeValueTheorem(Day1)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises1-7,findtheabsoluteextremaofthefunctionontheclosedinterval.1. 𝑓 𝑥 = 𝑥$ + 5𝑥,[−4, 0] 3. 𝑓 𝑥 = 𝑥 − 2,[0, 4]
5. 𝑓 𝑥 = 4𝑥𝑥2+9
,[−4, 4] 7. 𝑓 𝑥 = 2𝑥 + 5 cos 𝑥 ,[0, 2𝜋]
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Topic2:Rolle’sTheorem(Day1)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises9&11,DeterminewhetherRolle’stheoremcanbeappliedtofontheclosedinterval[a,b].IfRolle’sTheoremcanbeapplied,findallvaluesofcintheopeninterval(a,b)suchthat𝑓4 𝑐 = 0.IfRolle’sTheoremcannotbeapplied,explainwhynot.9. 𝑓 𝑥 = 𝑥6 − 3𝑥 − 6,[−1, 2]
11. 𝑓 𝑥 = 𝑥21−𝑥2
,[−2, 2]
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Topic3:MeanValueTheorem(Day1)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises13-19,DeterminewhethertheMeanValueTheoremcanbeappliedtofontheclosedinterval[a,b].IftheMeanValueTheoremcanbeapplied,findallvaluesofcintheopeninterval(a,b)suchthat𝑓4 𝑐 = : ; <: =
;<=.IftheMeanValueTheoremcannotbe
applied,explainwhynot.
13. 𝑓 𝑥 = 𝑥$ 6,[1, 8] 15. 𝑓 𝑥 = 1𝑥,[1, 4]
17. 𝑓 𝑥 = 𝑥 − cos 𝑥 , − ?$, ?$ 19. 𝑓 𝑥 = 1
𝑥2,[−2, 1]
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Topic4:IntervalsonWhichaFunctionisIncreasingorDecreasing(Day2)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises21-25:Findtheopenintervalsonwhichthefunctionisincreasingordecreasing.21. 𝑓 𝑥 = 𝑥$ + 3𝑥 − 12 23. 𝑓 𝑥 = 𝑥 − 1 $ 2𝑥 − 5 25. ℎ 𝑥 = 𝑥 𝑥 − 3 , 𝑥 > 0
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Topic5:Applyingthe1stDerivativeTest&Linearization(Day2)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises27-33:Findthecriticalnumbersoff,ifany.Thenfindtheopenintervalsonwhichthefunctionisincreasingordecreasing.Finally,applythe1stDerivativeTesttoidentifyallrelativeextrema.27. 𝑓 𝑥 = 𝑥$ − 6𝑥 + 5
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29. 𝑓 𝑡 = CD𝑡D − 8𝑡
31. 𝑓 𝑥 = 𝑥+4𝑥2
33. 𝑓 𝑥 = cos 𝑥 − sin 𝑥 , (0, 2𝜋)
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TheseproblemswereselectedfromtheExercisesonpage240.Inexercises5-9,findthetangentlineapproximationLtothegraphoffatthegivenx=2.UseLtoapproximatethegivenvalues.Thenidentifyifthisapproximationisanunderoroverapproximation.5. 𝑓 𝑥 = 𝑥$,𝑓(1.9) ≈7. 𝑓 𝑥 = 𝑥K,𝑓(2.01) ≈9. 𝑓 𝑥 = sin 𝑥,𝑓(2.1) ≈
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Topic6:MotionAlongaLine(Day2)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.Inexercises35&36,thefunction𝑠(𝑡)describesthemotionofaparticlealongaline.Findthevelocityfunctionoftheparticleforanytime𝑡 ≥ 0.Identifythetimeinterval(s)onwhichtheparticleismovinginapositivedirection.Identifythetimeinterval(s)onwhichtheparticleismovinginanegativedirection.Identifythetime(s)atwhichtheparticlechangesdirection.35. 𝑠 𝑡 = 3𝑡 − 2𝑡$36. 𝑠 𝑡 = 6𝑡6 − 8𝑡 + 3
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Topic7:ConcavityandPointsofInflection(Day3)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises37-41,findthepointsofinflectionanddiscusstheconcavityofthegraphofthefunction.37. 𝑓 𝑥 = 𝑥6 − 9𝑥$ 39. 𝑔 𝑥 = 𝑥 𝑥 + 541. 𝑓 𝑥 = 𝑥 + cos 𝑥 , [0, 2𝜋]
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Topic8:2ndDerivativeTestforExtrema(Day3)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.
Inexercises43-47,findallrelativeextremaofthefunction.UsetheSecondDerivativeTestwhereapplicable.43. 𝑓 𝑥 = 𝑥 + 9 $ 45. 𝑔 𝑥 = 2𝑥$(1 − 𝑥$)
47. 𝑓 𝑥 = 2𝑥 + 18𝑥
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Topic9:CurveSketching(Day3)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.Inexercises49&50,sketchthegraphofafunctionfhavingthegivencharacteristics.49.
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50.
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Topic10:Optimization(Day4)TheseproblemswereselectedfromtheReviewExercisesonpages242-246.79. Findtwopositivenumberssuchthatthesumoftwicethefirstnumberandthreetimes
thesecondnumberis216andtheproductisamaximum.80. Findthepointonthegraphof𝑓 𝑥 = 𝑥thatisclosesttothepoint(6,0).
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81. Arancherhas400feetoffencingwithwhichtoenclosetwoadjacentrectangularcorrals(seefigure).Whatdimensionsshouldbeusedsothattheenclosedareawillbeamaximum?
