# 4 examples find a lim x e x 1 sin 2 x b lim x e x x 2

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4.Examples.Find(a)limx0ex-1sin 2x(b)limx→∞exx2+x(c)limx→∞lnxx
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.4L’HOSPITALSRULE1325.Indeterminate Form0· ∞.Findlimx→∞xlnx-1x+ 1.6.Indeterminate Form∞ - ∞.Findlimx01x-1sinx.7.Indeterminate Form00,0,1.8.Examples.Find(a)limx0(cosx)1/x2
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.4L’HOSPITALSRULE1339. Additional Notes
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.5SUMMARY OFCURVESKETCHING1344.5Summary of Curve Sketching(This lecture corresponds to Section 4.5 of Stewart’s Calculus.)1.Puzzle.Connect the nine dots with four (only four) straight lines without ever lifting your pen orpencil from the paper.2.Guidlines(a)Domain(b)Intercepts:For thex-intercepts sety= 0and solve forx. For they-intercept calculatef(0).(c)Symmetry:i. Even function - symmetric about they-axis.ii. Odd function - symmetric about the origin.iii. Periodic functions.(d)Asymptotes:i. Horizontal Asymptotes:y=Liflimx→∞f(x) =Lor iflimx→-∞f(x) =L.ii. Vertical Asymptotes:x=aif at least one of the following is truelimxa+f(x) =limxa-f(x) =limxa+f(x) =-∞limxa-f(x) =-∞.iii. Slant Asymptotes:y=mx+biflimx→∞(f(x)-(mx+b)) = 0.(e)Intervals of Increase and Decrease:f0(x)>0on an intervalImeansfincreasing%onI.f0(x)<0on an intervalImeansfdecreasing&onI.(f)Local Maximum and Minimum Values:- First Derivative Test- Second Derivative Test(g)Concavity and Points of Inflection:f00(x)>0on an intervalImeansfconcave up(onI.f00(x)<0on an intervalImeansfconcave down)onI.(h)Sketch the curve.
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.5SUMMARY OFCURVESKETCHING1353.Examples.Sketch graphs of the following functions.(a)f(x) =2 +x-x2(x-1)2
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.5SUMMARY OFCURVESKETCHING136(b)f(x) =x2ex
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.5SUMMARY OFCURVESKETCHING1374. In this example we will focus just on asymptotes in the guidelines outlined in (2).Determine the asymptotes of the functionf(x) =x2+x-1x-1.
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.5SUMMARY OFCURVESKETCHING1385. Additional Notes
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.6OPTIMIZATIONPROBLEMS1394.6Optimization Problems(This lecture corresponds to Section 4.7 of Stewart’s Calculus.)(This lecture corresponds to Section 4.7 of Stewart’sCalculus.)1.Quote.“There is no branch of mathematics, however abstract, which may not someday be applied tothe phenomena of the real world.”(Nikolai Lobachevski, Russian mathematician, 1792 - 1856)2.Examples.(a) Find the dimensions of the right circular cylinder with greatest volume that can be inscribed ina right circular cone of radius8cm and height 12 cm.
PART4: APPLICATIONS OF THEDERIVATIVELECTURE4.6OPTIMIZATIONPROBLEMS140(b) A painting in an art gallery has height3m and is hung so that its lower edge is about1m abovethe eye of an observer. How far from the wall should the observer stand to get the best view?
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