Maximum and Minimum Values
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- Milo Peters
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1 Sec 4.1 Maimum ad Miimum Values A. Absolute Maimum or Miimum / Etreme Values A fuctio Similarly, f has a Absolute Maimum at c if c f f has a Absolute Miimum at c if c f f for every poit i the domai. f for every poit i the domai. B. Local/Relative Maimum or Miimum Values A fuctio A fuctio f has a Local Maimum at c if c f f has a Local Miimum at c if c f f for every poit that is ear c. f for every poit that is ear c. C. The Etreme Value Theorem If f is cotiuous o a closed iterval b a, b a, the f attais both a maimum ad a miimum value o D. Fermat s Theorem If f has a local maimum or miimum at c, ad f c eists, the c 0 f. E. Critical Number A critical umber of a fuctio If f, is a umber c i the domai such that f c 0 or f c f has a local maimum or miimum at c, the c is critical umber of f DNE E. Closed Iterval Method To fid Absolute Maimum or Miimum of a cotiuous fuctio 1. Fid the values of 2. Fid the values of f at the critical umbers of f at the edpoits a ad b of the iterval. 3. The largest of the values of step 1 ad 2 is the Absolute Maimum 4. The smallest of the values of step 1 ad 2 is the Absolute Miimum f o a closed iterval b f i a,b. a, :
2 Eamples: 1.) Fid the critical umbers for the followig fuctios a. b.
3 c. d.
4 2.) Cosider the fuctio. The absolute maimum value of (o the give iterval) is ad this occurs at equals ad the absolute miimum of (o the give iterval) is ad this occurs at equals ) Cosider the fuctio f o the iterval 9 4. Fid the critical umbers ad absolute miimum ad maimum values.
5 4.) Cosider the fuctio f 2cos miimum of the fuctio. o the iterval 0. Fid the absolute maimum ad ) Choose the best reaso that the fuctio f 13 miimum. (a) The fuctio f() is always positive. (b) The derivative f () is always egative. (c) The derivative f () is always positive. (d) The highest power of i f() is odd. has either a local maimum or a local
6 Sec 4.2 The Mea Value Theorem A. Rolle s Theorem Let f be a fuctio such that f is cotiuous o a, b, f is differetiable o a, b ad f a f b The there is a umber c i b c a, such that f 0 B. The Mea Value Theorem Let f be a fuctio such that f is cotiuous o a, bad f is differetiable o b The there is a umber c i b a, such that f c Or equivaletly f b f a f c b a f b f b a a a,. C. Costat Theorem If f 0 for all i a iterval a, b the f costat o a,b. D. Corollary If f g for all i a iterval b a, the g f is costat o a, b (i.e. f g c ) Eamples 1.) Cosider the fuctio o the iterval. Verify that this fuctio satisfies the three hypotheses of Rolle's Theorem o the iterval. is o ; is o ; ad The by Rolle's theorem, there eists a such that. Fid the value. 2.) Cosider the fuctio o the iterval. Fid the average or mea slope of the fuctio o this iterval, i.e.
7 By the Mea Value Theorem, we kow there eists a i the ope iterval such that is equal to this mea slope. For this problem, there is oly oe that works. 3.) By applyig Rolle's Theorem, check whether it is possible that the fuctio has two real roots. Possible or impossible? Your reaso is that if f has two real roots the by Rolle's theorem: f must be at certai value of betwee these two roots, but f is always egative,positive, or zero 4.) Suppose f is cotiuous o [2,8] ad 2 8 estimate f 8 f 2. f for all i (2,8). Use the Mea Value Theorem to f 8 f 2
8 Sec 4.3 Derivatives ad the Shape of Graphs A. The First Derivative Icreasig/Decreasig Test If f 0 o a iterval, the If f 0 o a iterval, the A critical umber of a fuctio If f is icreasig o that iterval f is decreasig o that iterval f, is a umber c i the domai such that f c 0 or f c f has a local maimum or miimum at c, the c is critical umber of f DNE The First Derivative Test f If f chages from positive to egative at c, the If f chages from egative to positive at c, the If f does ot chage sig at c, the Suppose c is a critical umber of a cotiuous fuctio f has a local maimum at c f has a local miimum at c f has o local maimum or miimum at c B. The Secod Derivative Cocavity If f 0 o a iterval, the If f 0 o a iterval, the f is cocave up o that iterval f is cocave dow o that iterval Cocave Up Cocave Dow Icreasig Slope Decreasig Slope
9 A iflectio poit of a fuctio f, is a poit at which the curvature (secod derivative) chages sig. The curve chages from beig cocave upwards (positive curvature) to cocave dowwards (egative curvature), or vice versa. The Secod Derivative Test Suppose that is cotiuous at c f If f 0 ad f 0 the f If f 0 ad f 0 the f Eample: 3 2 f ) has a local miimum at c. has a local maimum at c. a.) Fid the critical poits ad the itervals o icrease ad decrease. b.) State whether each critical poit is a maimum or a miimum. c.) Fid the iflectio poits ad the itervals o cocavity. d.) Sketch the graph ad verify your results.
10 2.) g 2cos o 0 2 a.) Fid the critical poits ad the itervals o icrease ad decrease. b.) State whether each critical poit is a maimum or a miimum. c.) Fid the iflectio poits ad the itervals o cocavity. d.) Sketch the graph ad verify your results.
11 3.) h e e 8 a.) Fid the critical poits ad the itervals o icrease ad decrease. b.) State whether each critical poit is a maimum or a miimum. c.) Fid the iflectio poits ad the itervals o cocavity. d.) Sketch the graph ad verify your results..
12 4.) Suppose that f is cotiuous o a.) If f 5 0 ad 5 6,. f, the f has a local at 5. b.) If f 19 0 ad 19 6 f, the f has a local at ) Give the graph of f, determie whether the followig coditios are true. 6.) Give the graph of f, determie whether the followig coditios are true.
13 3 2 7.) Fid a cubic fuctio f a c d value of 6 at 0. that has a local maimum value of 8 at 2 ad a local miimum
14 Sec 4.4 Curve Sketchig A. Guidelies for sketchig a curve 1. Domai 2. Itercepts (-itercepts ad y-itercepts) 3. Symmetry (Odd, eve or periodic fuctios) 4. Asymptotes 5. Itervals of Icrease ad Decrease 6. Maimum ad Miimum Values 7. Itervals of Cocavity Eample: Sketch the curve usig the guidelies ) f Domai 2. Itercepts (-itercepts ad y-itercepts) 3. Asymptotes 4. Itervals of Icrease ad Decrease 5. Maimum ad Miimum Values 6. Itervals of Cocavity 7. Iflectio Poits
15 o ) f 3cos cos 1. Domai 2. Itercepts (-itercepts ad y-itercepts) 3. Asymptotes 4. Itervals of Icrease ad Decrease 5. Maimum ad Miimum Values 6. Itervals of Cocavity 7. Iflectio Poits
16 B. Guidelies for sketchig a fuctio give a sketch of it s derivative. 1. Fid all itervals where the fuctio is icreasig ad decreasig 2. Fid all itervals where the fuctio is cocave up ad cocave dow 3. Sketch a fuctio that has these characteristics (there are may graphs possible)
17 Sketch the graph of a fuctio, f(), that satisfies all of the give coditios. 3. f (0) = f (2) = f (4) = 0 4. f(0) = 0, f (-2) = f (1) = f (9) = 0 f () > 0 if < 0 or 2 < < 4 lim f( ) 0 lim f( ) f () < 0 if 0 < < 2 or > 4 f () < 0 o (-,-2),(1,6) ad (9, ) f () > 0 if 1 < < 3 f () > 0 o (-2,1) ad (6,9) f () < 0 if < 1 or > 3 f () > 0 o (-,0) ad (12, ) f () < 0 o (0,6) ad (6,12) 6
18 Sec 4.5 Optimizatio Eamples 1.) Farmer Brow has 1200 ft of fece to create a rectagular pe that will be adjacet to a river. If he does ot eed to put ay fece o the side that borders the river, what dimesios will maimize the area of the pe, ad what is the maimum area? (Do ot forget uits!) 2.) Fid two umbers ad (with ) whose differece is 42 ad whose product is miimized.
19 3.) A bo is to be made out of a 10 by 18 piece of cardboard. Squares of equal size will be cut out of each corer, ad the the eds ad sides will be folded up to form a bo with a ope top. Fid the legth, width, ad height of the resultig bo that maimizes the volume. (Assume that ).
20 4.) A cylidrical oatmeal cotaier has a capacity of 3 liters. Fid the dimesios that will miimize the cost of productio material to costruct the cotaier.
21 5.) Fid the area of the largest rectagle that ca be iscribed i a semicircle with a radius 4 6.) Fid the dimesios of the rectagle of largest area that has its base o the -ais ad its other two vertices 2 above the -ais ad lyig o the parabola y 4.
22 7.) Fid the poit o the lie 4 7 y which is closest to the poit, ) If 2000 square cetimeters of material is available to make a bo with a square base ad a ope top, fid the largest possible volume of the bo.
23 9.) A piece of wire 12 m log is cut ito two pieces. Oe piece is bet ito the shape of a circle of radius ad the other is bet ito a square of side. How should the wire be cut so that the total area eclosed is: a.) Maimized b.) Miimized
24 10.) A Norma widow has the shape of a semicircle atop a rectagle so that the diameter of the semicircle is equal to the width of the rectagle. What is the area of the largest possible Norma widow with a perimeter of 45 feet?
25 11.) A ruig track has the shape of a rectagle with a semicircle o each ed. If the legth of the track is 400 meters, fid the dimesios so that a.) the rectagular (shaded)regio is maimized. b.) The etire regio is maimized.
26 Sec 4.6 Newto s Method A. The Newto s Method Formula 1 f f Eamples: 1.) Startig with 0 2 fid the third approimatio 3 3 to the root of the equatio f f 1 f f ) Startig with 0 1 fid the third approimatio 3 to the root of the equatio ta 1 1 f f 1 f f
27 To fid these approimatios usig the calculator: Let Y1 = f() ad let Y2 = f () The i the HOME SCREEN type i 0 ad press ENTER Type i immediately after you hit ENTER: Y1(As) / Y2(As) ad press ENTER (Each time you press eter you will get the et approimatio of the root.) 3. a.) Fid the equatio f that results i a solutio of 4 9 b.) Fid the secod, third ad fourth approimatios of the root to this fuctio if ) Fid the fourth approimatio 2 to the root of the equatio e 2 f f 1 f f 0 1 2
28 Sec 4.7 Ati Derivatives A fuctio Fis called the ati-derivative of f if F f Basic rules of ati differetiatio I geeral: Reverse basic rules of differetiatio. Importat: *Always use proper otatio! *Do t forget +C Eamples: Fid the ati derivative for each of the followig: ) f g ) 5 4
29 3.) h k 8 4.) 2 5.) m 10si 6cos. 8 k 1 6.) 2 7.) k 2 1 2
30 7 3 8.) Fid the fuctio Fgive that f 2 4 ad F ) Fid the fuctio f give that f , f 0 5 ad f 1 2
31 10.) A particle is movig with acceleratio t 12t 2 time t 0 is v 0 9. What is its positio at time 6 a. Its positio at time 0 t? t is s0 11 ad its velocity at
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