Study guide for exam 3
Study guide for exam 3 Math 1610-090
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This 3 page Study Guide was uploaded by Nikki Bee on Wednesday June 10, 2015. The Study Guide belongs to Math 1610-090 at Purdue University taught by in Spring 2015. Since its upload, it has received 109 views. For similar materials see Calculus 1 in Math at Purdue University.
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Date Created: 06/10/15
Spring 2015 MA 16100 Study Guide Exam 3 Related Rates Word Problems Method Read problem carefully several times to understand what is asked Draw a picture if possible and label Write down the given rate write down the desired rate Find an equation relating the variables Use Chain Rule to differentiate equation wrt to time and solve for desired rate The Linear Approscimation or tangent line approximation to a function f at 1 a is the function L1 fa f aa a Approximation formula fa f aa a for 1 near a if y f 13 the differential of y is dy f De nitions of absolute maximum absolute minimum local relative maxmium and local relative minimum 0 is a critical number of f if c is in the domain of f and either f c 0 or f c DNE Extreme Value Theorem If f is continuous on a closed interval a b then f always has an absolute maximum value and an absolute minimum value on a b Method to Find Absolute Max Min of f over Closed Interval a b 1 Find all admissible critical numbers in a b ii Find endpoints of interval iii Make table of values of f at the points found in and ii The largest value z abs max value of f and the smallest value z abs min value of f E Rolle s Theorem If f is continuous on ab and differentiable on a b and f a f b then f c 0 for some 0 E a b Mean Value Theorem If f is continuous on a b and differentiable on a b then there fb Ja lt I lt gt Z 1 CL is a number 0 where a lt c lt b such that slope f c v Yfx mfwn a a p v a a a a v gt fbf a slope ba a fa 197 fb fa f C b a If something about f is known then something about the sizes of fa and fb can be found M Useful for integration theory later a Iff a 0for alleI then 0 for alleI b If f 1 g a for all a E I then 95 C for all 1 E I Increasing functions f gt 0 gt f decreasing functions f lt 0 gt f l IncreasingDecreasingl XI 00 0 39 39 39 39 quot f39x I I I I I I I I I x 4 3 2 1 0 1 2 3 4 I I I critical not local max or min First Derivative Test Suppose c is a critical number of a continuous function f a If f changes from to at 0 gt f has local max at c b If f changes from to at 0 gt f has local min at c c If f does not change sign at 0 gt f has neither local max nor local min at 0 Displaying this information on a number line is much more e icient see above gure f concave up f 1 gt 0 gt fU and f concave down f 1 lt 0 gt f in ection point ie point Where concavity changes I Concave UpDownl U KW U quotI 0 00 fquotx I I I I I I I I I x 4 3 2 1 0 1 2 3 4 I I I inflection Pt inflection pt not inflection pt Displaying this information on a number line is much more e icient see above gure Second Derivative Test Suppose f is continuous near critical number 0 and f c 0 a If f c gt 0 gt f has a local min at c b If f c lt 0 gt f has a local max at 0 Note If f c 07 then 2quot Derivative Test cannot be used7 so then use 1SI Derivative Test INDETERMINATE FORMS 0 00 a Indetermlnate Form Types 6 00 000 00 00 00 000 1 b L Hopital s Rule Let f and g be differentiable and g 7E 0 on an open interval I containing a except possibly at a If lim f 0 and if lim g1 0 Type 3 m gta m gta or if lim 2 00 or oo m gta and if lim gx z 00 or oo Type g then gta hm fr13 2 Has ac m 95 ac m g a 7 provided the limit on the right side exists or is in nite Use algebra to convert the different Indetermine Forms in a into empressions where the above formula can be used Important Remark L Hopital s Rule is also valid for one sided limits7 1 gt a 7 1 gt a7 and also for limits When 1 gt 00 or 1 gt oo 14 Curve Sketching Guidelines a Domain of f b Intercepts if any 0 Symmetry f 13 for even function f 33 f1 for odd function f1 p f for periodic function d Asymptotes 1 a is a Vertical Asymptote if either lim f or lim f is in nite m m m gta y L is a Horizontal Asymptote if either lim f L or lim f L ac gt oo gtOO e Intervals Where f is increasing and decreasing local max and local min f Intervals Where f is concave up U and concave down in ection points 15 Optimization Min Word Problems Method Read problem carefully several times Draw a picture if possible and label it Introduce notation for the quantity7 say Q7 to be extremized as a function of one or more variables Use information given in problem to express Q as a function of only one variable say x Write the domain of Q Use Max Min methods to determine the absolute maximum value of Q or the absolute minimum of Q7 Whichever was asked for in problem
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