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164 Note for MA 16100 at Purdue

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This 3 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Purdue University taught by a professor in Fall. Since its upload, it has received 28 views.

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Date Created: 02/06/15
Fall 2011 MA 16200 Study Guide Exam 3 Method to solve Related Rates problems Read problem carefully several times Draw a picture if possible and label Write down given rate write down desired rate Find an equation relating the variables Use Chain Rule to differentiate equation wrt to time and solve for desired rate The linear approximation or tangent line approximation to a function f at z a is L fa f a x 7 a Approximation formula fz fa f a x 7 a 7 for z near a if y x7 the differential of y is dy f dx 3 H erbolic Tri Functions yp g a De nitions em 6 H em 7 6 sinhz em 7 64 7 11 sinhz 7 111 tanhz 7 2 coshx em 6 i cosh z b Derivatives i cosh x sinh z ii sinh x coshx iii tanh x sech2 z c Basic ldentites i cosh 7s coshx ii sinh 7z 7sinhz iii cosh2 7 sinhzx 1 De nitions of absolute maximum7 absolute minimum7 local maxmium7 and local minimum 0 is a critical number if c is in the domain of f and f c 0 or c DNE Extreme Value The orem method for computing absolute extrema for continuous functions over closed intervals Mean Value Theorem lf f is continuous on 17 and differentiable on 177 then there is a number 07 where7 a lt c lt b such that M big 4a Y slope 39C bfb yfx afa 512 fb fa 39 I x Fact lf f g z for all z in I then f g C for all z in I lncreasing and decreasing functions7 First Derivative Test concave up and concave down Concavity Test in ection point Second Derivative Test Using number line with f to determine where f is increasing or decreasing7 local max and min using number line with f z to determine where f is concave up or concave down and in ection points For example IncreasingDecreasing 00 o f39x I I I I I I I x 4 3 2 1 0 1 2 3 4 local min local max critical pt not local max or min Concave UpDown U m U U 0 0 0 fquotx I I I I I I I I I x 4 3 2 1 0 1 2 3 4 inflection pt inflection pt not inflection pt Indeterminate Forms a lndeterminate Form Types g 0 oo oo oo 00 000 1 b L7Hopita175 Rule Let f and g be differentiable and g 31 0 on an open interval I containing 1 except possible at a If lim fx 0 and limgx 0 or if lim f ioo and limgx ioo7 then man man lim La lim ff mam 9z 14m 9 provided the limit on the right exists or is in nite Curve sketching guidelines a Domain of f b lntercepts if any c Symmetry is f for even functions f7x 7fx for odd functions d Asymptotes x a is a vertical asymptote if lim f or lim f is in nite y L is a mac maa horizontal asymptote if lim f L or lim f L e lntervals where f is increasing and decreasing local max and local min f lntervals where f is concave up and concave down iln ection points Optimization MaxMin Problems Method Read problem carefully several times Draw a picture if possible and label it lntroduce 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 variable7 say x Write the domain of Q Use maxmin methods to determine the absolute maximum value of Q or the absolute minimum of Q that was asked for in problem

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