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# Study guide for exam 2 Math 1610-090

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This 4 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 107 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 2 A Average rate of change of y f over the interval 511 512 A y M this is also a 172 171 f a h f a the average velocity De nition of derivative of y f at a f a Illinti h or equivalently f a l M interpretation of derivative gta x a slope of tangent line the graph of y f at a f a velocity at time a instantaneous rate of change of f at a h Derivative as a function f Illin a i differentiable functions ie f exists 3 2 y or equivalently etc 5133 higher order derivatives y or equivalently p a Basic Differentiation Rules If f and g are differentiable functions and c is a constant d39v dc du U du d39v du U du b Z a d3 0 d3 d3 d3 C d3 d3 d3 d n c Power Rule dag n33quot1 a d v du u do d d d d PIOdUCt 31119 I u i U i Quotient Rule T W sin6 cos6 1 Speczal ng Lzmzts Ann 6 1 Ann Sin 6 1 ll 131 T 0 Hence also lim SIM kg 1 and lim 1 Note that sin k6 7E ksin6 0 gt0 k6 0 gt0 s1nk6 CHAIN RULE If g is differentiable at 1 and f is differentiable at 95 then the composite function f o g is differentiable at 1 and its derivative is f o ggt39ltxgt fgw f ltgltxgtgtg39ltxgt dy dy du h 1e1fy Wandu 9x ten d3 duda E Implicit Differentiation If an equation de nes one variable as a function of the other inde pendent variable then to nd the derivative of the function Wrt the independent variable Step 1 Differentiate both sides of equation Wrt independent variable Step 2 Solve for the desired derivative Logarithmic Differentiation Step 1 Take natural log of both sides of y simplify using Law of Logarithms Step 2 Differentiate implicitly Wrt 1 Step 3 Solve the resulting equation for d a Inverse Trig Functions Note for example sin 1 1 is same as arcsin a but sin 1 1 7E sin 3 a De nitions y sin 1 ycos1altgtcosya OS ytan1xltgttanya b Basic Derivatives xltgtsinyaj dsin1 3 1 d3 V1 12 Hyperbolic Trig Functions a De nitions 6 6 9 2 cosh 1 b Basic Derivatives dcosh 13 d3 sinh 1 1 F 3x39 1 3 7r E ltxlt9 dlttar1rgt 1 1 dw 1 x2 dx W Z 39 1 1 d If u is a differentiable function of a then by the Chain Rule Sm u u7 d3 11 U2 d3 em 6 90 sinha e 6 s1nha tanha 2 cosha em 600 h h M 2 cosh w sech2 d3 d3 d h If u is a differentiable function of a then by the Chain Rule COS u a G Basic Identities cosh 1 cosh 1 10 APPLICATIONS Model 1 Exponential GrowthDecay sinh a sinha k a y d sinh u i etc 2 2 cosh x sinh 331 where k 2 relative growthdecay rate If the rate of change of y is proportional to y then the above differential equation holds a If k gt 0 this is the law of Natural Growth for example population growth a If k lt 0 this is the law of Natural Decay for example radioactive decay All solutions to this differential equation have the form M0M Jt Usually need two pieces of information to determine both constants y0 and k unless they are given explicitly Half life time it takes for radioactive substance to lose half its mass Model 2 Newton s Law of Cooling If Tt 2 temperature of an object at time t and T5 2 temperature of its surrounding environment then the rate of change of Tt is pro portional to the difference between Tt and T5 The solution to this particular differential equation is always dT a k M Tt 2 T5 Cek t Usually need two pieces of information to determine both constants C and k unless they are given explicitly Related Rates Method to Solve Read problem carefully seVera1 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 USEFUL FORMULAs FOR RELATED RATES i Pythagorean Theorem 02 a2 2 mlib ii Similar Triangles iii Formulas from Geometry Circle of radius 7 Sphere of radius 7 4 A 7TT2 V 7r7quot3 3 C 27139 S 47TT2 surface area of sphere Cylinders and Cones A J V7r39r2h V 7TT2h ADDITIONAL DIFFERENTIATION FORMULAS u is a differentiable function of 13 d n1 du d 6 u du d a39 u du dCL nu 5 dm 6 5 dm a 111603 d In u l du d loga u 1 du dCL u dCL dCL ulna dCL d u cos u 2 d u sin u 2 d t3 u 2 see2 u 2 d 380 u du d see u du d cot u 2 du dcc eseueot u dcc seeutan u dcc 80 u

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