Class Note for MATH 124 at UA 2
Class Note for MATH 124 at UA 2
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This 15 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 17 views.
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Date Created: 02/06/15
A Section 34 The Chain Ruie Section 35 The Trigonometric Functions Section 36 The Chain Rule and Inverse Functions Math 124 Section 023 Fall 2007 Instructor Paul Dostert p1 We wish to take the derivative of composite functions such as 6902 Since 69 e is f 2 6902 Why or why not Let y fg2 where f and g are two differentiable functions Write z 993 y fz A change in 2 leads to a change in 2 which leads to a change in y Thus Ay Ay A2 A9 A2 A93 If we think of this in terms of derivatives we have dy dy dz d3 Edit p2 The Chain Rule If f and g are differentiable functions then gm we g39ltmgt Here are some quick shortcut rules 1 If a is a constant and y fz then dy a l d9 afv f x 2 If y em then dy f00 I div 6 f 93 Ex Consider y u2 u 1 where u 2 1 d a Find d y by substituting u Into y CC b Find ll y by using the chain rule CC p3 Ex Find the derivative of f2 for each of the following a we 63m b V932 1 c fv 2w2 d fv 21 Ex Suppose hc fg2 with f3 f 3 f 6 93 9 3 9 6 1 2 5 6 1 1 Find h 3 Ex Suppose the distance 5 of a moving body from a fixed point is given as a function of time by 3IH 875 2 206 where t is measured in seconds and s is measured in meters a Find the velocity v of the body as a function of t b Find the velocity aftert 5 seconds c Find the acceleration a of the body as a function of t d Find the acceleration aftert 5 seconds p5 Ex Find the derivative of f2 for each of the following a 69 15 b 2I21e5 c m 6232 52 92 55 dfltxgt 32 Ex Find an equation of the tangent line to y 26 30 ata 1 p6 Guess what sina is from the following graphs Of sin2 and COSCC What about COSCI 15 05 p p7 A Derivative of sine and cosine We have sin2 30833 and d 893 811193 CO d9 Ex Find the derivative of each function a 2C08lt5CCgt p8 Ex A company s monthly sales St in dollars are seasonal and given as a function of time t in months by St 2000 600 sin a Sketch a graph of St fort O to t 12 b What is the maximum monthly sales minimum monthly sales c lft O is January 1 when during the year are sales highest cl Find 82 and S 2 and interpret each in terms of sales p9 Use the product rule quotient rule chain rule and the fact that sinz 30833 and cosz sina to define each of the following Ex Find the derivative of each function a g6 08026 b ta21 If we know the chain rule and the derivative of 69 how might we find the derivative of 11133 Write 61W 23 Then from the chain rule we have d d 1nltgt o 1I1I 2 1n2 1 d ln00 d3 6 d3 d3 Derivative of natural log Using the above derivation we have d 1 1 d3 Maj 1 Ex Find the derivative of each of the following a 1n 27r332 623 Derivative of arcsin We have d 1 dCC arcsm CC Derivative of arccos We have d 1 dx arCCOS CC T 2 Ex Find the derivative of each of the following a COS 1 sin2 b W arcsin c gt arCCOS 1nt Derivative of arctan We have d 1 E arotan2 2 1 Ex Find the derivative of each of the following a tan 1 423 b tan 1 COSZ Ex Find the equation of the line tangent to CtaD 1CC ata1 If we are given a function f2 how do we find the derivative of f 12 We have f f 12 23 thus if f 1v f f 1v f 1ltagt 1 The derivative of an inverse function For a function f2 we have d 1 1 f gm rum provided that the denominator is nonzero Ex Find g 4 where 933 f1a is the inverse function of f2 3 2 69 Ex If f2 2 oos2 and 993 is the inverse of f2 then what is g 1 Ex Let P ft be the US population in millions in year t a What does the statement f 2000 281 tell you about the US population b Find and interpret f 1281 Give units c What does the statement f 2000 3476 tell you about the population Give units cl Evaluate and interpret f 139281 Give units
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