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# Calculus for Life Sciences Students MATH 3A

UCLA

GPA 3.55

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This 5 page Study Guide was uploaded by Kaylin Wehner on Friday September 4, 2015. The Study Guide belongs to MATH 3A at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 43 views. For similar materials see /class/177839/math-3a-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.

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Date Created: 09/04/15

Math 3A 2 Midterm 2 Study Guide Updated 22307 Written by Akemi Kashiwada Midterm March 27 2007 Disclaimer This is just a guide to help you study for the midterm The material in this study guide will not necessarily be on the midterm and Vice versa This is just what Iwould knowstudy if I were a student in Math 3A There may be some typosmistakes so do not just use this to study ie be sure to look at old homeworks7 notes7 book7 etc Material Covered since rst midterm Section 42 Section 52 Section 42 Power Rule Power rule d n n71 Ezinz Theorem 1 For any a E R and di erentiable functions and 91 d d 011690 aleI El l 9Il fI 57 Example 1 Find all points on the curve y 213 7 41 1 whose tangent line is parallel to the line y 7 21 1 Section 43 Product and Quotient Rule Product Rule If hz where are differentiable then hI f I9I yEl Quotient Rule If hz where are differentiable and 91 y 0 then hm WA l2 NOTE You do not have to memorize quotient rule because hz so product rule will work on the latter equation Example 2 Assume and 91 are di erentiable Find an expression for the derivative of y 2f 1 1 139 y 73 Math 3A Discussions 20 A 2D 2 y fryr Section 44 Implicit DifferentiationRelated Rates and Higher Order Derivatives Implicit Di erentiation Use when dif cult to write dependent variable as a function of the independent variable eg it would be extremely difficult to write ygz 7 Q 39 zy 1n the formy f 1 Using implicit differentiation is just an application of chain rule remember that we can think of y as a function of I rather than a variable Example 3 Find 3 of ygz 7 1 We also use implicit differentiation to solve related rate problems Remember for these problems most of our variables are now functions of time t Example 4 Suppose we re having a party after the midterm so we order a 5ft tall 6ft diameter cylindrical barrel completely lled with ll in your favorite beverage We can calculate the volume of the barrel by V 7rr2h If people are really thirsty so the drink is being consumed at giftsmin at what rate is the height of the liquid changing when h 4ft Higher order derivatives Finding second derivative and higher Since is just another function of I we can nd the second derivative of f denoted f by differentiating the rst derivative ie Mr gm Example 5 Find the rst and second derivatives of I2 32 Application lf is the position of an object at time t then f t is the velocity of the object at time t and f t is the acceleration of the object at time t Math 3A Discussions 20 A 2D 3 Example 6 You re outside playing catch with your pet dogcatsquirrel and you re so perfect that every time you throw the ball you can calculate its position by 4t2 t 4t 7 7 a 3 Find the acceleration of the ball when the velocity is zero Section 45 Derivative of Trig Functions d d d 2 s1nz 7 cos I Ecosz 7 781111 Etanz 7 sec 1 3 gt7 t 3 gt7 t ilttwi 2 dx secz 7secz anz7 dr cscz 77csczco x dx co 1 77csc I Remember trig functions are functions7 not variables7 so many problems that ask to you nd the derivative of something With a trig function involves chain rule Example 7 Find the derivative of fr 1 Section 46 Derivative of Exponential Functions 1 i dze Example 8 Find the deriavtive of sinze d ex ia1axlna dz Section 47 Derivative of Inverse Functions and Log De nition Let a function f z A A B be onetoonel The inverse function f 1 z 7gt A is de ned by f 1y I if and only if y for all y E Note that f 1 o z d71 7 l Ef roemgt Math 3A Discussions 20 A 2D 4 Example 9 Let y arc cosx 71 S x S 1 he the inverse of cosx 0 S x S 7r Find are cos1 Noting that lnx is the inverse function of e7 7 we can use the above formula to nd the derivative of lnx d 1 1 1 i l 7 a 41 mm em an Example 10 Find the derivative of fx Section 48 Linearization Use to nd approximate solution to where x is really bad77i Given and we want to approximate f at x07 we can do this by lo W 1 fa10 a 111 if a is close to xoi We call this the tangent line approximation or linearization of f at xai Example 11 Use linearization to approximate ln199 Section 51 Extrema and Mean Value Theorem De nition A function f has a global maximum at x c if fc 2 fx for all x f has a global minimum at x c if fc S fx for all x These points are called global extremal Extreme Value Theorem If f is continuous on a7 12 and differentiable on a7b then f has a global min and max on a7 Example 12 Do the following functions have global extrema over the given intervals Why or why not 1 fxl71SxS1 x 239 fzlnxe 12Sxlt1 sin7rxx7 1S x S 2 Math 3A Discussions 20 A 2D 5 De nition The function f has a local maximum at z c if there is 6 gt 0 such that 2 for all z E c 7 67c 6 Similarly f has a local minimum at z c if there is 6 gt 0 such that S for all z E c76c6 Fermat s Theorem If f has a local extrema at c 6 ab and fc exists then fc 0 Notice that this does not mean every point With derivative zero is an extrema Example 13 Let I 7 23 Show that fQ 0 but I 2 is not a local extrema Mean Value Theorem If f is continuous on db and differentiable on a7b then there exists c 6 ab such that b f5 This theorem is very useful and gives rise to the following corollaries Rolle s Theorem If f is continuous on db differentiable on ab and fa 1617 then there exists c 6 ab such that fc 0 Corollary 1 If f is continuous on a z differentiable on ab and m S S M for all z 6 ab then m 7 Miv Corollary 2 If f is continuous on db differentiable on a z and 0 for all z 6 ab then f is constant on a7 Example 14 Pretend you are a bug and you want to travel the world which is the real line You start at 30 0 then travel until you nd yourself at 35 100 Is there a time t where you were going faster than 15 unitssec Once you get to 35 100 you decide to slow down and do some sightseeing so your velocity is between 5 unitssec and 5 unitssec What is the interval that 37 lies in Example 15 Let Find an interval that contains a number c so that fc 0 Section 52 Monotonicity and Concavity De nition f is strictly increasing on I if fzl lt f12 for all 1112 6 I with r1 lt 12 f is strictly decreasing on I if frl gt fI2 for all 11 12 E I with r1 lt 12 Testing for IncreasingDecreasing functions Suppose f is continuous on ab and differentiable on a7b lf f z gt 0 for all I then f is increasing lf lt 0 for all I then f is decreasing

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