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# BUSINESS &ECON CALC MATH 112

UW

GPA 3.76

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This 10 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 112 at University of Washington taught by Staff in Fall. Since its upload, it has received 21 views. For similar materials see /class/192104/math-112-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 112 Review 2 This review sheet gives a quick overview of Worksheets 12 18 for Math 112 This review sheet may help you remember some of the key points of the course so far However this review does not include everything You are expected to know ALL material from these worksheets for the exam It is a good idea to look at old exams online at http wwwmathwashingtonedu m112 Also make sure to study the problems at the end of each worksheet 1 Worksheet 12 and 13 More Derivative Rules and How to Combine Them 0 You should be comfortable with all the derivative rule General Power Rule y a y nfx 1 Exponential Rule y 6f a y efmf W Logarithmic Rule y lnfx H y f Product Rule y fxgx a y f9 Macy35 Quotient Rule y H y W 0 Recall that the first three rules above are collectively called the chain rule We also wrote the product and quotient rules as follows FS FS r r SF DIV3 And we discussed how to remember these 0 It s not always easy to combine these rules so please review Chapter 13 for practice 2 Worksheet 14 15 and 16 Global and Local Optima 0 GLOBAL MAX MIN Given a function f 25 how do you find the global max and min over the interval from x a to z b a Find the critical numbers That is take the derivative set it equal to 0 and solve for z b Plug the critical numbers into the original function and Plug the endpoints into the original function c The biggest output is the global maximum The smallest output is the global minimum 0 DEMAND CURVE PROBLEMS A demand curve is a formula for your price in terms of your quantity You need to be able to find when a demand curve is positive and de creasing this gives you the interval of values that make sense Remember TRq pg 50 to get TR you multiply the demand function by q If you have to find the maximum TR you use the global max min technique o SECOND DERIVATIVE TEST LOCAL MAX MIN Given a function f at how do you determine if a critical number is a local max min a Find the critical numbers b Find the second derivative c Plug in the critical numbers to the second derivative If f 25 lt 0 then the curve is concave down and the point is a local maximum If f z gt 0 then the curve is concave up and the point is a local minimum If f z 0 then the test is inconclusive 3 Worksheet 17 Multivariable Functions 0 Know how to do partial derivatives That is you should know what the following types of notations mean fmzy and fyzy or and 37 0 Be able to find candidates for local maximum and minimum Here s how this is done a Find each of the partial derivatives and set them all equal to zero b Solve all the equations together That is combine the equations c Any solution to ALL the equations is a candidate for local maximum and minimum 4 Worksheet 18 Linear Programming 0 Understand the process of linear programming a Write out the objective and the constraints b Graph the constraints This will involving graphing lines by finding two points c Find each vertex of the feasible region This will probably involve solv ing when lines intersection d Plug them all into your objective The biggest output gives the global maximum and the smallest output gives the global minimum of your objective Math 112 Exam 1 Review The exam Wlll cover material from WS 1 11 Worksheet 1 and 2 SpeedMRMC and Tangent Lines Slope of the Secant line from a to b fa a 1 Kb 0 2 T The Slope of the Diagonal Line 3 lnstantaneous Speed The Slope of a Tangent Line 4 MR Marginal Revenue Slope of the tangent line to the TR graph 5 MC Marginal Cost Slope of the tangent line to the TC graph WS 3 7 and 8 Tangents amp Secants 1 If h is small then Slope of the Tangent Line at x Slope of the Secan Line from 8 to 3 h f 1 h f 56 h 2 Be able to nd derivatives without shortcuts c c c f x h f x a First7 sirnphfy b Let it go to zero to get f x 3 Understand how to get information from equations such as w 4m2r 3 92h 92 1 h 212h1 Rq2 Rqi 7 C12 611 7 2612 Q1 539 4 You should be able to nd H m g 2 and R q from the information above WS 9 Derivative Shortcuts Become a derivative machine 1 Expand the expression 2 Rewrite the powers 3 Take the Derivative Now apply the power rule 21 Bring down the power b Subtract one from the exponent 4 SimplifyDone PRACTICE PRACTICE PRACTICE WS 4 5 6 10 and 11 Derived Graphs 1 Connections between Derived and Original Graphs ORIGINAL GRAPH DERIVED GRAPH FLAT horizontal tangent ZERO crosses x axis INCREASING uphill POSITIVE above x axis DECREASING downhill NEGATIVE below x axis SLOPE OF TANGENT HEIGHT or y value DISTANCE SPEED TR MR TC MC ltgt ltgt ltgt ltgt ltgt ltgt ltgt ltgt HEIGHT OF A BALLOON RATE OF ASCENT 2 Major Applications of these Connections a To nd the Quantity that gives Mart Pro t i Compute MR R q ii Compute MC C q iii Solve MR MC b To nd the Locations of Horizontal Tangents for some function fx i Find f the derivative ii Solve f a 0 Math 112 Review 1 This review sheet gives a quick overview of Worksheets 1 11 for Math 112 This review sheet may help you remember some of the key points of the course so far However this review does not include everything You are expected to know ALL material from these worksheets for the exam It is a good idea to look at old exams online at http wwwmathwashingtonedu m112 Also make sure to study the problems at the end of each worksheet 1 Worksheet 1 2 3 Slopes of Secants Slopes of Tangents and Terminology 0 You should know what all of the following are total revenue TR total cost TC profit P marginal revenue MR marginal cost MC 0 The slope of the secant line from x a to z b I1 IUSI o The slope of the secant line from x m to z m h If h is close to zero then we get an approximation for the slope of the tangent line We experimented with this idea by plugging in numbers like h 001 or h 00001 0 Basic applications the instantaneous speed of an object 2 the slope of the tangent line to the dis tance graph MR 2 the change in TR from q to q one unit39 If one unit39 is small39 on the graph then we can say M R m the slope of the tangent line to the TR graph M O m the slope of the tangent line to the TC graph 2 Worksheet 4 5 6 Derived Graphs o In these sections we measured the slopes of tangent lines in order to fill in tables We plotted the slopes of these tangent lines on a different graph and we connected the dots We called the resulting graph the derived graph 0 The Derived Graph of the Distance Functions 2 The Speed Graph The Derived Graph of Total Revenue 2 The Marginal Revenue Graph The Derived Graph of Total Cost 2 The Marginal Cost Graph 0 We discussed the following general principles f crosses the z axis f 0 precisely when f has a horizontal tangent f is positive f gt 0 precisely when f is increasing f is negative f lt 0precisely when f is decreasing You need to be able to answer questions about f z or f x by looking at their graphs This is very important take the time to really understand these ideas 3 Worksheet 7 The Precise Value for the Slope of the Tangent 0 We found the precise value of f as follows a Simplify the expression b Let h go to zero f what39s left over as h a 0 0 Be able to work out these types of problems even when you are given different infor mation about W see Problems 311 3111 411 51 71 711 7111 4 Worksheet 9 Derivative Shortcuts 0 Please Please Please Practice Taking Derivatives 0 Power Rule lf fz x then f z nzn l o Sum Rule me WW 7 W we 0 Coe icient Rule cfz cf z o Becoming a Derivative Machine a Expand b Rewrite Powers c Use the Power Rule on each term d Simplify your answer 0 Here is a quick example Let y z 7 10 4 77 Find 27 y 72 7 lOz 4x 77 Expand y 3 7 lOzlzlZ 4x1 77 Rewrite Powers y 3x1 7 10z112 4x1 77 Still Rewriting Powers y 3x4 7 lOzl5 4x1 77 Finished Rewriting the Powers Now we start to take the derivative 7 371z 2 7 1015z0395 41z0 Use the Power Rule on each term 7 73z 2 7 15z0395 4 Simplify your answer i 71 7 15 4 Either of the last two expressions would receive full points on an exam 5 Worksheet 10 11 Using Derivatives in Business and Finding Max s and Min s 0 You should be able to use derivatives to find quantity that gives the maximum profit a Find formulas for MR and MC by taking the derivatives of TR and TC b Solve MR 2 MC for q c If you are also asked for the maximum profit then you need to plug q into the profit function 0 To find the locations where f has horizontal tangents we compute f Then we solve f z 0 0 Once again you should be able to go back and forth between f and f You need to know how they are connected and how to use these connections MATH 112 REVIEW FOR EXAM II OPTIMIZATION I Derivative Rules 0 There will be a page of derivatives on the exam Know how to apply all the derivative rules WS 12 and 13 II Functions of One Variable 0 Be able to distinguish between local and global optima WS 14 0 Be able to nd the global maximum and minimum of a function y fz on the interval from x a to z b7 using the fact that optima may only occur where fz has a horizontal tangent line and at the endpoints of the interval Step 1 Compute f z Step 2 Find all values of x at which f z 0 ie the critical numbers Step 3 Plug all the values of z from Step 2 that are m the interval from a to b and the endpoints of the interval into the function Step 4 Plot the points from Step 37 then use them to sketch a rough graph of fz and pick off the global max and min 0 Be familiar with the following two applications Maximizing TRq starting with a demand curve WS 15 7 Optimizing the slope of a diagonal line through a given curve WS 16 Understand how to use the Second Derivative Test WS 16 If f a 0 and f a gt 07 then fz has a local min at z a f a lt 07 then fz has a local max at z a f a 07 then the test tells you nothing IMPORTANT For the Second Derivative Test to work7 you must have f a 0 If f a gt 0 but f a 31 07 then the graph of fz is concave up at z a but fz does not have a local min there III Functions of Two Variables 0 Be able to compute overall7 incremental7 and instantaneous rates of change of a function of two variables WS 17 0 Be able to compute partial derivatives using all the derivative rules 0 Know how to nd the candidates for maxima and minima of a function of two variables Take both partial derivatives7 set them equal to 07 and solve the resulting system of equations MATH 112 Review of Exponents 1 x 17 Examples a 2 z3 5 b 4 5x3 2x7 5x7 2x11 a m 2 7 1717 12 Examples 2 7 32 a mi 5x3 2x7 7 5x3 2x7 b 4 5x 12z3 b b a 3 7 iz ex 0 Examples 1 71 a E 4 72 b P 4 3 71 3 C W3z l l d 7 7 71 4x 495 3 3 2 8 Q 59 3 3 3 d 7 714 11w 11z14 11

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