CALCULUS IN BUSINESS AND ECONOMICS

1 CALCULUSINBUSINESS AND ECONOMICSG. S. MonkMathematics 112 Revised Winter 2010 Published by Professional Copy & Print1414 NE 42nd , WA 98105(206-634-2689)Copyrightc 1981 by Monk. All rights r k s h e e t # 1 . T h e Ve r t i c a l S p e e d o f a S h e l l .. 2 Worksheet #2. Two Versions of Marginal Revenue and Cost .. 7Wo r k s h e e t # 3 . Ta n g e n t s a n d S e c a n t s .. 12Wo r k s h e e t # 4 . D r aw i n g S p e e d G r a p h s .. 18 Worksheet #5. Drawing and Using a Marginal Revenue Graph .. 22Wo r k s h e e t # 6 . D e r ive d G r a p h s .. 262 Tangents and Formulas; The Derivative33Wo r k s h e e t # 7 . T h e Fo r m u l a f o r t h e S l o p e o f t h e Ta n g e n t .. 34Wo r k s h e e t # 8.

2 T h e Fo r m u l a s f o r Two S p e e d G r a p h s .. 40Wo r k s h e e t # 9 . Ta k i n g D e r iva t ive s .. 45 Worksheet #10. Using the Marginal Revenue and Marginal Cost Formulas .. 51 Worksheet #11. Two Balloons ..553 DifferentiationTechnique61 Worksheet #12. A Second Set of Rules .. 62 Worksheet #13. Combining Differentiation Rules .. 684 Optimization73Wo r k s h e e t # 1 4 . L o c a l a n d G l o b a l O p t i m a .. 74 Worksheet #15. Demand Curves ..79Wo r k s h e e t # 1 6 . T h e S e c o n d D e r iva t ive Te s t .. 825 FunctionsofSeveralVariables89Wo r k s h e e t # 1 7 . M u l t iva r i a b l e F u n c t i o n s .. 90Wo r k s h e e t # 1 8 . L i n e a r P r o g r a m m i n g.

3 976 Going Backwards: Integration103Wo r k s h e e t # 1 9 . D i s t a n c e s f r o m S p e e d G r a p h s ..104 Worksheet #20. Total Revenue and Cost from Marginal Revenue andCost .. 111Wo r k s h e e t # 2 1 . A r e a U n d e r G r a p h s ..117iiiivCONTENTSWo r k s h e e t # 2 2 . A r e a s b y Fo r m u l a ..126Wo r k s h e e t # 2 3 . A n t i - D e r iva t ive s ..130 Worksheet #24. The Fundamental Theorem of Integral to Selected Exercises ..143 Graphs and Tables .. 163 PrefaceLike Math 111, Math 112 does not follow the example/problem/example/problem model whereyou simply work problems exactly like given examples, only with different numbers. There arevery few drill problems. You will often be asked questions that need to be answered with wordsinstead of numbers or formulas.

4 You will be challenged to think about ideas rather than pluggingnumbers into formulas. This makes some people uncomfortableatfirst,butmostpeoplefindth atthey get used to it with to Use This TextThe heart of this text consists of twenty-four worksheets. Most worksheets begin with one ormoreKey u s h o u l d b e g i n e a c h w o r k s h e e t b y r e a d i n g a n d t h i n k i n g a b o uttheKeyQuestions. Youwillnot always be able to answer the Key Questions right away; will lead you to an understanding of the issues involvedand, we hope, to answers to the exercise is numbered, like8Yo u s h o u l d w o r kallof the exercises carefully. Sometimes,you might not see right away what it is that they are getting at, up your solutions to ALL of the involve drawing on given graphs, there is an appendixofGraphsandTablesatthebackofthe book that you may draw on and rip out to turn in with your with the exercises, you will find bits of exposition.

5 They will be formatted like this para-graph. They do not require answers, but you will certainly want to read the end of most worksheets, you ll find one or more problems which (a) provide some review of the material covered in that portion of the course,(b) add a little bit ofnew material,and(c) Study Center (MSC)Because of the challenging nature of this course, the Mathematics Department offers a StudyCenter for the students in Math 111 and 112. The Math Study Centerprovidesasupposrtiveatmosphere for you to work on your math either individually oringroups. Detailsregardingthetime and location of the Math Study Center will be announced inclass and on your instructor , Suggestions, Typos,..This text is a work in progress.

6 It probably contains mistakes. We will keep an updated list ofcorrections posted on the web m112 you find any mistakes that are not already mentioned there,please report them via email 1 Tangents and Derived Graphs2 CHAPTER1. TANGENTS ANDDERIVEDGRAPHSW orksheet #1 The Vertical Speed of a altitude as a function of that there is a tiny speedometer in the shell that records the shell s velocity just as thespeedometer in your car records the car s velocity. The function describing the shell s altitude(A) (t),whereAis in feet andtis in QuestionsI. What is the reading on the shell s speedometer at timet=2?II. What is the reading on the shell s speedometer when it reaches the highest point of its path?III. Does the shell stop at the top?

7 IV. What is the reading on the shell s speedometer at timet=0,justasitisbeingfired?1 Use the altitude graph to find the following quantities:a) The distance covered from timet=2tot= )f( ) f( )2 WORKSHEET#1 The Vertical Speed of a Shell3c) The average speed of the shell from timet=0to timet= ) The average speed of the shell from timet=1to timet= )f(5) f(2)3f) The time required for the shell to reach the altitude 300 that the following quantities can be read from the altitude ofheightchange insecantslope ofFirst read the following quantities from the graph and then tell what this quantity is in termsof a shell being ) The height of the graph att=3seconds;t= ) The change in the height of the graph fromt=2tot= ) The slope of the secant to the graph fromt=3tot= formula that corresponds to the altitude graph isA=f(t) = 160t use the formula to get the information you have been getting from the graph,you have torecallThe Substitution Rule.

8 To findf(4)from the formula you substitute 4 in forteverywhere in theformula:f(4) = 160(4) 16(4)2= 384feet. To findf(5) f(4), you substitute and followthe recipe:f(5) f(4) = [160(5) 16(5)2] [160(4) 16(4)2] = 400 384 = findf(5) f(3)2, you substitute and again follow the recipe:f(5) f(3)2=[160(5) 16(5)2] [160(3) 16(3)2]2= 32 the formula forA=f(t)to determine the quantities in questions 1(a) (f) and 2. Checkyour answers against the answers you got from the TANGENTS ANDDERIVEDGRAPHSThus far the questions have required that you get information about average speeds and dis-tances covered from the graph or formula. There is an accuracy problem with the graph, butthere should not be any great confusion about how these quantities are tobe found.

9 However,we have not yet dealt with the issue of the readings on the imaginary speedometer in the can be great confusion about this issue in fact, at the beginning, itis a matter of personalopinion and approach to the question of what is a speedometer reading is the one we used in Math 111:Math 111 version. The speedometer reading is the average speed over a very tiny timeinterval for instance, one that is or seconds average speed is measured on the graph by the slope of the secant over the time your best approximation to the secant line to the altitude graph fromt=2tot= a nice long line with a plastic ruler and compute its slope. Would you say that thisnumber is a good approximation of the speedometer reading att=2,andthereforeananswer to Key Question I?

10 5By measuring the slope of the secant to the graph over a very tinytimeinterval,makeyourbest estimate of the speed of the shell when it reaches the peakofitspath,attimet= you have a graphical answer to Key Question I answer Question 5 with my clear plastic ruler I draw a linethatisabsolutelyhori-zontal. Its slope is 0, so I conclude that the speed of the shellatt=5seconds is 0. Do youagree? Does this mean that the shell stops at timet=5?Asyouanswerthisquestion,thinkcar efully about the way the word stops is used in everyday people would say that at timet=0,justastheshellisfired,ithasnospe ed,orthatitsspeedometer is not reading, or that its speedometer readingis 0. However, if you adopt theMath 111 version of speedometer readings that they are approximated by secants over tinytime intervals what would you say the speedometer reading oftheshellisatt=0?