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1 Calculus for Business an Social Sciences - Prof D Yuen Eam Review version /9/01 Check website for any poste typos an upates Eam is on Sections, 5, 6,, 1,, Derivatives Rules Know how to fin the formula for the erivative f '( when given a formula for f ( by using the erivative rules o Builing blocks: c 0, m m, [power rule] n n1 [general power rule] g( ng( g' ( or u nu o Algebra rules: n n1 n u n ( f g f ' g' ( f g f ' g' ( cf cf ' [prouct rule] ( fg f ' g fg' f f ' g fg' [quotient rule] ( g g [chain rule] ( f ( g( f '( g( g'(, or y y u for any three variables y, u, u where y epens on u an u epens on, we have Be comfortable with using ifferent variables Pay attention to which is the erivative variable when there are multiple variables, an pay attention what variable epens on what variable Sketching a graph using of erivative to information f is increasing when f ' 0 an f is ecreasing when f ' 0 The 1 st erivative iagram is probably the most important tool to sketch a graph so that the relative an absolute maima an minima are sketche correctly The 1 st erivative iagram tells you where the function is increasing an where it is ecreasing Know how to use this tool! f is concave up when f '' 0 an f is concave own when f '' 0 For the occasions where you are require to fin inflection points, use the n erivative iagram to fin the intervals of concave up an concave own Ma-Min (Optimization problems given an objective function of one variable The optimum (either ma or min if it eists will occur where the erivative is zero or at an enpoint if the omain has an enpoint The stanar technique is: (0 Make sure you have a function of one variable! (1 Fin the erivative of the objective function ( Set the erivative equal to 0 to fin the caniates for relative (local etrema (also calle critical points Only caniates insie the omain matter ( [Very important] You must use the 1 st erivative iagram to unerstan how the function increases an ecreases This will tell you where the absolute ma/min is It is often at where the erivative is 0, but on some occasions it coul be at an enpoint n1
2 Ma-Min (Optimization wor problems involving fences an boes Rea the problem carefully, raw a iagram an assign variables Do not assign variables to constant quantities Set up an objective function (to be maimize or minimize, which might at first involve more than one input variable Give this objective function a name, such as a generic Q (for quantity or a more escription letter such as A for area or L for length If there is more than one input variable that the objective function epens on, write own relations (constraints among the input variables If the objective function epens on more than one input variable, then you must use the constraints to eliminate own to one input variable This is usually one by solving for one input variable in terms of the other an then plugging this into the objective function Note the omain of the objective function in this one input variable You usually fin the omain by consiering for what inputs oes the function make sense This often involves looking at any constraints, if there are any Now that you have an objective function of one variable, following the stanar steps of: (1 Take a erivative ( Set erivative to zero ( Analyze the 1 st erivative iagram Minimizing inventory cost size of orer, r size of orer Constraint is r total to be orere Inventory cost is I r( cost per orer (carrying cost per unit Follow the steps of using the constraint to eliminate a variable an then follow the stanar steps of: (1 Take a erivative ( Set erivative to zero ( Analyze the 1 st erivative iagram Maimizing profit an revenue eman/units prouce/units sol p price Sometimes price is constant an sometimes price epens on the eman via a relation calle the eman equation (eman curve Sometimes the eman equation is given an sometimes you have to fin it base on two ata points an an assumption that the eman curve is linear Revenue is R( p Profit is P( R( C( The cost function is often given Once you have the relevant function, then maimizing it involves the stanar steps of: (1 Take a erivative ( Set erivative to zero ( Analyze the 1 st erivative iagram Implicit Differentiation Differentiate both sies of the equation with respect to the input variable, making sure that the (other variable erivative involving any other variable woul generate a Then solve for your (input variable esire erivative Typically the erivative result will have both the input an output variables in it To evaluate the erivative at a point, you will typically plug in both coorinates of the point Relate Rates Differentiate both sies of the equation with respect to the time variable t even if t oes not appear in the equation, again making sure that the erivative involving any variable that is not t (other variable woul generate a t Some problems ask you to solve for one rate in terms of the other rates Some problems as you for the value of a rate given values of the other rates an values of all the variables (ecept for the time variable In this case, it is easier to plug in all the values into the variables an known rates before solving for the unknown rate Caution: you must ifferentiate implicitly with respect to the time variable first before plugging in the numbers
3 Practice Problems 1 Differentiate the following functions using erivative rules (a f ( (b f ( ( (c y ( y ( ( 5 (e f ( (f f ( 1 Fin an epression for y Assume f ( an u are some unspecifie functions of (a y f ( f ( (b y ( 5 f ( 10 (c y u 10u u ( y u Consier the curve y y 0 (a Fin y (b Fin the tangent line at the point ( 1, y Given the relation y y 1, fin t in terms of, y, t 5 Suppose eman an price p are relate by p 10 p 80 (a At the moment when the eman is 10 an the price is $, if the price increases at a rate of $1 per month, how is the eman changing? (b At the moment when the eman is 10 an the price is $, if the eman increases at a rate of per month, how is the price changing? 6 Suppose eman an price p are relate by 10 p 50 (a At the moment when the eman is 10 an the price is $5, if the price ecreases at a rate of $1 per month, how is the eman changing? (b At the moment when the eman is 10 an the price is $5, if the eman ecreases at a rate of per month, how is the price changing? For f (, use the first erivative test to fin its relative etrema an sketch its graph, making sure to label the -coorinate(s of any relative etrema 8 For f ( 6, use the first erivative test to fin its relative etrema (give the - coorinate(s 9 For f ( 1, fin its inflection points (give the -coorinate(s Be sure to show your work using the n erivative test 10 Suppose we wish to make sie by sie ientical rectangular pig pens We have 10 ft of fence Fin the imensions that woul maimize the total area 11 Suppose we wish to make sie by sie ientical rectangular pig pens Each pig pen shoul be 600 square feet Fin the imensions that woul minimize the length of fence use 1 Suppose we wish to make bo of volume cubic inches Suppose the length must be twice the with Suppose further that the bottom of the bo costs twice as much because it is twice as thick What imensions woul minimize the cost? 1 Suppose we sell 100 units uring a year Suppose the elivery charge is $60 per orer an the carrying cost per item per year is $10 An suppose we moel the carrying cost base on the average number of units in the inventory, how many units per orer woul minimize the total inventory cost? 1 Suppose we sell 600 units uring a year Suppose the elivery charge is $60 per orer an the carrying cost per item per year is $10 An suppose we moel the carrying cost base on the maimum number of units in the inventory, how many units per orer woul minimize the total inventory cost? 15 Suppose the cost of proucing units is C( (a If the price per unit is fie at 80, fin that woul maimize profit
4 (b If the price is base on eman by the eman equation p 80, (b1 fin that woul maimize revenue, an (b fin that woul maimize profit (c Suppose we can sell 10 units at a price of 180 an we can sell 0 units at a price of 150 (c1 Assuming a linear relation between the price an eman, fin the eman equation (the price as a function of the eman (c Fin that woul maimize revenue (c Fin that woul maimize profit 1 (a Solutions to Practice Problems 6 1 1/ / 6 1 f '( 1 0 (b f ( ( ( 1/ 1 ( ( 6 1 1/ (c y ' ( y ' ( ( 5 ( ( ( ( ( (e '( 1 f ( 1/ ( ( ( ( (f f '( ( 1 / ( 1 ( (a (c y y y 9 f '( f '( (b 10(5 f ( (5 f '( u (100 0u u y 6 u u u ( y y (a y 1 y t t y (b y 5 ( 1 p 5 ( p t ( 10 t 0 (a eman is ecreasing at /month (b price is ecreasing at $05/month p 6 t 0 p t 0 (a eman increasing at 5/month (b price increasing at $0/month Relative minimum at / Sketch is a parabola with absolute min at ( /, 9 / 8 Relative ma at, an relative min at 9 Inflection point at 1/ 10 Maimum area when the pig pens are each 10 ft by 15 ft with the common sies being 15 ft long 11 Minimum fence length when the pig pens are 0 ft by 0 ft with the commons being 0 ft long 1 The bo shoul be in by in by in high The objective function is Cost 6 6( where the bo is wie an long Cost 60( 10 Minimum at Cost 60( 10 Minimum at (a P ( Maimum profit at 15 (b1 R( (80 Maimum revenue at 0 (b P ( Maimum profit at 5 (c1 p 10 (c R( (10 Maimum revenue at 5 (c P ( Maimum profit at 0
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