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# Calculus ConceptApplications MATH 108

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This 16 page Class Notes was uploaded by Miss Hillary Grady on Thursday October 29, 2015. The Class Notes belongs to MATH 108 at College of William and Mary taught by David Lutzer in Fall. Since its upload, it has received 14 views. For similar materials see /class/231138/math-108-college-of-william-and-mary in Mathematics (M) at College of William and Mary.

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Math 108 Notes on Elasticity of Demand Percentage increases Percentage rate of increase A few months ago Coca cola in the second oor vending room of this building cost 75 cents per can Starting in September the price was 1 per can The wrong way to look at this situation is Twenty ve cents is not much money so the increase was small The right way to look at the situation is Dividing the increase by the former price shows a 33 price increase and that s big It s the percentage increase not the actual increase that should worry consumers Whenever you have a function y f x relating two quantities we know that fx gives the rate of change of 2 compared to x The f x and percentage rate of change 100 gtk f x f x f 96 Note that these quantities probably depend on the x value you started with and should really be called the relative and percentage rates of change starting at x relative rate of change Elasticity of Demand Demand for gizmos is sensitive to unit price An increase in price causes a drop in demand Suppose we have a demand equation x f p where p is unit price and x is the number of gizmos that can be sold at price p The percentage rate of change in demand is 100 f p f P This predicts the percentage change in demand corresponding to a price increase of 1 and it is always negative Starting at a given price p raising prices by 1 will result in a percentage increase in price of 100 gtk 1 Economists de ne elasticity of demand starting at price level p to be the ratio E 100 fP E lpercentage change in demand fp l 17 f p l percentage change in price I 100gtk 17 l p Why use the absolute value Because otherwise every entry in a table of demand elasticities would be negative Note that because our f p lt 0 this is exactly the same as the book s de nition E w Why Care About Elasticity of Demand Recall that our revenue 2 income for selling gizmos is given by R Revenue unit price gtk number sold 17 f p where p is unit price and f p is the demand function Our real interest is in the question Will our revenue rise if we increase prices slightly starting at price level p In other words is R an increasing function near p In still other words is Rp gt 0 We use the product rule to nd R p and throw in some algebraic trickery to show how elasticity of PfP fP T VHFP Wi 1fl7 E1 demand E is the key At one point we will use the fact that 7E Here is the calculation Rpfp1fp fP 1 Math 108 Introduction to R3 Pictures for this handout will be drawn in class I Coordinates special planes and surfaces in R3 Recall two dimensional graphing We start with two mutually perpendicular axes called the z and y axes that meet at origin denoted 00 We locate a point in the plane by giving two coordinates telling us how to get to the point from origin Consequently 2 3 is the point that we reach by going 2 units in the positive direction along the z axis and then 3 units in the positive direction vertically The plane is often called R2 The x axis is the set of all points m0 where z is allowed to be any real number Similarly the y axis is the set of all points 0 y where y is allowed to be any real number In three dimensional space we have three mutually perpendicular axes called the m y and z axes that meet at origin denoted 000 We locate a point in space by giving three coordinates telling us how to get to the point from origin Consequently we reach the point 23 4 by going 2 units is the positive direction along the z axis then 3 units in the positive y direction then 4 units in the positive 2 direction Three dimensional space is often called R3 The x axis is the set of all points x 00 where z is allowed to be any real number The y axis is the set of all points 0 y where y is any real number The 2 axis is the set of all points 002 where z is any real number The m y plane is the set of all points z y0 where z and y are allowed to be any real numbers The m 2 plane is the set of all points z 0 z where z and z are allowed to be any real numbers The y 2 plane is similarly de ned Drawing things in R3 requires perspective drawing which is a challenge In R2 we graph functions y fz whose domains are pieces of the z axis We get curves oating above or below the z axis and the formula y fz tells us the height of the curve above or below the point z 0 on the z axis In R3 we graph functions 2 fm y whose domains are pieces of the m y plane We get surfaces oating above or below the m y plane and the formula 2 fm y tells us the height of the surface above or below the point z y 0 in the m y plane The plane R2 contains special lines For a constant a the line z a is a line through the point a 0 on the m axis that is parallel to the y axis and contains all points a y for all possible values of y Therefore the graph of z 0 is the y axis For a constant I the line y b is a line through 01 that is parallel to the z axis and contains all points z b for all possible values of m Therefore the graph of y 0 is the z axis Three dimensional space R3 contains special planes For a constant a the graph of z a is a plane through a0 0 containing all points a yz for all possible values of y and 2 Therefore the graph of z 0 is the plane containing origin and the entire y and z axes This is the y 2 plane For a constant I the graph of y b is the plane consisting of all points z bz for all possible values of z and 2 In particular the graph of y 0 is the plane containing origin and the z and z axes namely the m 2 plane The graph of z c is the set of all points z y c and is parallel to the z and y axes Homework 1 In R3 draw the special planes z 2 and y 3 Describe the intersection of these two special planes II Slicing surfaces with special planes General Principle 1 If we intersect a surface S with a plane P we get a curve lying in P We study the surface by studying the intersections of the surface S with the special planes z constant and y constant and z constant The process of intersecting the surface S with the plane P is called slicing the surface S with the plane P Imagine the surface 2 2 y2 1 oating above the zy plane We slice it with special plane z 3 and the points we get must have both z 3 and z 2 y2 1 so we get points satisfying 2 9 y2 1 10 yz We recognize this curve as being a parabola opening upwards In fact for every slice z a we get a curve 2 1 a2 y2 that is a parabola opening upwards Next imagine slicing that same surface with the special plane y 2 We get the curve 2 2 4 1 which is a parabola opening upwards In fact slicing with any special plane y b we get the curve 2 2 b2 1 and that is a parabola opening upwards We could also slice with the horizontal plane 2 c where c is a constant We would get 2 y2 1 c which is 2 y2 c 7 1 In case 0 lt 1 no points satisfy that equation because 2 y2 is always positive while 0 7 1 lt 0 In case 0 1 only z O y 0 satisfy the equation and we conclude that the surface hits the slicing plane 2 1 in the single point 001 In case 0 gt 1 we get 2 y2 1 c which is 2 y2 c 7 1 Because 0 7 1 gt 0 this is a circle with radius xc 7 1 Now the problem is to put together all of those slices to get the surface 2 2 y2 1 This is not easy at rst We will draw the picture in class Homework 2 Describe the curves obtained by slicing the surface 2 3 7 2 7 y2 with the special planes z a where a is a constant What if we slice with the special planes y b What if we slice with the special planes 2 constant Homework 3 Describe what you get if you slice the surface 2 2x 1 3y with the special planes z a where a is any constant What if you slice with the plane y b Pasting together slices was all the ancients could do Today we have software that can graph in R3 so we can get rough ideas of what the surface 2 m y will look like Probably the surface 2 fzy will have hilltops and valley bottoms aka local maxima or local minima or relative maxima or relative minima Our job is to nd them exactly General Principle 2 If the surface 2 fzy has a hilltop directly above the point ab0 and if we slice the surface with the plane y b then we get a curve 2 fzb with variables 2 and z and the value x a must give a local maximum for that function Similarly if we slice our surface with z a we get a curve 2 fay which must have a local maximum at y b There is an analogous principle for valley bottoms on the surface Homework 4 Consider the surface 2 2 2x 7 y2 1 4y and the point 71 2 a Slice the surface with z 71 What curve do you get Does it have a local maximum where y 2 b Slice the surface with the point y 2 What curve do you get Does is have a local maximum where z 71 c Could the given surface have a hilltop directly above 71 2 0 General Principle 3 If the surface surface 2 fzy has a hilltop directly above the point ab0 and if we slice the surface with the plane y b then we get a curve 2 fzb with variables 2 and z and the derivative of z fzb with respect to z must have value zero when x a There is an analogous principle for valley bottoms on the surface Homework 5 State an analog of General Principle 3 if the surface surface 2 fzy has a valley bottom directly above the point a b 0 and we slice the surface with the special plane z a III Partial derivatives Consider the surface 2 m y 3 myy2 and the point a b 0 of its domain Slicing that surface with the plane z a gives us the formula 2 a3 ay yz That is the formula of a curve and we know how to nd if 0 a 2y Alternatively we may slice with the vertical plane y b to get the curve 2 m b 3 bx b2 and we know how to nd 3x2 b As noted in General Principal 3 if a hill top or valley bottom of the surface lies directly above a b 0 then both of these derivatives when evaluated at z a and y I must have value zero But what if we do not know what point a b 0 to consider Suppose our job is to nd the point a b 0 We are now ready to talk about partial derivatives of m y We will do two arti cial looking things that are usually described as Hold z constant and nd the derivative of m y with respect to the variable y or Hold y constant and nd the derivative of fz y with respect to the variable m The rst is denoted by or and the second by or 37 We already know all of the derivative rules that we will ever need The new and tricky part is to be able to say to ourselves Even though z looks like a variable it is held constant or Even though y looks like a variable it is really constant Consider some examples With 2 3 my yz hold y constant Then 3x2 y 0 Next hold z constant in the z formula Then Ox2y Let 2 mezy Then 1 because y is constant and therefore so is 523 Also 37 0 2521 because with z constant its derivative with respect to y is zero Let 2 Then 5 3x2 because with y held constant 21 is also constant so that z is just the constant 21 times the variable part as Also 37 3 71y 2 because with z held constant 2 is just the constant 3 times the variable part y l Let 2 mzemy Hold z constant Then the only variable part of z is 50mmth so we have 2 3 z emy z x cm Next hold y constant Then 2 is the product of two functions of the variable x namely 2 and ec msmm so we need the product rule and we have 37 2x 51y x2 emy y Suppose z mylnz 1 Then has y constant so 2 constant z lnm 1 and we use the product rule to get 82 l 7 35 8m y m 1 With z held constant we get 2 the constant z lnm 1 y so that 1lnmlgt 32 a mlnz 1 Relation to slicing surfaces with vertical planes The process Hold y constant and treat x as if it s the only variable is the same as slicing surfaces with special vertical planes y constant to get curves and then taking derivatives of the resulting curves with respect to z Look at the following example Start with z 3342 First compute 3z2y2 because we hold y constant Evaluating 37 at the point z ay I gives ab 3a2b2 Second slice the given surface with the vertical plane y b to get the curve z 3 2 Its derivative with respect to the variable x is 3x2 b2 and when we evaluate that derivative at z a we get 3a2b2 which is what we got before An economic application Often the number 2 of gizmos produced is determined by the amount of labor used and the amount of capital invested so 2 fzy where z is the number of hours of labor used and y is the amount of capital used You have heard your economics professors talk about the marginal productivity of labor77 They mean the rate of change of gizmo production if we hold capital expenditure constant and allow the labor variable to change77 In other words then mean 37 The economic term marginal productivity of capital77 is analogously de ned Homework 6 p 405 13579 1119 plus lfz fm y gives the number 2 of gizmos produced if we use z hours of labor and y units of capital what do we mean by the marginal productivity if capital77 IV Candidates for high and low points on surfaces Theorem Suppose z fzy If ab is a high or low point on the surface then ab 0 and ab 0 Therefore to nd candidates for hilltops or valley bottoms on the surface 3 7 3f 7 solve the system of equations E 7 0 and g 7 Example 4 1 Let 2 2 7 y2 7 4x 1 6y 7 5 Then we must solve 0 2x7 4 and 0 72y 1 6 Solving 2x 7 4 0 and 72y 1 6 0 shows that the point 23 is the only possible place where a hilltop or a valley bottom might occur As in single variable calculus it can happen that not every candidate is elected77 In particular we can see that the point 23 gives neither a hilltop nor a valley bottom for the surface Slicing the surface with the vertical plane z 2 gives the curve 2 4 7 y2 7 8 1 6y 7 5 79 1 6y 7 yz which is a parabola opening downwards Therefore if 23 gives a local max or min of the surface it must be a maximum point Next slice with y 3 We get 2 2 7 9 7 4x 18 7 5 2 7 4x 4 which is a parabola opening upwards so if 2 3 is a local max or min it must be a local minimum Those two conclusions contradict each other and we are forced to conclude that 23 does not give a maximum or a minimum on the surface The fact that different slices have different concavity is the problem Example 4 2 lf 2 2 y2 my 1 z 1 2y then the candidates for hill tops and valley bottoms on the surface are found by solving the two equations 0 2x y 1 and 0 2y 1 z 2 The rst gives y 72m 7 1 which we substitute into the second to get 02yx2272x71m274x72m23m and so z 0 Putting z 0 into y 72m 7 1 gives y 71 Therefore our only candidate for max or min ofm is 071 ls the candidate 071 elected Slicing the surface with the vertical planes z 0 and y 71 gives the curves 2 y2 1 2y and z 2 1 7 z z 7 2 and both are parabolas opening upwards so we do not have the same contradiction the we discovered in Example 4 1 But there are other vertical planes through 0 71 that we could have used to slice the surface for example the vertical plane z 7 y 1 ls it true that every possible vertical slice of the surface gives a curve opening upwards That is for tomorrow Homework 7 page 414 13579 plus nd all candidates for 17 19 21 V Second partial derivatives Recall from single variable calculus that for a curve y x we used f z to study the rate of change of y and we used f z to study concavity of the curve Something similar happens with 2 fzy but there are complications Recall that the symbol means Hold y constant and take the derivative with respect to m and the symbol 3 has an analogous meaning Recall that 37 is a formula often involving both z and y That allows us to ask about both 3 a and 3 a 3m 3m 3y 3m 39 Similarly 37 is a formula involving z and y so we can ask about both 3 32 d 3 32 7 7 an 7 7 3m 3y 3y 3y In other words the function 2 m g has two rst derivatives and four second derivatives We compress the notations for second partial derivatives eg 322 3 32 322 3 32 7 7 7 and 7 7 7 3ny 3m 3y 3x2 3m 3m i i i 322 322 i i 322 322 The second partial derivatives W and 3722 are called pure second partials while away and 321 are called mixed second partials Example 5 1 Let 2 Then i 3x2 so that 8273 3 L32 76 32m73x 3m 73m y Ty 322 3 32 3 l 2 2 2 mltiylt 1gty 3 37 71 3 344 so 3532 71 3 y z 73m2y Example 5 2 Still with 2 my we have 2 Note that for the function 2 33 the two mixed second partials were equal That is no accident The following theorem is proved in Math 403 Theorem Suppose m g has continuous second partial derivatives Then the mixed partials of f are equal What good are second partials Recall that is the derivative of the curve obtained by slicing the surface 2 m y with the vertical plane y constant Therefore the pure second partial 373 is just the second derivative of that curve so presumably allows us to study concavity of the curve A similar comment applies to the other pure second partial The use of the mixed second partials is harder to explain but they will appear in the next section Homework 8 page 406 19 23 plus Consider the surface 2 fzy 2 4x 7 3342 12y How do we know that 72 2 is the only point where m y might have a high or low point Now slice the surface with the vertical plane z 72 Discuss the concavity of the resulting curve at the point where z 72y 2 Finally slice the surface with the vertical plane y 2 and discuss the concavity of the resulting curve at the point where z 72 y 2 VI Second partial derivative tests Recall the three basic shapes of parts of curves y fz from single variable calculus In each of the following pictures f 1 0 so that z 1 is a candidate for max or min There are three basic shapes of parts of surfaces 2 fm y that we will encounter We will need some way to distinguish between them The second partial derivatives will be the key Theorem Suppose fzy has continuous partial derivatives and that 1 1 is known to be a candidate for maxmin of z fzy ie suppose 1 1 is a solution of 0 and 0 Find the formula D l 822 822 822 2 w 8727 My Compute Dval1 1 i if Dval1 1 lt 0 then neither max nor min occurs at 1 1 ii if Dval1 1 0 this test will not determine what happens at 1 1 iii if Dval1 1 gt 0 then all slices by vertical planes through 1 1 have the same concavity so that the surface has either a max or a min at 1 1 and you determine which is it by looking at the concavity of the curve obtained by slicing the surface with either z 1 or y 1 ie by looking at the sign of 11 In this case iii a positive pure second partial derivative gives a concave up slice curve and therefore the surface has a valley bottom above 11 and a negative pure second partial derivative gives concave down slice curve and therefore the surface has a hilltop above 1 1 Example 6 1 Consider z 6x312 7 2x3 7 3314 Suppose we have used the rst partial derivative test to determine that the points 1 1 171 and 00 are candidates for maxmin of the surface Use Dval to determine what really happens at these candidates We have 37 6y 7 612 712m 333 12y 2 37 12xy 7 12 121 7 3612 and Dual 712m12z 7 36342 7 12y2 Therefore Dval1 1 71212 7 36 7 144 288 7144 gt 0 so that 11 gives either a max or a min To determine which we note that 1 1 712 so our slices at 1 1 are all concave down and we have a hilltop at 11 Math 108 Notes on Elasticity of Demand Percentage increases Percentage rate of increase Some time ago Coca cola in the second oor vending room of this building cost 75 cents per can Then the price jumped to 1 per can The wrong way to look at this situation is Twenty ve cents is not much money so the increase was small The right way to look at the situation is Dividing the increase by the former price shows a 33 price increase and that s big It s the percentage increase not the actual increase that should worry consumers Whenever you have a function y f x relating two quantities we know that fx gives the rate of change of 2 compared to x The x x f lt gt and percentage rate of change 100 gtk f lt f x f 00 Note that these quantities probably depend on the x value you started with and should really be called the relative and percentage rates of change starting at x relative rate of change Elasticity of Demand Demand for gizmos is sensitive to unit price An increase in price causes a drop in demand Suppose we have a demand equation x f p where p is unit price and x is the number of gizmos that can be sold at price p The percentage rate of change in demand is 100 f p f p This predicts the percentage change in demand corresponding to a price increase of 1 and it is always negative Starting at a given price p raising prices by 1 will result in a percentage increase in price of 100 gtk 1 Economists de ne elasticity of demand starting at price level p to be the ratio E l 100 M E i lpercentage change in demand 7 fp i l 17 f p 7 l percentage change in price 7 I 100 17 I i l p Why use the absolute value Because otherwise every entry in a table of demand elasticities would be negative Note that because our f p lt 0 this is exactly the same as the book s de nition E W Why Care About Elasticity of Demand Recall that our revenue 2 income for selling gizmos is given by R Revenue unit price gtk number sold 17 f p where p is unit price and f p is the demand function Our real interest is in the question Will our revenue rise if we increase prices slightly starting at price level p In other words is R an increasing function near p In still other words is R p gt 0 We use the product rule to nd R p and throw in some algebraic trickery to show how elasticity of demand E is the key At one point we will use the fact that pig 7E Here is the calculation pert17 m 1 fPE1 Rpfp1fpf17 1 Math 108 Notes on Elasticity of Demand Percentage increases Percentage rate of increase A few months ago Cocacola in the second oor vending room of this building cost 75 cents per can Starting in September the price was 1 per can The wrong way to look at this situation is Twenty ve cents is not much money so the increase was small The right way to look at the situation is Dividing the increase by the former price shows a 33 price increase and that s big It s the percentage increase not the actual increase that should worry consumers Whenever you have a function y f x relating two quantities we know that fx gives the rate of change of y compared to x The x x f gt and percentage rate of change 100 gtk f f x f 96 Note that these quantities probably depend on the x value you started with and should really be called the relative and percentage rates of change starting at x relative rate of change Elasticity of Demand Demand for gizmos is sensitive to unit price An increase in price causes a drop in demand Suppose we have a demand equation x f p where p is unit price and x is the number of gizmos that can be sold at price p The percentage rate of change in demand is KP 100 gtllt f f p This predicts the percentage change in demand corresponding to a price increase of 1 and it is always negative Starting at a given price p raising prices by 1 will result in a percentage increase in price of 100 gtk Economists de ne elasticity of demand starting at price level p to be the ratio ercentage change in demand f 100fL IPfP EP l I 100 I fp percentage change in price Why use the absolute value Because otherwise every entry in a table of demand elasticities would be negative Note that because our f p lt 0 this is exactly the same as the book s de nition E W Why Care About Elasticity of Demand Recall that our revenue income for selling gizmos is given by R Revenue unit price gtk number sold pgtkfp where p is unit price and f p is the demand function Our real interest is in the question Will our revenue rise if we increase prices slightly starting at price level p In other words is R an increasing function near p In still other words is Rp gt 0 We use the product rule to nd Rp and throw in some algebraic trickery to show how elasticity of demand E is the key At one point we will use the fact that P3 p f p f p 7E Here is the calculation Rpf p1fpfp 1fpE1 1 Math 108 Introduction to R3 Pictures for this handout will be drawn in class I Coordinates special planes and surfaces in R3 Recall two dimensional graphing We start with two mutually perpendicular axes called the z and y axes that meet at origin denoted 00 We locate a point in the plane by giving two coordinates telling us how to get to the point from origin Consequently 2 3 is the point that we reach by going 2 units in the positive direction along the z axis and then 3 units in the positive direction vertically The plane is often called R2 The x axis is the set of all points m0 where z is allowed to be any real number Similarly the y axis is the set of all points 0 y where y is allowed to be any real number In three dimensional space we have three mutually perpendicular axes called the m y and z axes that meet at origin denoted 000 We locate a point in space by giving three coordinates telling us how to get to the point from origin Consequently we reach the point 23 4 by going 2 units is the positive direction along the z axis then 3 units in the positive y direction then 4 units in the positive 2 direction Three dimensional space is often called R3 The x axis is the set of all points x 00 where z is allowed to be any real number The y axis is the set of all points 0 y where y is any real number The 2 axis is the set of all points 002 where z is any real number The m y plane is the set of all points z y0 where z and y are allowed to be any real numbers The m 2 plane is the set of all points z 0 z where z and z are allowed to be any real numbers The y 2 plane is similarly de ned Drawing things in R3 requires perspective drawing which is a challenge In R2 we graph functions y fz whose domains are pieces of the z axis We get curves oating above or below the z axis and the formula y fz tells us the height of the curve above or below the point z 0 on the z axis In R3 we graph functions 2 fm y whose domains are pieces of the m y plane We get surfaces oating above or below the m y plane and the formula 2 fm y tells us the height of the surface above or below the point z y 0 in the m y plane The plane R2 contains special lines For a constant a the line z a is a line through the point a 0 on the m axis that is parallel to the y axis and contains all points a y for all possible values of y Therefore the graph of z 0 is the y axis For a constant I the line y b is a line through 01 that is parallel to the z axis and contains all points z b for all possible values of m Therefore the graph of y 0 is the z axis Three dimensional space R3 contains special planes For a constant a the graph of z a is a plane through a0 0 containing all points a yz for all possible values of y and 2 Therefore the graph of z 0 is the plane containing origin and the entire y and z axes This is the y 2 plane For a constant I the graph of y b is the plane consisting of all points z bz for all possible values of z and 2 In particular the graph of y 0 is the plane containing origin and the z and z axes namely the m 2 plane The graph of z c is the set of all points z y c and is parallel to the z and y axes Homework 1 In R3 draw the special planes z 2 and y 3 Describe the intersection of these two special planes II Slicing surfaces with special planes General Principle 1 If we intersect a surface S with a plane P we get a curve lying in P We study the surface by studying the intersections of the surface S with the special planes z constant and y constant and z constant The process of intersecting the surface S with the plane P is called slicing the surface S with the plane P Imagine the surface 2 2 y2 1 oating above the zy plane We slice it with special plane z 3 and the points we get must have both z 3 and z 2 y2 1 so we get points satisfying 2 9 y2 1 10 yz We recognize this curve as being a parabola opening upwards In fact for every slice z a we get a curve 2 1 a2 y2 that is a parabola opening upwards Next imagine slicing that same surface with the special plane y 2 We get the curve 2 2 4 1 which is a parabola opening upwards In fact slicing with any special plane y b we get the curve 2 2 b2 1 and that is a parabola opening upwards We could also slice with the horizontal plane 2 c where c is a constant We would get 2 y2 1 c which is 2 y2 c 7 1 In case 0 lt 1 no points satisfy that equation because 2 y2 is always positive while 0 7 1 lt 0 In case 0 1 only z O y 0 satisfy the equation and we conclude that the surface hits the slicing plane 2 1 in the single point 001 In case 0 gt 1 we get 2 y2 1 c which is 2 y2 c 7 1 Because 0 7 1 gt 0 this is a circle with radius xc 7 1 Now the problem is to put together all of those slices to get the surface 2 2 y2 1 This is not easy at rst We will draw the picture in class Homework 2 Describe the curves obtained by slicing the surface 2 3 7 2 7 y2 with the special planes z a where a is a constant What if we slice with the special planes y b What if we slice with the special planes 2 constant Homework 3 Describe what you get if you slice the surface 2 2x 1 3y with the special planes z a where a is any constant What if you slice with the plane y b Pasting together slices was all the ancients could do Today we have software that can graph in R3 so we can get rough ideas of what the surface 2 m y will look like Probably the surface 2 fzy will have hilltops and valley bottoms aka local maxima or local minima or relative maxima or relative minima Our job is to nd them exactly General Principle 2 If the surface 2 fzy has a hilltop directly above the point ab0 and if we slice the surface with the plane y b then we get a curve 2 fzb with variables 2 and z and the value x a must give a local maximum for that function Similarly if we slice our surface with z a we get a curve 2 fay which must have a local maximum at y b There is an analogous principle for valley bottoms on the surface Homework 4 Consider the surface 2 2 2x 7 y2 1 4y and the point 71 2 a Slice the surface with z 71 What curve do you get Does it have a local maximum where y 2 b Slice the surface with the point y 2 What curve do you get Does is have a local maximum where z 71 c Could the given surface have a hilltop directly above 71 2 0 General Principle 3 If the surface surface 2 fzy has a hilltop directly above the point ab0 and if we slice the surface with the plane y b then we get a curve 2 fzb with variables 2 and z and the derivative of z fzb with respect to z must have value zero when x a There is an analogous principle for valley bottoms on the surface Homework 5 State an analog of General Principle 3 if the surface surface 2 fzy has a valley bottom directly above the point a b 0 and we slice the surface with the special plane z a III Partial derivatives Consider the surface 2 m y 3 myy2 and the point a b 0 of its domain Slicing that surface with the plane z a gives us the formula 2 a3 ay yz That is the formula of a curve and we know how to nd if 0 a 2y Alternatively we may slice with the vertical plane y b to get the curve 2 m b 3 bx b2 and we know how to nd 3x2 b As noted in General Principal 3 if a hill top or valley bottom of the surface lies directly above a b 0 then both of these derivatives when evaluated at z a and y I must have value zero But what if we do not know what point a b 0 to consider Suppose our job is to nd the point a b 0 We are now ready to talk about partial derivatives of m y We will do two arti cial looking things that are usually described as Hold z constant and nd the derivative of m y with respect to the variable y or Hold y constant and nd the derivative of fz y with respect to the variable m The rst is denoted by or and the second by or 37 We already know all of the derivative rules that we will ever need The new and tricky part is to be able to say to ourselves Even though z looks like a variable it is held constant or Even though y looks like a variable it is really constant Consider some examples With 2 3 my yz hold y constant Then 3x2 y 0 Next hold z constant in the z formula Then Ox2y Let 2 mezy Then 1 because y is constant and therefore so is 523 Also 37 0 2521 because with z constant its derivative with respect to y is zero Let 2 Then 5 3x2 because with y held constant 21 is also constant so that z is just the constant 21 times the variable part as Also 37 3 71y 2 because with z held constant 2 is just the constant 3 times the variable part y l Let 2 mzemy Hold z constant Then the only variable part of z is 50mmth so we have 2 3 z emy z x cm Next hold y constant Then 2 is the product of two functions of the variable x namely 2 and ec msmm so we need the product rule and we have 37 2x 51y x2 emy y Suppose z mylnz 1 Then has y constant so 2 constant z lnm 1 and we use the product rule to get 82 l 7 35 8m y m 1 With z held constant we get 2 the constant z lnm 1 y so that 1lnmlgt 32 a mlnz 1 Relation to slicing surfaces with vertical planes The process Hold y constant and treat x as if it s the only variable is the same as slicing surfaces with special vertical planes y constant to get curves and then taking derivatives of the resulting curves with respect to z Look at the following example Start with z 3342 First compute 3z2y2 because we hold y constant Evaluating 37 at the point z ay I gives ab 3a2b2 Second slice the given surface with the vertical plane y b to get the curve z 3 2 Its derivative with respect to the variable x is 3x2 b2 and when we evaluate that derivative at z a we get 3a2b2 which is what we got before An economic application Often the number 2 of gizmos produced is determined by the amount of labor used and the amount of capital invested so 2 fzy where z is the number of hours of labor used and y is the amount of capital used You have heard your economics professors talk about the marginal productivity of labor77 They mean the rate of change of gizmo production if we hold capital expenditure constant and allow the labor variable to change77 In other words then mean 37 The economic term marginal productivity of capital77 is analogously de ned Homework 6 p 405 13579 1119 plus lfz fm y gives the number 2 of gizmos produced if we use z hours of labor and y units of capital what do we mean by the marginal productivity if capital77 IV Candidates for high and low points on surfaces Theorem Suppose z fzy If ab is a high or low point on the surface then ab 0 and ab 0 Therefore to nd candidates for hilltops or valley bottoms on the surface 3 7 3f 7 solve the system of equations E 7 0 and g 7 Example 4 1 Let 2 2 7 y2 7 4x 1 6y 7 5 Then we must solve 0 2x7 4 and 0 72y 1 6 Solving 2x 7 4 0 and 72y 1 6 0 shows that the point 23 is the only possible place where a hilltop or a valley bottom might occur As in single variable calculus it can happen that not every candidate is elected77 In particular we can see that the point 23 gives neither a hilltop nor a valley bottom for the surface Slicing the surface with the vertical plane z 2 gives the curve 2 4 7 y2 7 8 1 6y 7 5 79 1 6y 7 yz which is a parabola opening downwards Therefore if 23 gives a local max or min of the surface it must be a maximum point Next slice with y 3 We get 2 2 7 9 7 4x 18 7 5 2 7 4x 4 which is a parabola opening upwards so if 2 3 is a local max or min it must be a local minimum Those two conclusions contradict each other and we are forced to conclude that 23 does not give a maximum or a minimum on the surface The fact that different slices have different concavity is the problem Example 4 2 lf 2 2 y2 my 1 z 1 2y then the candidates for hill tops and valley bottoms on the surface are found by solving the two equations 0 2x y 1 and 0 2y 1 z 2 The rst gives y 72m 7 1 which we substitute into the second to get 02yx2272x71m274x72m23m and so z 0 Putting z 0 into y 72m 7 1 gives y 71 Therefore our only candidate for max or min ofm is 071 ls the candidate 071 elected Slicing the surface with the vertical planes z 0 and y 71 gives the curves 2 y2 1 2y and z 2 1 7 z z 7 2 and both are parabolas opening upwards so we do not have the same contradiction the we discovered in Example 4 1 But there are other vertical planes through 0 71 that we could have used to slice the surface for example the vertical plane z 7 y 1 ls it true that every possible vertical slice of the surface gives a curve opening upwards That is for tomorrow Homework 7 page 414 13579 plus nd all candidates for 17 19 21 V Second partial derivatives Recall from single variable calculus that for a curve y x we used f z to study the rate of change of y and we used f z to study concavity of the curve Something similar happens with 2 fzy but there are complications Recall that the symbol means Hold y constant and take the derivative with respect to m and the symbol 3 has an analogous meaning Recall that 37 is a formula often involving both z and y That allows us to ask about both 3 a and 3 a 3m 3m 3y 3m 39 Similarly 37 is a formula involving z and y so we can ask about both 3 32 d 3 32 7 7 an 7 7 3m 3y 3y 3y In other words the function 2 m g has two rst derivatives and four second derivatives We compress the notations for second partial derivatives eg 322 3 32 322 3 32 7 7 7 and 7 7 7 3ny 3m 3y 3x2 3m 3m i i i 322 322 i i 322 322 The second partial derivatives W and 3722 are called pure second partials while away and 321 are called mixed second partials Example 5 1 Let 2 Then i 3x2 so that 8273 3 L32 76 32m73x 3m 73m y Ty 322 3 32 3 l 2 2 2 mltiylt 1gty 3 37 71 3 344 so 3532 71 3 y z 73m2y Example 5 2 Still with 2 my we have 2 Note that for the function 2 33 the two mixed second partials were equal That is no accident The following theorem is proved in Math 403 Theorem Suppose m g has continuous second partial derivatives Then the mixed partials of f are equal What good are second partials Recall that is the derivative of the curve obtained by slicing the surface 2 m y with the vertical plane y constant Therefore the pure second partial 373 is just the second derivative of that curve so presumably allows us to study concavity of the curve A similar comment applies to the other pure second partial The use of the mixed second partials is harder to explain but they will appear in the next section Homework 8 page 406 19 23 plus Consider the surface 2 fzy 2 4x 7 3342 12y How do we know that 72 2 is the only point where m y might have a high or low point Now slice the surface with the vertical plane z 72 Discuss the concavity of the resulting curve at the point where z 72y 2 Finally slice the surface with the vertical plane y 2 and discuss the concavity of the resulting curve at the point where z 72 y 2 VI Second partial derivative tests Recall the three basic shapes of parts of curves y fz from single variable calculus In each of the following pictures f 1 0 so that z 1 is a candidate for max or min There are three basic shapes of parts of surfaces 2 fm y that we will encounter We will need some way to distinguish between them The second partial derivatives will be the key Theorem Suppose fzy has continuous partial derivatives and that 1 1 is known to be a candidate for maxmin of z fzy ie suppose 1 1 is a solution of 0 and 0 Find the formula D l 822 822 822 2 w 8727 My Compute Dval1 1 i if Dval1 1 lt 0 then neither max nor min occurs at 1 1 ii if Dval1 1 0 this test will not determine what happens at 1 1 iii if Dval1 1 gt 0 then all slices by vertical planes through 1 1 have the same concavity so that the surface has either a max or a min at 1 1 and you determine which is it by looking at the concavity of the curve obtained by slicing the surface with either z 1 or y 1 ie by looking at the sign of 11 In this case iii a positive pure second partial derivative gives a concave up slice curve and therefore the surface has a valley bottom above 11 and a negative pure second partial derivative gives concave down slice curve and therefore the surface has a hilltop above 1 1 Example 6 1 Consider z 6x312 7 2x3 7 3314 Suppose we have used the rst partial derivative test to determine that the points 1 1 171 and 00 are candidates for maxmin of the surface Use Dval to determine what really happens at these candidates We have 37 6y 7 612 712m 333 12y 2 37 12xy 7 12 121 7 3612 and Dual 712m12z 7 36342 7 12y2 Therefore Dval1 1 71212 7 36 7 144 288 7144 gt 0 so that 11 gives either a max or a min To determine which we note that 1 1 712 so our slices at 1 1 are all concave down and we have a hilltop at 11 Math 108 Notes on Elasticity of Demand Percentage increases Percentage rate of increase Some time ago Coca cola in the second oor vending room of this building cost 75 cents per can Then the price jumped to 1 per can The wrong way to look at this situation is Twenty ve cents is not much money so the increase was small The right way to look at the situation is Dividing the increase by the former price shows a 33 price increase and that s big It s the percentage increase not the actual increase that should worry consumers Whenever you have a function y f x relating two quantities we know that fx gives the rate of change of 2 compared to x The x x f lt gt and percentage rate of change 100 gtk f lt f x f 00 Note that these quantities probably depend on the x value you started with and should really be called the relative and percentage rates of change starting at x relative rate of change Elasticity of Demand Demand for gizmos is sensitive to unit price An increase in price causes a drop in demand Suppose we have a demand equation x f p where p is unit price and x is the number of gizmos that can be sold at price p The percentage rate of change in demand is 100 f p f p This predicts the percentage change in demand corresponding to a price increase of 1 and it is always negative Starting at a given price p raising prices by 1 will result in a percentage increase in price of 100 gtk 1 Economists de ne elasticity of demand starting at price level p to be the ratio E l 100 M E i lpercentage change in demand 7 fp i l 17 f p 7 l percentage change in price 7 I 100 17 I i l p Why use the absolute value Because otherwise every entry in a table of demand elasticities would be negative Note that because our f p lt 0 this is exactly the same as the book s de nition E W Why Care About Elasticity of Demand Recall that our revenue 2 income for selling gizmos is given by R Revenue unit price gtk number sold 17 f p where p is unit price and f p is the demand function Our real interest is in the question Will our revenue rise if we increase prices slightly starting at price level p In other words is R an increasing function near p In still other words is R p gt 0 We use the product rule to nd R p and throw in some algebraic trickery to show how elasticity of demand E is the key At one point we will use the fact that pig 7E Here is the calculation pert17 m 1 fPE1 Rpfp1fpf17 1

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