Math 1314 Lesson 3 Partial Derivatives When we are asked to ind the derivative o a unction o a single variable, (x), we know exactly what to do However, when we have a unction o two variables, there is some ambiguity With a unction o two variables, we can ind the slope o the tangent line at a point P rom an ininite number o directions We will only consider two directions, either parallel to the x axis or parallel to the y axis When we do this, we ix one o the variables Then we can ind the derivative with respect to the other variable So, i we ix y, we can ind the derivative o the unction with respect to the variable x And i we ix x, we can ind the derivative o the unction with respect to the variable y These derivatives are called partial derivatives We will use two dierent notations: First-Order Partial Derivatives x y x y In this case, you consider y as a constant In this case, you consider x as a constant We can use GGB to determine the irst-order partial derivatives The command is: derivative[<unction>,<variable>] Example 1: Suppose ( x, y) x 3xy + 4y Enter the unction into GGB a Find x x b Find y y Lesson 3 Partial Derivatives 1
We can also evaluate the irst partial derivatives at a given point Example : Find the irst-order partial derivatives o the unction 3 3 ( x, y) 4x y + x y 1x + 3y + 10 evaluated at the point (-1, 3) Enter the unction into GGB a x ( 1,3) b y ( 1,3) Second-Order Partial Derivatives Sometimes we will need to ind the second-order partial derivatives To ind a second-order partial derivative, you will take respective partial derivatives o the irst partial derivative There are a total o 4 second-order partial derivatives There are two notations, but we will only use one o them x xx x x x y xy y x y yy y y y x yx x y Lesson 3 Partial Derivatives
3 3 3 Example 3: Evaluate the second-order partial derivatives o ( x, y) 3x x y + 5xy + 6y at the point (1, ) Enter the unction into GGB Then produce the irst-order partials a xx ( 1,) b xy ( 1,) c yy ( 1,) d yx ( 1,) Lesson 3 Partial Derivatives 3
A unction o the orm ( x, y) b 1 b ax y where a and b are positive constants and 0 b 1 < < is called a Cobb-Douglas production unction In this unction, x represents the amount o money spent or labor, and y represents the amount o money spent on capital expenditures such as actories, equipment, machinery, tools, etc The unction measures the output o inished products The irst partial with respect to x is called the marginal productivity o labor It measures the change in productivity with respect to the amount o money spent or labor In inding the irst partial with respect to x, the amount o money spent on capital is held at a constant level The irst partial with respect to y is called the marginal productivity o capital It measures the change in productivity with respect to the amount o money spent on capital expenditures In inding the irst partial with respect to y, the amount o money spent on labor is held at a constant level Example 4: A country s production can be modeled by the unction ( ) x, y 50x y /3 1/3 where x gives the units o labor that are used and y represents the units o capital that were used a Find the irst-order partial derivatives and label each as marginal productivity o labor or marginal productivity o capital Enter the unction into GGB b Find the marginal productivity o labor and the marginal productivity o capital when the amount expended on labor is 15 units and the amount spent on capital is 7 units Lesson 3 Partial Derivatives 4
Math 1314 Lesson 4 Maxima and Minima o Functions o Several Variables We learned to ind the maxima and minima o a unction o a single variable earlier in the course We had a second derivative test to determine whether a critical point o a unction o a single variable generated a maximum or a minimum, or possibly that the test was not conclusive at that point We will use a similar technique to ind relative extrema o a unction o several variables Since the graphs o these unctions are more complicated, determining relative extrema is also more complicated At a speciic critical number, we can have a max, a min, or something else That something else is called a saddle point The method or inding relative extrema is very similar to what you did earlier in the course 1 Find the irst partial derivatives and set them equal to zero You will have a system o equations in two variables which you will need to solve to ind the critical points Apply the second derivative test To do this, you must ind the second-order partial derivatives Let D x, y) ( ) You will compute D( a, b) or each critical point ( xx yy xy ( a, b) Then you can apply the second derivative test or unctions o two variables: I D(a, b) > 0 and ( a, b) < 0, then has a relative maximum at (a, b) xx I D(a, b) > 0 and ( a, b) > 0, then has a relative minimum at (a, b) xx I D(a, b) < 0, then has neither a relative maximum nor a relative minimum at (a, b) (ie, it has a saddle point, which is neither a max nor a min) I D(a, b) 0, then this test is inconclusive Lesson 4 Maxima and Minima o Functions o Several Variables 1
Example 1: Find the relative extrema o the unction Begin by entering the unction into GGB a Find the irst-order partials x y x xy y x y (, ) 3 + + 14 + 8 b Set each irst-order partial equal to zero and enter each into GGB c Find the point o intersection o the equations in part B These points o intersection are the critical points o the unction d Determine D ( x, y ) ( ) xx yy xy e Apply the second derivative test to classiy each critical point ound in step C For any maxima point and minima point ound in part E, calculate the maxima and minima, respectively Lesson 4 Maxima and Minima o Functions o Several Variables
3 Example : Find the relative extrema o the unction ( x, y) x + y 9x 4y + 1x Begin by entering the unction into GGB a Find the irst-order partials b Set each irst-order partial equal to zero and enter each into GGB c Find the point o intersection o the equations in part B These points o intersection are the critical points o the unction d Determine D ( x, y ) ( ) xx yy xy e Apply the second derivative test to classiy each critical point ound in step C For any maxima point and minima point ound in part E, calculate the maxima and minima, respectively Lesson 4 Maxima and Minima o Functions o Several Variables 3
Example 3: Suppose a company s weekly proits can be modeled by the unction P( x, y) 0x 05y 0xy + 100x + 90y 4000 where proits are given in thousand dollars and x and y denote the number o standard items and the number o deluxe items, respectively, that the company will produce and sell How many o each type o item should be manuactured each week to maximize proit? What is the maximum proit that is realizable in this situation? Begin by entering the unction into GGB a Find the irst-order partials b Set each irst-order partial equal to zero and enter each into GGB c Find the point o intersection o the equations in part B These points o intersection are the critical points o the unction d Determine D ( x, y ) ( ) xx yy xy Lesson 4 Maxima and Minima o Functions o Several Variables 4
e Apply the second derivative test to classiy each critical point ound in step C How many o each type o item should be manuactured each week to maximize proit? g What is the maximum proit that is realizable in this situation? Lesson 4 Maxima and Minima o Functions o Several Variables 5