Math 1314 Lesson 23 Partial Derivatives

Transcription

1 Math 1314 Lesson 3 Partial Derivatives When we are asked to ind the derivative o a unction o a single variable, (x), we know exactl what to do However, when we have a unction o two variables, there is some ambiguit 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 onl consider two directions, either parallel to the x axis or parallel to the 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, 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 These derivatives are called partial derivatives We will use two dierent notations: First-Order Partial Derivatives x In this case, ou consider as a constant In this case, ou 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, ) x 3x + 4 Enter the unction into GGB a Find x b Find Lesson 3 Partial Derivatives 1

2 3 Example : Suppose ( x, ) 5x x + 9x Enter the unction into GGB a Find x b Find We can also evaluate the irst partial derivatives at a given point Example 3: Find the irst-order partial derivatives o the unction 3 3 ( x, ) 4x + x 1x evaluated at the point (-1, 3) Enter the unction into GGB a x ( 1,3) b ( 1,3) Lesson 3 Partial Derivatives

3 Second-Order Partial Derivatives Sometimes we will need to ind the second-order partial derivatives To ind a second-order partial derivative, ou 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 onl use one o them xx x x x Example 4: Find the second-order partial derivatives o the unction ( x, ) 3x 5x + 10 Enter the unction into GGB a Find x b Find xx c Find x d Find e Find Find x Lesson 3 Partial Derivatives 3

4 3 3 3 Example 5: Evaluate the second-order partial derivatives o ( x, ) 3x x + 5x + 6 at the point (1, ) Enter the unction into GGB Then produce the irst-order partials a xx b x c d x Lesson 3 Partial Derivatives 4

5 b 1 b A unction o the orm ( x, ) ax 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 mone spent or labor, and represents the amount o mone spent on capital expenditures such as actories, equipment, machiner, tools, etc The unction measures the output o inished products The irst partial with respect to x is called the marginal productivit o labor It measures the change in productivit with respect to the amount o mone spent or labor In inding the irst partial with respect to x, the amount o mone spent on capital is held at a constant level The irst partial with respect to is called the marginal productivit o capital It measures the change in productivit with respect to the amount o mone spent on capital expenditures In inding the irst partial with respect to, the amount o mone spent on labor is held at a constant level Example 6: A countr s production can be modeled b the unction /3 1/3 (, ) 50 x x x gives the units o labor that are used and represents the units o capital that were used a Find the irst-order partial derivatives and label each as marginal productivit o labor or marginal productivit o capital Enter the unction into GGB where b Find the marginal productivit o labor and the marginal productivit o capital when the amount expended on labor is 15 units and the amount spent on capital is 7 units Lesson 3 Partial Derivatives 5