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- Math 116 - Chapter 8 - Calculus of Several Variables

Jake L.

Function of 2 variables

A function that consists of:

1) A set A of ordered pairs of real numbers (x,y) called the domain.

2) A rule that associates with each ordered pair in the domain of ? one and only one real number, denoted by z=f(x,y)

1) A set A of ordered pairs of real numbers (x,y) called the domain.

2) A rule that associates with each ordered pair in the domain of ? one and only one real number, denoted by z=f(x,y)

Domain

A set A of ordered pairs of real numbers (x,y)

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3-D Cartesian Coordinate System

a coordinate system with 3 axes which are mutually perpendicular

Level Curve

If c is some value of the function ?, then the equation f(x,y)=c describes a curve lying on the plane z = c called the trace of the graph of ? in the plane z=c. If this trace is projected onto the xy-plane, the resulting curve in the xy-plane is called this.

First partial Derivative

?f/?x=lim h-> f(x+h, y)-f(x,y)/h

and

?f/?y=lim h-> f(x, y+k)-f(x,y)/h

and

?f/?y=lim h-> f(x, y+k)-f(x,y)/h

Cobb-Douglas Production Function

?(x,y)=ax^b y^(1-b) where a and b are positive constants with 0<b<1. x: amount of money expended for labor

y: cost of capital equipment

function ?: output of the finished product.

y: cost of capital equipment

function ?: output of the finished product.

Marginal Productivity of Labor

Measures the rate of change of production w/respect to amount of money spent on labor.

Marginal Productivity of Capital

Measures the rate of change of production w/respect to amount of money spent on capital.

Substitute Commodities

A decrease in the demand for one results in an increase in the demand for the other.

Complementary Commodities

A decrease in the demand for one results in a decrease in the demand for the other as well.

Second-order Partial Derivative

The derivative of the first partial derivatives.

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Relative Maximum

A point (a,b) of the function ? where f(x,y)?f(a,b) for all points (x,y).

Relative Maximum Value

The number f(a,b)

Relative Minimum

A point (a,b) of the function ? where f(x,y)?f(a,b) for all points (x,y).

Relative Minimum Value

The number f(a,b) which the minimum occurs at.

Absolute Maximum

The point (a,b) where a,b,f(a,b) is the highest point for all points (x,y) in the domain of ?.

Absolute Minimum

The point (a,b) where a,b,f(a,b) is the lowest point for all points (x,y) in the domain of ?.

Absolute Maximum Value

The value f(a,b) of the absolute maximum

Absolute Minimum Value

The value f(a,b) of the absolute minimum

Saddle Point

A point (a,b,f(a,b)) that is neither a relative maximum nor a relative minimum at the point (a,b)

Critical Point

a point (a,b) in the domain of ? such that both ??/?x(a,b)=0 and ??/?y(a,b)=0 or at least one of the partial derivatives does not exist.

Second Derivative Test

D(x,y)=?xx?yy-?xy^2

If D(a,b)>0;fxx(a,b)<0 then f(x,y) is a relative max

If D>0;fxx>0, f(x,y) is a rel. min

If D<0 then f(x,y) is a saddle point

If D=0, the test is inconclusive

If D(a,b)>0;fxx(a,b)<0 then f(x,y) is a relative max

If D>0;fxx>0, f(x,y) is a rel. min

If D<0 then f(x,y) is a saddle point

If D=0, the test is inconclusive

Method of Least Squares

A general method for determining a straight line that best fits a set of data points when the points are scattered about a straight line.

Scatter Diagram

A plot of data points used to illustrate the method of least squares

Least Squares Line (Regression Line)

The line L obtained by minimizing the sum of the squares of the errors in the least squares method

Normal Equation

The equations:

(1) (x1^2+x2^2)m+(x1+x2)b = x1y1+x2y2+xnyn

(2) (x1+x2+..xn)m+nb = y1 + y2 +...+yn

(1) (x1^2+x2^2)m+(x1+x2)b = x1y1+x2y2+xnyn

(2) (x1+x2+..xn)m+nb = y1 + y2 +...+yn

Constrained Relative Extremum

The relative extrema of a function (x,y) whose independent variables x and y are required to satisfy one or more consraints of the form g(x,y)=0.

Method of Lagrange Multipliers

Used to find the relative extremum of a function subject to a constraint g(x,y)=0.

Total Differential

let z=f(x,y) define a differentiable function of x and y.

1. The differentials of the independent variables x and y are dx=?x and dy=?y

2. The differential of the dependent variable z is dz=(?f/?x)dx+(??/?y)dy

1. The differentials of the independent variables x and y are dx=?x and dy=?y

2. The differential of the dependent variable z is dz=(?f/?x)dx+(??/?y)dy

Riemann Sum

S=?(p1)h+?(p2)h+...?(pn)h

Double Integral

??f(x,y)dA

Volume of a Solid Under a Surface

V=??f(x,y)dA

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