In the late 1900s, the relationship among the domestic...

In the late 1900s, the relationship among the domestic market shares of three major manufacturers of breakfast cereal was $$ x_{1}=-0.4+1.2 x_{2}+2 x_{3} $$ where $x_{1}, x_{2}$, and $x_{3}$ are, respectively, the fractions of the market held by Kellogg, General Mills, and General Foods. ${ }^{4}$ Thinking of Kellogg's market share as a function of shares of the other two manufacturers, describe the graph of the resulting function. How are the different slices by $x_{2}=$ constant related to one another? What does this say about market share?

In the late 1900s, the relationship among the domestic market shares of three major manufacturers of breakfast cereal was
$$
x_{1}=-0.4+1.2 x_{2}+2 x_{3}
$$
where $x_{1}, x_{2}$, and $x_{3}$ are, respectively, the fractions of the market held by Kellogg, General Mills, and General Foods. ${ }^{4}$ Thinking of Kellogg's market share as a function of shares of the other two manufacturers, describe the graph of the resulting function. How are the different slices by $x_{2}=$ constant related to one another? What does this say about market share?

Key Concepts

Parallel Slices and Consistent Relationships

Given the linearity of the function, slices created by setting different constant values for one variable result in lines that are parallel to each other. The parallelism indicates that the rate of change of the dependent variable with respect to the other independent variable remains the same regardless of the fixed value. This reflects a consistent additive effect within the system and implies that the influence of each independent variable on the outcome is stable across different scenarios.

Multivariable Linear Function

A multivariable linear function describes a relationship in which a dependent variable is expressed as a linear combination of two or more independent variables plus a constant. Graphically, when plotted with two independent variables on the horizontal axes and the dependent variable on the vertical axis, the result is a plane. This concept is foundational in linear algebra and multivariable calculus, as it captures how changes in the independent variables affect the outcome in a predictable, additive way.

Graphical Representation of Planes

In the context of functions with two independent variables, the resulting graph is a plane in three-dimensional space. The coefficients of the independent variables determine the slopes of the plane along each axis, while the constant term sets the plane's intercept with the dependent variable axis. Understanding this plane is crucial because it models how the underlying system behaves over a continuous range of inputs.

Slicing or Cross-Section Analysis

Slicing refers to the technique of fixing one independent variable at a constant value, which reduces the multivariable function to a single-variable linear function. Each fixed slice appears as a line in the plane, showing how the dependent variable changes with the remaining independent variable. This method is useful to analyze the marginal effect of one variable while keeping others constant.

Transcript

00:01 In this problem of functions of several variables we have given a problem based on market share in 90s.

00:07 So we can say this is market share in 1900s.

00:12 So this is 1900s and between kellogg's, general mills and general foods.

00:18 So this is between kellogg's and then general mills.

00:26 So we can say this is general meals and then general foods.

00:32 So, this is general foods.

00:37 In the late 90s, the relationship amongst the domestic market share of three major manufacture of the breakfast cereal was, say we can say x1 is equals to minus 0 .4 plus 1 .2 multiplied with x2 plus twice of x3, where x1, x2 and x3 are respectively the fractions of the market held by calox so calox is represented by x1 general meals so this is x2 and general food by x3 thinking of calox market share as a function of share of the other two manufacturers describe the graph of the resulting function so this is the equation that is x1 is for galox x2 is for general meals and x3 is for general food so as you can see this is the you can see the degree of the polyneumon is equal to one that means this would represent a plane so we can say the graph would be so we can say graph is a plane and if this is a plane so we can find x1 intercept x2 intercept and x3 intercept so x1 intercept so we can say x1 intercept so for this we have to put x2 and x3 both are equal to 0 so putting x2 and x3 equals to 0 we get x1 intercept as xg equals to minus 0 now we can find x2 intercept, so x2 intercept.

02:14 So for this we have to put x3 and x1 both equals to 0.

02:17 So x1 and x3 putting both equals to 0.

02:20 So we get x2 is equal to 0 .4 divided with 1 .2 which is equals to 0 .33.

02:31 Now x3 intercept, so x3 intercept.

02:35 So for this we have to put x1 and x3.

02:38 X2 both equals to 0.

02:40 So putting x1 and x2 both equals to 0 so we get x3 is equals to 0 .4 divided with 2 so this is equals to this is 0 .2.

02:52 So we have graph is a plane with intercept or we can say having say we can say x1 intercept say x1 intercept as x1 is equal to minus 0 .4.

03:10 Now x2 intercept...