Foundations of Economics and Linear Functions - Study Notes

Foundations in Mathematics: Number Lines, Points, and Coordinates

• Representations of a number on a number line:

  • Three representations: (i) as a number; (ii) as a point on the number line; (iii) as a coordinate/location on the line.

• Basic components of a one-dimensional number line:

  • Zero (origin), one (a unit distance to the right), and the positive direction (to the right).

  • Any number can be located on the line by these three pieces of information.

• Points on a line and coordinates:

  • A point on the line (denoted P) has a coordinate (label) and is located at a specific distance from the origin.

  • The distance from origin to P is measured in units; if the distance is one unit, the coordinate is 1; a diachronous labeling is possible (e.g., P, Q).

• Two-dimensional coordinates:

  • A point on a plane is represented by a pair (x, y) obtained by the intersection of a vertical line (x = constant) and a horizontal line (y = constant).

  • The idea generalizes to three dimensions with three coordinates (x, y, z).

• Three-number representation for space:

  • In three dimensions, a point requires three numbers: (x, y, z).

• Implication for representing points:

  • To locate any point, you need a pair of coordinates in 2D, or three coordinates in 3D, derived from intersecting vertical and horizontal reference lines.

Slope, Rate of Change, and Lines

• Two-point representation and the slope:

  • Given two points, the slope of the line through them is defined as

  • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

  • The numerator is the change in y ($\Delta y$) and the denominator is the change in x ($\Delta x$).

• Delta notation:

  • $\Delta x = x<em>2 - x</em>1, \quad \Delta y = y<em>2 - y</em>1$

  • The slope is the ratio of the changes: $m = \frac{\Delta y}{\Delta x}$

• Rise over run and the idea of rate of change:

  • Rise = change in y; Run = change in x; slope is a measure of how y changes as x changes.

  • The slope represents the rate of change: how much y changes per unit change in x.

• Independence of point labeling:

  • If you switch which point you call $(x<em>1,y</em>1)$ and $(x<em>2,y</em>2)$, the slope remains the same (the calculation is invariant to labeling).

  • Algebraically, the expression is equivalent if you rewrite as $(y<em>1 - y</em>2)/(x<em>1 - x</em>2)$; simplifying shows the same result.

• Constant slope and linearity:

  • For a straight line, the slope is constant regardless of which two points on the line you choose.

  • This constancy is what makes linear functions predictable and is the foundation for later calculus of rate-of-change concepts.

• Real-world interpretation of rate of change:

  • Example: pricing and cost. If a computer costs $500$, the total cost for n computers is

  • $\text{Cost}(n) = 500 n$

  • For $n = 10$, $\text{Cost}(10) = 500 \times 10 = 5000$

• Velocity as rate of change:

  • If you drive at a constant $70$ miles per hour, the distance after t hours is

  • $d = v t = 70 t$

  • After $2$ hours, the distance is $d = 70 \times 2 = 140$ miles.

• Non-constant rates and calculus:

  • In many real-world scenarios, rates of change are not constant; calculus studies changing rates over time.

  • Stock prices, for example, can rise and fall unpredictably, making future prediction more complex.

• What the slope tells you:

  • The slope is a measure of change per unit of x; in a function, it captures the rate at which y responds to changes in x.

• Important takeaway about linear functions:

  • For a linear function, knowing two points is enough to determine the full line, because the slope is constant.

Equations of Lines: Forms and Interrelationships

• Slope-intercept form:

  • The equation of a line with slope m and y-intercept b is

  • $y = m x + b$

  • The y-intercept b is the value of y when $x = 0$, i.e., the point where the line crosses the y-axis.

• Intercept concept:

  • The x-intercept is the point where the line crosses the x-axis ($y = 0$).

• Point-slope form:

  • If you know a slope m and a single point $(x<em>1, y</em>1)$ on the line, you can write the line as

  • $y - y<em>1 = m (x - x</em>1)$

• Deriving the slope from point coordinates (slope from two points):

  • Given two points on a line, the slope is

  • $m = \frac{y - y<em>1}{x - x</em>1}$ or equivalently using $(x<em>2, y</em>2)$ and $(x<em>1, y</em>1)$:

  • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Conceptual note: two-point form and slope-intercept form are equivalent ways to describe the same line; choosing between them depends on available information.

• Special cases:

  • Vertical line: $\Delta x = 0 \Rightarrow m \text{ is undefined}$

  • Horizontal line: $m = 0$, y is constant along the line.

• Practical use: to sketch or analyze a line you can:

  • pick two points on the line and connect them, or

  • use the slope with a known point, or

  • use the slope-intercept form if you know m and b, or

  • use the point-slope form if you know m and a single point.

• Parallelism and slope equivalence:

  • Lines with the same slope are parallel (and do not intersect unless they are the same line, i.e., have the same intercept).

• Practical memorization and understanding:

  • A key takeaway is that many problems reduce to identifying the slope (rate of change) and a point or intercept to determine the line.

Special Case: Putting It All Together

• Two points determine a line with a constant slope; different pairs of points on the same line give the same slope.

• The two most commonly used forms for a line are:

  • Slope-intercept: $y = m x + b$

  • Point-slope: $y - y<em>1 = m (x - x</em>1)$

• When given a slope and a single point, you can construct the full equation of the line (even if that point is not on the y-axis).

• When given two points, you can compute the slope with

  • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Then either plug into slope-intercept form or use point-slope form to obtain the equation.

Recap: Rate of Change and the Path to Calculus

• The central concept is rate of change: how quickly one quantity changes with respect to another.

• For a straight line, the rate of change is constant; for nonlinear relationships, the rate of change can vary, which motivates calculus.

• The connection to real life:

  • In finance (price, cost) and physics (velocity), rate of change is central to modeling and predictions.

• The course emphasizes the algebraic tools (slope, line equations) as building blocks for calculus, where rates of change become functions that can vary over domains.

Quick Memoranda and Key Takeaways

• Always remember the core slope formula:

  • $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• For a line, you can describe it by either:

  • Slope-intercept form: $y = m x + b$, with $b = y|_{x=0}$

  • Point-slope form: $y - y<em>1 = m (x - x</em>1)$

• Special slopes:

  • Horizontal lines: slope = $0$

  • Vertical lines: slope undefined (since $\Delta x = 0$)

• The slope is a visual and numerical measure of the rate of change; it remains constant along a straight line.

• Real-world analogy examples help intuition:

  • Cost grows linearly with quantity: $\text{Cost}(n) = 500n$

  • Distance grows linearly with time at constant velocity: $d = v t$

• Understanding two points is enough to define a line; the line’s equation uniquely captures the relationship between x and y on that line.

• The educational aim: Mastery of these linear concepts lays groundwork for studying non-linear relationships and the calculus of rates of change.