P.1 Graphs and Models

The Graph of an Equation

Rene Descartes (1596-1650) - Descartes made many contributions to philosophy, science,

and mathematics.

• The idea of representing points in the plane by pairs of real numbers and representing curves in the plane by equations was described by Descartes in his book La Géométrie, published in 1637.

• With Descartes’s coordinate plane, geometric concepts could be formulated analytically and algebraic concepts could be viewed graphically.

• The power of this approach was such that within a century of its introduction, much of calculus had been developed.

Perspectives in Calculus

• The same approach can be followed in your study of calculus. That is, by viewing calculus from multiple perspectives—graphically, analytically, and numerically—you will increase your understanding of core concepts.

Example

Consider the equation: 3x + y = 7

Analytic Approach

The point is a solution point of the equation because the equation is satisfied (is true) when 2 is substituted for and 1 is substituted for This equation has many other solutions, such as and To find other solutions systematically, solve the original equation for y.

y = 7 - 3x

Numerical Approach

Graphical Approach

P.2 Linear Models and Rates of Change

P.3 Functions and Their Graphs

P.4 Fitting Models to Data