Course Notes for Calculus I

Analytic geometry was a huge breakthrough for mathematics. Prior to the 17th century, algebra and geometry were haphazardly connected branches of mathematics. Various attempts had been made to give algebraic descriptions to geometric objects (including some primitive versions of cartesian coordinates long before Descartes), but none of these ideas and systems had managed to produce a systematic and thorough connection. In the 17th century, Descartes proposed the coordinate system which now bears his name: the cartesian coordinates. Assigning numerical values to points in the plane (2 dimensional) or space (3 dimensional) allowed geometric problems to be interpreted algebraically, and vice-versa. Moreover, this connection was complete and systematic: in theory, any geometry in the plane or space could be described by Descartes’ coordinates. This breakthrough is fundamental to the use of mathematics in the sciences, starting in the early modern age and continuing into the present.

The basic idea of cartesian coordinates should be familiar to you from your previous mathematical experience. In the plane, cartesian coordinates are formed by choosing a center point, drawing two perpendicular directions called axes (usually, but not necessarily, labelled \(x\) and \(y\)), and describing each point in the plane by two real numbers, written \((a,b)\text{,}\) representing how far along each axis the point falls. By convention, moving right and upward are considered positive directions on the axes.

In the definition I’ve given, the coordinates \(a\) and \(b\) are real numbers. Though sometimes only integer or rational coordinates are required, the system allows for any real numbers to represent coordinates. Since there are two dimensions in the plane, I can refer to the plane with cartesian coordinates as \(\RR^2\text{.}\) Similarly, the space with cartesian coordinates \((a,b,c)\text{,}\) is written \(\RR^3\text{.}\) The axes in \(\RR^3\) are typically called \(x\text{,}\) \(y\) and \(z\text{.}\)

Though I won’t deal with higher dimensions in this course, the cartesian coordinate system generalizes to any number of dimensions. I can no longer visualize or draw the geometry in higher dimensions, but I can still work with the algebraic representation according to the same principles. For example, in \(\RR^5\text{,}\) there are five axes, all perpendicular to each other, and points are represented by five real numbers \((a,b,c,d,e)\text{,}\) showing the distance along each of the axes. I can write \(\RR^n\) for a cartesian system in general \(n\) dimensions. This is one of the most powerful aspects of cartesian coordinates — I can now do geometry in as many dimensions as I wish, transcending the limitations of three-dimensional vision and perception.

Cartesian coordinates are useful for giving algebraic definitions for various geometric shapes and objects. For an equation in \(x\) and \(y\text{,}\) the coresponding shape consists of all points \((a,b)\) in the plane which satisfy the equation when \(x\) is replaced by \(a\) and \(y\) is replaced by \(b\text{.}\) Such an collection of points is called locus of the equation. The plural of locus is loci. In Figure 1.1.2, the familiar circle of radius \(1\) is the locus of the equation \(x^2 + y^2 = 1\text{.}\)

The two most important classes of loci are lines (see Section 1.2) and conics (see Section 1.3). Lines are first both because their geometry is approachable and because they are loci of the most basic kind of equations. Conics follow as loci of the next most approachable kind of equation.

Lines and conics are just two of many classes and families of loci in \(\RR^2\text{.}\) The term ‘curve’ is a general term for loci of equations in \(\RR^2\text{.}\) Many equations give rise to complicated and unpredicable curves. Future calculus courses will study the calculus of curves. For now, three curves are shown in Figure 1.1.3.

Curves and loci can be very complicated: they can double back, self intersect, or have multiple disconnected components. In addition, several other strange situations can occur; consider the locus of the equation \(x^2 + y^2 =0\text{.}\) Though this is a reasonable equation in the coordinate variables \(x\) and \(y\text{,}\) this locus only has one point: \((0,0)\text{.}\) Since \(x^2\) and \(y^2\) are always positive, no other values satisfy. Worse, consider \(x^2 + y^2 = -1\text{.}\) This locus has no points at all, since the left side cannot be negative and the right side certainly is.

Graphs of functions are also loci. If \(f(x)\) is a real-valued function, then the locus \(y = f(x)\) is the graph of that function. Note that the graph itself is not the function, but just a geometric picture or representation of what the function does. Figure 1.1.4 shows the graphs of four functions.

Since each value \(x\) leads to at most one function value \(f(x)\text{,}\) the graph of a function has the important property that for any fixed value \(x\text{,}\) there is only one point of the locus with that \(x\) coordinate. This is usually refered to a a vertical line test — a vertical line can cross the graph of a function at most once. Many of the loci seen above do not satisfy this; therefore, they cannot be graphs of functions. In addition, graphs never double back and never self-intersect.