1.0 Motivation

Author’s note: Much of the discussion in this section is repeated in section 1.8, and this content can be saved until then. The purpose of this section is to motivate the use of a new type of number.

The concept of a tangent line to a curve has many uses in the study of calculus. What is a tangent line?

A tangent line to a circle is a line that intersects the circle in exactly one point (figure 1). While this description of a tangent line works well on some curves that are not circles, for others it does not. Consider the curves in figure 2. A line that just touches the curve in

Figure 1: (above) a line tangent to a circle

Figure 2: (left) tangents to curves

one point might or might not pass through the curve in some other point. It might even cut through the curve altogether. We need a different description of the term “tangent.”

One way to think of a tangent line is the direction of momentum when traveling along the curve. If you are on the edge of a spinning merry-go-round and release a ball from your hand, it will not keep going in a circle along with you. Ignoring the fact that the ball drops toward the ground, it travels on a straight line in the direction it was heading just at the time of release; that is the tangent line to the curve (figure 3). The same would be true for any curve; the tangent line is the direction of momentum.

Another way to think of a tangent line is the direction of travel if you are traveling along the curve in a vehicle (figure 4). The head-

light beam of the vehicle does not go uphill and downhill following the curve; it goes straight ahead. That is the direction of the tangent line.

Figure 3: (above) a ball released at the edge of a merry-go-round

Figure 4: (left) a vehicle traveling on a curve. The headlights shine in the direction of the tangent.

Tangent line to a curve

Suppose we wish to find the equation of the tangent line to the curve f (x) = 4 − x2 at x = 1. By “at x = 1” we mean the point whose x-coordinate is 1. The y-coordinate of that point would be f (1) = 4 − 12 = 3, so the point at which we want the tangent line is (1, f (1)) = (1, 3). The tangent is pictured in figure 5.

One formula used for the equation of a line is the point-slope equation of a line, y − y1 = m(x − x1). All we need is a point (which we have) and the slope of the line (which we need to find). There is also a formula for slope,

m =

y2 − y1 x2 − x1

The slope formula needs two points on the line, (x1, y1) and (x2, y2). We only have one point, (x1, y1) = (1, 3). What should we use for the second point?

Idea #1: The first idea we might try is to use a second point from the curve. To illustrate, try using the point with x = 2 as the second point. Since f (2) = 4 − 22 = 0, the point is (x2, y2) = (2, 0). We can find the slope using the slope formula, m = 0 2 − − 3 1 = −3, but it would not be the slope of the tangent line; it would be the slope of a secant line (a line through two points of the curve) instead. See figure 6. Idea #1 does not work.

f (x)

(1, 3)

x

Figure 5: The curve f (x) = 4 − x2 (blue) with tangent line at x = 1 (orange). The point (1, 3) is called the point of tangency.

f (x)

(1, 3)

Idea #2: Since we only want the curve to go through the point x (1, 3) and not some other point on the curve, perhaps we could use the point (1, 3) as both points in the slope formula. Then we would Figure 6: The curve f (x) = 4 − x2 have m = 3 1 − − 3 1 = 0 0 , which is undefined. Idea #2 does not work. with tangent line (orange) at x = 1 and

To summarize, we need two points to use the slope formula, but secant line (green) through (1, 3) and (2, 0). The slopes of the tangent linewe cannot use a second point away from the point of tangency with- and secant line are not the same. out changing the slope.

One way to solve this dilemma is to introduce a new type of number. Your concept of “number ” has expanded before, for instance when you were introduced to fractions or to negative numbers. This time we will introduce a type of number that will help us overcome the problem of idea #2, division by zero. No, we will not actually divide by zero, but we will learn to do the next best thing: divide by a number that is infinitely close to zero. This will also overcome the problem of idea #1, by allowing the second point to be infinitely close to the point of tangency, making the slope of the line between the two points infinitely close to the slope of the tangent line.

It turns out that all of the major concepts of calculus can be understood with the use of these new, infinitely small numbers. Bring on By “new” we mean new to the reader the infinitesimals!

that has not used them before.