Calculus, by Smith and Minton

2.1 Tangent Lines and Velocity

A traditional slingshot is essentially a rock on the end of a string, which you rotate around in a circular motion and then release. When you release the string, in which direction will the rock travel? A simple illustration of this situation is shown in Figure 2.1.

Figure 2.1
Path of rock.

Many people mistakenly believe that the rock will follow a curved path, but Newton's first law of motion tells us that the path is straight. In fact, the rock follows a path along the tangent line to the circle, at the point of release.

If we wanted to determine the path followed by the rock, we could do so, as tangent lines to circles are relatively easy to find. (Recall from elementary geometry that a tangent line to a circle is a line that intersects a circle in exactly one point.) Our aim in this section is to extend the notion of tangent line to more general curves. Among other things, we will see that such tangent lines can give us important information about the motion of objects in general.

To make our discussion more concrete, suppose that we want to find the tangent line to the curve y = x²+1 at the point (1, 2) (see Figure 2.2). Unfortunately, the notion that a tangent line is a line that intersects the curve at only one point does not work here. For instance, in Figure 2.3, we show two different straight lines, both of which intersect the curve y = x²+1 only at the point (1, 2). Are these both tangent lines? Certainly not; only one of these acts like a tangent line to a circle by hugging the curve near the point of tangency. In other words, like the tangent line to a circle this tangent line has the same direction as the curve at the point of tangency.

Figure 2.2
y = x²+1.

Figure 2.3
y = x²+1 and lines intersecting the curve only at the point (1, 2).

That is, if you were standing on the curve at the point of tangency and had to take a small step and try to stay on the curve, you would step in the direction of the tangent line. Another way to think of this is to observe that this curve appears to be locally linear. That is, if we zoom in sufficiently far, the graph appears to approximate that of a straight line. In Figure 2.4, we show the graph of y = x²+1 zoomed in on the small rectangular box indicated in Figure 2.3.

Figure 2.4
y = x²+1.

(Be aware that the “axes” indicated in Figure 2.4 do not intersect at the origin. We provide them only as a guide as to the scale used to produce the figure.) We now choose two points from the curve: for example, (1, 2) and (3, 10), and compute the slope of the line joining these two points. Such a line is called a secant line and we denote its slope by m_sec:

An equation of the secant line is then determined by

so that

y = 4(x-1)+2.

As can be seen in Figure 2.5a, the secant line doesn't look very much like a tangent line.

Now, let's take the second point a little closer to the point of tangency, say (2, 4). In this case, the slope of the secant line is

so that an equation of the tangent line is y = 2 (x-1)+2. As seen in Figure 2.5b, this looks much more like a tangent line, but it's still not quite there. Choosing our second point much closer to the point of tangency, say (1.05, 2.1025) should give us an even better approximation to the tangent line. In this case, we have that the slope of the secant line joining these two points is given by

An equation of the secant line is then given by y = 2.05(x-1)+2. As can be seen in Figure 2.5c, the secant line looks very much like a tangent line, even when zoomed in quite far, as in Figure 2.5d.

Figure 2.5a
Secant line joining (1, 2) and (3, 10).

Figure 2.5b
A secant line.

Figure 2.5c
A secant line.

Figure 2.5d
Closeup of secant line.

Rather than continue this process by choosing additional points close to (1, 2), we compute the slope of the secant line joining (1, 2) and the unspecified point (1+h, f (1+h)), for some value of h close to 0. The slope of this secant line is

Notice that as h approaches 0, the slope of the secant line approaches 2, which we define to be the slope of the tangent line.

1.1

Figure 2.6
Tangent line intersecting a curve at more than one point.

The General Case

We have now developed a graphical process for getting better and better approximations to the slope of a tangent line. The general problem is to find an equation of the tangent line to y = f (x) at x = a. The first step is to pick two points on the curve. It is convenient to choose the point of tangency, (a, f (a)) as one of the points. We will name the x - coordinate of the second point x = a+h , for some small number h. The corresponding y - coordinate is then f (a+h). It is natural to think of h as being positive, as shown in Figure 2.7a, although h can also be negative, as shown in Figure 2.7b.

Figure 2.7a
Secant line (h > 0).

Figure 2.7b
Secant line (h < 0).

The slope of the secant line through the points (a, f (a)) and (a+h, f (a+h)) is given by

(1.1)

Notice that the expression in (1.1) (called a difference quotient) gives the slope of the secant line for any second point we might choose (i.e., for any h

0 ). Recall that in order to obtain better and better approximations to the tangent line, we zoom in closer and closer toward the point of tangency. This makes the two points closer together, which in turn makes h closer to 0. Just how far should we zoom in? The farther, the better; this means that we want h to approach 0. This process is illustrated in Figure 2.8, where we have plotted a number of secant lines for h > 0.

Figure 2.8
Secant lines approaching the tangent line at the point P.

Notice that as the point Q approaches the point P (i.e., as h

0) , the secant line approaches the tangent line at P.

We define the slope of the tangent line to be the limit of the slopes of the secant lines as h tends to 0, whenever this limit exists. From (1.1), we obtain the following definition.

1.1

The slope m_tan of the tangent line to y = f (x) at x = a is given by

(1.2)

provided the limit exists.

The tangent line is then the line passing through the point (a, f (a)) with slope m_tan and so, is determined from

so that the equation of the tangent line is

y = m_tan(x-a)+f (a).

1.1

Finding the Equation of a Tangent Line

Find an equation of the tangent line to y = x²+1 at x = 1.

We compute the slope using (1.2):

m_tan
		Multiply out and cancel.

		Factor out common h and cancel.

Notice that the point corresponding to x = 1 is (1, 2) and the line with slope 2 through the point (1, 2) has equation

y = 2(x-1)+2 or y = 2x.

Note how closely this corresponds to the secant lines computed earlier. We show a graph of the function and this tangent line in Figure 2.9.

Figure 2.9
y = x²+1 and the tangent line at x = 1.

1.2

Tangent Line to the Graph of a Rational Function

Find an equation of the tangent line to

at x = 2.

From (1.2), we have

m_tan		Since
		Add fractions and multiply out.
		Cancel h's.

The point corresponding to x = 2 is (2, 1), since f (2) = 1. An equation of the tangent line is then

y = -

(x-2)+1.

We show a graph of the function and this tangent line in Figure 2.10.

Figure 2.10
and tangent line.

Velocity

Finding the slope of a tangent line was an interesting adventure, but you might ask why anyone would care to do that. It turns out that the slopes of tangent lines have many applications. As you may have gathered from our brief discussion of the slingshot, one of these is in computing velocity. The term velocity is certainly familiar to you, but can you say precisely what it is? We often describe velocity as a quantity determining the speed and direction of an object. Alright, so what exactly is speed? If your car did not have a speedometer, how would you determine your speed? You would probably use the familiar formula

distance = rate

time.

(1.3)

Using (1.3), if you want to know the rate (speed), you simply divide the distance by the time. Unfortunately, this is a little too simple. The rate in (1.3) refers to average speed over a period of time. We are interested in the speed at a specific instant. The following story should indicate the difference.

Suppose that you are stopped on the highway by a police officer. The officer steps up to your car and asks that dreaded question, “Do you know how fast you were going?” An overzealous student might answer, “Absolutely. ” I've been driving for 3 years, 2 months, 7 days, 5 hours and 45 minutes. During that time I've driven exactly 45,259.7 miles. Therefore, my speed was

Of course, most police officers would not be impressed with this analysis. To be sure, it's ridiculous. But, why is it wrong? Certainly there's nothing wrong with formula (1.3) or the arithmetic. However, the police officer could reasonably argue that you were not even in your car during most of the 3 - year period and, hence, the results of your computation are invalid.

Suppose that you substitute the following argument instead, “I left my house at precisely 6:17 p.m. and by the time you pulled me over at 6:43 p.m., I had driven exactly 17 miles. Therefore, my speed was

well under the posted 45 mph speed limit. ”

Certainly, this is a much better estimate of your velocity than the 1.6 mph computed previously, but you are still computing an average velocity using too long of a time period. Since cars can speed up and slow down very quickly, we must compute the velocity using a much smaller time interval. But, how small is small enough?

As we did with finding slopes of tangent lines, it is helpful to rework our arguments symbolically. Suppose that the function f (t ) gives the position of an object moving along a straight line, at time t. That is, f (t ) gives the displacement (signed distance) from a fixed reference point, so that f (t ) < 0 means that the object is located |f (t )| away from the reference point, but in the negative direction. Then, for two times a and b (where a < b), f (b)-f (a) gives the signed distance between positions f (a) and f (b). The average velocity v_avg is then given by

(1.4)

1.3

Finding Average Velocity

Suppose that you are driving in a straight line and that your position after t minutes is given by the function

s(t ) =

t ²-

t ³, 0

Approximate your velocity at time t = 2.

As a start, we might find the average velocity over the first 2 minutes (i.e., from t = 0 to t = 2 ). From (1.4), we have

Of course, the average velocity of a car over a 2 - minute-long interval is not a particularly good approximation of its velocity at an instant, since cars can speed up and slow down a great deal in 2 minutes. An improved approximation is found by averaging over the second minute. Again, from (1.4), we have

Of course, this latest estimate is considerably better than the first one, since the time interval is so much shorter, but is it accurate enough? As we make the time interval shorter and shorter, the average velocity should be getting closer and closer to the velocity at the instant t = 2. It stands to reason that for h > 0 , if we compute the average velocity over the time interval [2-h, 2] , and then let h

0 , the resulting average velocities should be getting closer and closer to the velocity at the instant t = 2. On the interval [2-h, 2] , we have

In the accompanying table, we compute a sequence of these values for successively smaller values of h > 0.

h
1.0	0.9166666667
0.1	0.9991666667
0.01	0.9999916666
0.001	0.999999916
0.0001	1.0
0.00001	1.0

Of course, it also makes sense to average the velocity over intervals beginning at t = 2 and going slightly to the right, that is, over intervals of the form [2, 2+h] , for small values of h > 0. We get

A sequence of these average velocities are displayed in the accompanying table.

h
1.0	0.9166666667
0.1	0.9991666667
0.01	0.9999916667
0.001	0.999999917
0.0001	1.0
0.00001	1.0

Notice that the first formula for vavg is equivalent to the second one if we allow h to be negative. From both sets of results, it appears as if the average velocity is approaching 1 mile/minute (60 mph), as h 0. This limiting value is what we refer to as instantaneous velocity.

This leads us to make the following definition.

1.2

If f (t ) represents the position of an object relative to some fixed location at time t as it moves along a straight line, then the instantaneous velocity at time t = a is given by

(1.5)

provided the limit exists.

Notice that if (for example) t is measured in seconds and f (t ) is measured in feet, velocity (average or instantaneous) is measured in feet per second (ft/s). The term velocity is always used to refer to instantaneous velocity.

1.4

Finding Average and Instantaneous Velocity

Suppose that the height of a falling object t seconds after being dropped from a height of 64 feet is given by f (t ) = 64-16t ² feet. Find the average velocity between times t = 1 and t = 2 ; the average velocity between times t = 1.5 and t = 2 ; the average velocity between times t = 1.9 and t = 2 and the instantaneous velocity at time t = 2.

The average velocity between times t = 1 and t = 2 is

The average velocity between times t = 1.5 and t = 2 is

The average velocity between times t = 1.9 and t = 2 is

The instantaneous velocity is the limit of such average velocities. From (1.5), we have

v(2)

		Multiply out and cancel.
		Factor out common h and cancel.

Notice that the average velocities in this example appear to be approaching the instantaneous velocity as the time interval gets shorter and shorter. But, what does the negative sign on the velocity indicate? We noted earlier that velocity indicates both speed and direction. In this problem, f (t ) measures the height above the ground. A negative velocity thus indicates that the object is moving in the negative (or downward) direction. The speed of the object at the 2 - second mark is then 64 ft/s. (Speed is simply the absolute value of velocity.)

You should observe that the formulas for instantaneous velocity (1.5) and for the slope of a tangent line (1.2) are identical. We want to make this connection as strong as possible. Let's first illustrate example 1.4 graphically. We graph the position function f (t ) = 64-16t ² for 0

2. The average velocity between t = 1 and t = 2 is the slopeof the secant line between the points at t = 1 and t = 2 (see Figure 2.11a). Similarly, the average velocity between t = 1.5 and t = 2 is the slope of the corresponding secant line (see Figure 2.11b). Finally, the instantaneous velocity at time t = 2 is the slope of the tangent line at t = 2 (see Figure 2.11c).

Figure 2.11a
Secant line between t = 1 and t = 2.

Figure 2.11b
Secant line between t = 1.5 and t = 2.

Figure 2.11c
Tangent line at t = 2.

We often think of velocity as a rate (in the present case, the instantaneous rate of change of position with respect to time). Further, for any quantity, we can generalize this notion of instantaneous rate of change. In general, we have that the average rate of change of a given function f (x) between x = a and x = b ( a b ) is given by

The instantaneous rate of change of f (x) at x = a is given by

provided the limit exists. The units of the instantaneous rate of change are the units of f divided by (or “per”) the units of x.

1.5

Finding Average and Instantaneous Rates of Change

Suppose that the population of a city is estimated to be

million people t years from now. Find the average rate of change of the population for the next 2 years and the instantaneous rate of change of the population 2 years from now.

The average rate of change is

million people per year. The instantaneous rate of change is then

Since the instantaneous rate of change at t = 2 is less than the average rate of change from t = 0 to t = 2 , we might conjecture that the rate of population growth is slowing down, a fact not at all evident in the graph of f (t ) (see Figure 2.12).

Figure 2.12

Finally, when discussing population change, we typically report the rate of change as a percentage of the current population. Since the population at time t = 2 is

million people, the percentage change at t = 2

or about 3.4% per year.

Additional applications of the slope of a tangent line are too numerous to list. These include the rate of a chemical reaction, the inflation rate in economics and learning growth rates in psychology. In short, rates of change in nearly any discipline you can name can be thought of as slopes of tangent lines. We explore many of these applications as we progress through the text.

You may have noticed that we tacked the phrase “provided the limit exists” onto the end of the definitions of the slope of a tangent line, the instantaneous velocity and the instantaneous rate of change. You might have already wondered whether these defining limits don't always exist. They do not, as the following example illustrates.

1.6

A Graph with No Tangent Line at a Point

Determine if there is a tangent line to y = |x| at x = 0.

We can look at this problem graphically, numerically and symbolically. The graph is shown in Figure 2.13.

Figure 2.13
y = |x|.

Our graphical technique is to zoom in on the point of tangency until the graph appears straight. But, no matter how far we zoom in on (0, 0) , the graph continues to look like Figure 2.13. Try this for yourself. (This is one reason why we left off the scale on Figure 2.13.) From this evidence alone, we would conjecture that the tangent line does not exist. Recall that the tangent line can be thought of as the limiting position of a sequence of secant lines, as the second point approaches the point of tangency. Numerically, we observe that the secant line through (0, 0) and (1, 1) has slope 1, as does the secant line through (0, 0) and (0.1, 0.1). In fact, if h is any positive number, the slope of the secant line through (0, 0) and (h, | h|) is 1. However, the secant line through (0, 0) and (-1, 1) has slope - 1, as does the secant line through (0, 0) and (h, | h|) for any negative number h. We therefore conjecture that the one-sided limits are different and hence, the limit (and also the tangent line) does not exist. To prove this conjecture, we must work symbolically. Taking our cue from the numerical work, we look at the one-sided limits: if h > 0 , then | h| = h , so that

On the other hand, if h < 0 then |h| = -h (remember that if h < 0, -h > 0),

Since the one-sided limits are different, we conclude that

and hence, the tangent line does not exist.