2.1 Limits, Rates of Change, and Tangent Lines

Rates of change play a role whenever we study the relationship between two changing quantities. Velocity is a familiar example (the rate of change of position with respect to time), but there are many others, such as

• The infection rate of an epidemic (newly infected individuals per month)
• Inflation rate (change in consumer price index per year)
• Rate of change of atmospheric temperature with respect to altitude

Roughly speaking, if y and x are related quantities, the rate of change should tell us how much y changes in response to a unit change in x. For example, if an automobile travels at a velocity of 80 km/hr, then its position changes by 80 km for each unit change in time (the unit being 1 hour). If the trip lasts only half an hour, its position changes by 40 km, and in general, the change in position is 80t km, where t is the change in time (that is, the time elapsed in hours). In other words,

However, this simple formula is not valid or even meaningful if the velocity is not constant. After all, if the automobile is accelerating or decelerating, which velocity would we use in the formula?

The problem of extending this formula to account for changing velocity lies at the heart of calculus. As we will learn, differential calculus uses the limit concept to define instantaneous velocity, and integral calculus enables us to compute the change in position in terms of instantaneous velocity. But these ideas are very general. They apply to all rates of change, making calculus an indispensable tool for modeling an amazing range of real-world phenomena.

In this section, we discuss velocity and other rates of change, emphasizing their graphical interpretation in terms of tangent lines. Although at this stage, we cannot define precisely what a tangent line is—this will have to wait until Chapter 3—you can think of a tangent line as a line that skims a curve at a point, as in Figure 2.1 (A) and (B) but not (C).

Figure 2.1: The line is tangent in (A) and (B) but not in (C).

2.1.1 Historical Perspective

The scientific revolution of the sixteenth and seventeenth centuries reached its high point in the work of Isaac Newton (1643–1727), who was the first scientist to show that the physical world, despite its complexity and diversity, is governed by a small number of universal laws. One of Newton’s great insights was that the universal laws are dynamical, describing how the world changes over time in response to forces, rather than how the world actually is at any given moment in time. These laws are expressed best in the language of calculus, which is the mathematics of change.

More than 50 years before the work of Newton, the astronomer Johannes Kepler (1571–1630) discovered his three laws of planetary motion, the most famous of which states that the path of a planet around the sun is an ellipse. Kepler arrived at these laws through a painstaking analysis of astronomical data, but he could not explain why they were true. According to Newton, the motion of any object—planet or pebble—is determined by the forces acting on it. The planets, if left undisturbed, would travel in straight lines. Since their paths are elliptical, some force—in this case, the gravitational force of the sun—must be acting to make them change direction continuously. In his magnum opus Principia Mathematica, published in 1687, Newton proved that Kepler’s laws follow from Newton’s own universal laws of motion and gravity.

For these discoveries, Newton gained widespread fame in his lifetime. His fame continued to increase after his death, assuming a nearly mythic dimension and his ideas had a profound influence, not only in science but also in the arts and literature, as expressed in the epitaph by British poet Alexander Pope: “Nature and Nature’s Laws lay hid in Night. God said, Let Newton be! and all was Light.”

2.1.2 Velocity

When we speak of velocity, we usually mean instantaneous velocity, which indicates the speed and direction of an object at a particular moment. The idea of instantaneous velocity makes intuitive sense, but care is required to define it precisely.

Consider an object traveling in a straight line (linear motion). The average velocity over a given time interval has a straightforward definition as the ratio

For example, if an automobile travels 200 km in 4 hours, then its average velocity during this 4-hour period is . At any given moment the automobile may be going faster or slower than the average.

We cannot define instantaneous velocity as a ratio because we would have to divide by the length of the time interval (which is zero). However, we should be able to estimate instantaneous velocity by computing average velocity over successively smaller time intervals. The guiding principle is: Average velocity over a very small time interval is very close to instantaneous velocity. To explore this idea further, we introduce some notation.

The Greek letter Δ (Delta) is commonly used to denote the change in a function or variable. If s (t) is the position of an object (distance from the origin) at time t and [t₀, t₁] is a time interval, we set

The change in position Δs is also called the displacement, or net change in position. For t₁ ≠ t₀,

2.1.3 Velocity: Example

Figure 2.2: Distance traveled by a falling object after t seconds s (t) = 4.9t² meters.

One motion we will study is the motion of an object falling to earth under the influence of gravity (assuming no air resistance). Galileo discovered that if the object is released at time t = 0 from a state of rest (Figure 2.2), then the distance traveled after t seconds is given by the formula

We express our conclusion in the previous example by saying that average velocity converges to instantaneous velocity or that instantaneous velocity is the limit of average velocity as the length of the time interval shrinks to zero.

2.1.4 Graphical Interpretation of Velocity

The idea that average velocity converges to instantaneous velocity as we shorten the time interval has a vivid interpretation in terms of secant lines. The term secant line refers to a line through two points on a curve.

Consider the graph of position s (t) for an object traveling in a straight line (Figure 2.3). The ratio defining average velocity over [t₀, t₁] is nothing more than the slope of the secant line through the points (t₀, s (t₀)) and (t₁, s (t₁)). For t₁ ≠ t₀,

Figure 2.3: The average velocity over [t₀, t₁] is equal to the slope of the secant line.

By interpreting average velocity as a slope, we can visualize what happens as the time interval gets smaller. Figure 2.4 shows the graph of position for the falling stone of Example 1, where s (t) = 4.9t². As the time interval shrinks, the secant lines get closer to—and seem to rotate into—the tangent line at t = 0.8.

Figure 2.4: The secant lines “rotate into” the tangent line as the time interval shrinks. Note: The graph is not drawn to scale.

And since the secant lines approach the tangent line, the slopes of the secant lines get closer and closer to the slope of the tangent line. In other words, the statement

has the graphical interpretation

We conclude that instantaneous velocity is equal to the slope of the tangent line to the graph of position as a function of time. This conclusion and its generalization to other rates of change are of fundamental importance in differential calculus.

2.1.5 Other Rates of Change

Velocity is only one of many examples of a rate of change. Our reasoning applies to any quantity y that depends on a variable x—say, y = f (x). For any interval [x₀, x₁], we set

For x₁ ≠ x₀, the average rate of change of y with respect to x over [x₀, x₁] is the ratio

The instantaneous rate of change at x = x₀ is the limit of the average rates of change. We estimate it by computing the average rate over smaller and smaller intervals.

In Example 1 above, we considered only right-hand intervals [x₀, x₁]. In the next example, we compute the average rate of change for intervals lying to both the left and the right of x₀.

Speed of Sound in Air

The formula provides a good approximation to the speed of sound υ in dry air (in m/s) as a function of air temperature T (in kelvins). Estimate the instantaneous rate of change of υ with respect to T when T = 273 K. What are the units of this rate?

Solution To estimate the instantaneous rate of change at T = 273, we compute the average rate for several intervals lying to the left and right of T = 273. For example, the average rate of change over [272.5, 273] is

Temperature interval	Average rate of change
[272.5, 273]	0.60550
[272.8, 273]	0.60534
[272.9, 273]	0.60528
[272.99, 273]	0.60523

Table 2.2: Left-Hand Intervals

Temperature interval	Average rate of change
[273, 273.5]	0.60495
[273, 273.2]	0.60512
[273, 273.1]	0.60517
[273, 273.01]	0.60522

Table 2.3: Right-Hand Intervals

Table 2 and Table 3 suggest that the instantaneous rate is approximately 0.605. This is the rate of increase in speed per degree increase in temperature, so it has units of m/s-K, or meters per second per kelvin. The secant lines corresponding to the values in the tables are shown in Figure 5 and Figure 6.

Figure 2.5: Secant lines for intervals lying to the left of T = 273.

Figure 2.6: Secant lines for intervals lying to the right of T = 273.

2.1.6 CalcClip Example

The video below shows another example of estimating the instantaneous rate of change, this time in a financial context.

You'll practice this process further in the tutorial that follows this eBook section, then more in the homework assignment for this unit.

2.1.7 Rate of change and slope

Figure 2.7: For a linear function f (x) = mx + b, the ratio Δf/Δx is equal to the slope m for every interval.

To conclude this section, we recall an important point discussed in Section 1.2: For any linear function f (x) = mx + b, the average rate of change over every interval is equal to the slope m (Figure 2.7). We verify as follows:

The instantaneous rate of change at x = x₀, which is the limit of these average rates, is also equal to m. This makes sense graphically because all secant lines and all tangent lines to the graph of f (x) coincide with the graph itself.