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HISTORICAL PERSPECTIVE
Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language…in which it is written. It is written in the language of mathematics…
—GALILEO GALILEI, 1623
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.”
THEOREM 1
For any constants k and c,
-
(a)
, -
(b)
.
Reasoning as in Example 1 but with arbitrary constants, we obtain the following simple but important results:
THEOREM 1
For any constants k and c,
-
(a),
-
(b).
To explore this question, we’ll experiment with the function
49
Notice that f(0) is not defined. In fact, when we set x = 0 in
The undefined expression 0/0 is referred to as an “indeterminate form.”
we obtain the undefined expression 0/0 because sin 0 = 0. Nevertheless, we can compute f(x) for values of x close to 0. When we do this, a clear trend emerges.
To describe the trend, we use the phrase “x approaches 0” or “x tends to 0” to indicate that x takes on values (both positive and negative) that get closer and closer to 0. The notation for this is x → 0, and more specifically we write
- •x → 0+ if x approaches 0 from the right (through positive values).
- •x → 0− if x approaches 0 from the left (through negative values).
Now consider the values listed in Table 1. The table gives the unmistakable impression that f(x) gets closer and closer to 1 as x → 0+ and as x → 0−.
This conclusion is supported by the graph of f(x) in Figure 1. The point (0, 1) is missing from the graph because f(x) is not defined at x = 0, but the graph approaches this missing point as x approaches 0 from the left and right. We say that the limit of f(x) as x → 0 is equal to 1, and we write
We also say that f(x) approaches or converges to 1 as x → 0.
| x |
|
x |
|
|---|---|---|---|
| 1 | 0.841470985 | −1 | 0.841470985 |
| 0.5 | 0.958851077 | −0.5 | 0.958851077 |
| 0.1 | 0.998334166 | −0.1 | 0.998334166 |
| 0.05 | 0.999583385 | −0.05 | 0.999583385 |
| 0.01 | 0.999983333 | −0.01 | 0.999983333 |
| 0.005 | 0.999995833 | −0.005 | 0.999995833 |
| 0.001 | 0.999999833 | −0.001 | 0.999999833 |
| x → 0+ | f (x) → 1 | x → 0− | f (x) → 1 |
.
