第3章微分

Now that we have both a conceptual understanding of a limit and the practical ability to compute limits, we have established the foundation for our study of calculus, the branch of mathematics in which we compute derivatives and integrals. Most mathematicians and historians agree that calculus was developed independently by the Englishman Isaac Newton $\text{(1643–1727)}$ and the German Gottfried Leibniz $\text{(1646–1716),}$ whose images appear in Figure 3.2. When we credit Newton and Leibniz with developing calculus, we are really referring to the fact that Newton and Leibniz were the first to understand the relationship between the derivative and the integral. Both mathematicians benefited from the work of predecessors, such as Barrow, Fermat, and Cavalieri. The initial relationship between the two mathematicians appears to have been amicable; however, in later years a bitter controversy erupted over whose work took precedence. Although it seems likely that Newton did, indeed, arrive at the ideas behind calculus first, we are indebted to Leibniz for the notation that we commonly use today.

We begin our study of calculus by revisiting the notion of secant lines and tangent lines. Recall that we used the slope of a secant line to a function at a point $(a,f\left(a\right))$ to estimate the rate of change, or the rate at which one variable changes in relation to another variable. We can obtain the slope of the secant by choosing a value of $x$ near $a$ and drawing a line through the points $(a,f\left(a\right))$ and $\left(x,f\left(x\right)\right),$ as shown in Figure 3.3. The slope of this line is given by an equation in the form of a difference quotient:

We can also calculate the slope of a secant line to a function at a value a by using this equation and replacing $x$ with $a+h,$ where $h$ is a value close to 0. We can then calculate the slope of the line through the points $(a,f\left(a\right))$ and $(a+h,f\left(a+h\right)).$ In this case, we find the secant line has a slope given by the following difference quotient with increment $h\text{:}$

These two expressions for calculating the slope of a secant line are illustrated in Figure 3.3. We will see that each of these two methods for finding the slope of a secant line is of value. Depending on the setting, we can choose one or the other. The primary consideration in our choice usually depends on ease of calculation.

In Figure 3.4(a) we see that, as the values of $x$ approach $a,$ the slopes of the secant lines provide better estimates of the rate of change of the function at $a.$ Furthermore, the secant lines themselves approach the tangent line to the function at $a,$ which represents the limit of the secant lines. Similarly, Figure 3.4(b) shows that as the values of $h$ get closer to $0,$ the secant lines also approach the tangent line. The slope of the tangent line at $a$ is the rate of change of the function at $a,$ as shown in Figure 3.4(c).

In Figure 3.5 we show the graph of $f\left(x\right)=\sqrt{x}$ and its tangent line at $(1,1)$ in a series of tighter intervals about $x=1.$ As the intervals become narrower, the graph of the function and its tangent line appear to coincide, making the values on the tangent line a good approximation to the values of the function for choices of $x$ close to $1.$ In fact, the graph of $f(x)$ itself appears to be locally linear in the immediate vicinity of $x=1.$

Formally we may define the tangent line to the graph of a function as follows.

Just as we have used two different expressions to define the slope of a secant line, we use two different forms to define the slope of the tangent line. In this text we use both forms of the definition. As before, the choice of definition will depend on the setting. Now that we have formally defined a tangent line to a function at a point, we can use this definition to find equations of tangent lines.