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 (1643–1727) and the German Gottfried Leibniz (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.

Tangent Lines 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(𝑎)) 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(𝑎)) and (𝑥,𝑓(𝑥)), as shown in Figure 3.3. The slope of this line is given by an equation in the form of a difference quotient: 𝑚sec=𝑓(𝑥)−𝑓(𝑎)𝑥−𝑎. 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(𝑎)) and (𝑎+ℎ,𝑓(𝑎+ℎ)). In this case, we find the secant line has a slope given by the following difference quotient with increment ℎ: 𝑚sec=𝑓(𝑎+ℎ)−𝑓(𝑎)𝑎+ℎ−𝑎=𝑓(𝑎+ℎ)−𝑓(𝑎)ℎ.