Chapter 3 | Derivatives
215
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: m sec = 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 : m sec = f ( a + h )− f ( a ) a + h − a = f ( a + h )− f ( a ) h .
Definition Let f be a function defined on an interval I containing a . If x ≠ a is in I , then
f ( x )− f ( a ) x − a
(3.1)
Q =
is a difference quotient . Also, if h ≠0 is chosen so that a + h is in I , then
f ( a + h )− f ( a ) h
(3.2)
Q =
is a difference quotient with increment h .
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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.
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