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In northern forests, where the sun is relatively low in the sky, trees tend to be tall and narrow to maximize their exposure to the available light. In contrast, in equatorial regions trees tend to have broad flat tops because the sun is mostly overhead during the day. Optimization is an important characteristic of natural systems, as well as a key factor in decision-making processes in applied problem solving. In this chapter, we will use the derivative to analyze functions and find optimal solutions in a variety of applied settings.
4 APPLICATIONS OF THE DERIVATIVE T
his chapter puts the derivative to work. The first and second derivatives are used to analyze functions and their graphs and to solve optimization problems (finding minimum and maximum values of a function). Newton’s Method in Section 4.8 employs the derivative to approximate solutions of equations.
4.1 Linear Approximation and Applications In this section, we introduce the process of linear approximation that uses the tangent line to the graph of a function f at x = a to approximate f (x) for x near a. These approximation methods are desirable because linear functions are usually easier to use and compute with than nonlinear ones. We introduce a few different formulas involving linear approximation. There are different settings and situations where each is useful. Keep in mind that they all come from the same basic idea that the tangent line approximates the function close to the point of tangency (Figure 1).
Linear Approximation In some situations, we are interested in the effect of a small change. For example, •
•
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How does a small change in angle affect the distance of a basketball shot? (Exercise 47) How are revenues at the box office affected by a small change in ticket prices? (Exercise 37) The cube root of 27 is 3. How much greater is the cube root of 27.2? (Exercise 7)
In each case, we have a function f and we’re interested in the change
y y = f (x)
f = f (a + x) − f (a)
Tangent line
where x is small. The Linear Approximation uses the slope of the tangent line (i.e., the derivative) to estimate f without computing it exactly. By definition, the derivative is the limit x
FIGURE 1 The tangent line approximates the graph of f near the point of tangency.
f (a) = lim
x→0
f (a + x) − f (a) f = lim x→0 x x
So when x is small, we have f / x ≈ f (a), and thus, f ≈ f (a) x
REMINDER The notation ≈ means “approximately equal to.” The accuracy of the Linear Approximation is discussed at the end of this section.
Linear Approximation of f
If f is differentiable at x = a and x is small, then f ≈ f (a) x
1
It is important to understand the different roles played by f and f (a) x. The quantity of interest is the actual change f . We estimate it by f (a) x, the change on the tangent line with slope f (a). The Linear Approximation tells us that up to a small error, f is approximately equal to f (a) x when x is small. 211
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