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Unformatted text preview: Chapter 5 Differentiation In this chapter, we will study important concepts such as Newton quotients, secant lines, tangent lines, and derivatives. These concepts will be necessary for the next chapter, where we will prove our third “value theorem”—the mean value theorem ; it is the most important result in differential calculus and we will see its use in various applications. 5.0 Some Definitions and Basic Results Definition 5.0.1. Suppose that f : S → R where S ⊂ R , and that some point a ∈ S is such that a ∈ I for some open interval I ⊂ S . For some x ∈ I , we call the quantity f ( x )- f ( a ) x- a a Newton quotient for f ( x ) centered at x = a . Geometrically, the Newton quotient f ( x )- f ( a ) x- a represents the slope of the secant line through ( a,f ( a )) and ( x ,f ( x )). On the other hand, for some x ∈ I , if we let y = f ( x ), Δ y = f ( x )- f ( a ) (the change in y ), and Δ x = x- a (the change in x ), then the Newton quotient f ( x )- f ( a ) x- a = Δ y Δ x can be viewed as the average rate of change in f ( x ) over the interval from a to x . For example, if f ( x ) is the distance travelled since time x = 0 and x is the current time, then Δ y Δ x = f ( x )- f ( a ) x- a is the average velocity. In this way, a natural question arises: What do we mean by the “instantaneous velocity” or the “instantaneous rate of change”? We shall mean the limit of the average velocity or the average rate of change as Δ x → 0. With this as our motivation, we formally define the derivative below. Definition 5.0.2 [Derivative] . Suppose that f : S → R . Suppose a ∈ S is such that an open interval I ⊂ S contains a . We say that f ( x ) is differentiable at x = a if lim x → a f ( x )- f ( a ) x- a 69 exists. If this limit exists, then we will denote this limit by f ( a ) and call it the derivative of f ( x ) at x = a . That is, f ( a ) := lim x → a f ( x )- f ( a ) x- a . Otherwise, we say that f ( x ) is not differentiable at x = a (notation: f ( a ) does not exist). Note that if h := x- a = Δ x , then h := x- a → 0 if and only if x → a . We have the following alternate form of the derivative. Remark 5.0.3. Suppose that f : S → R . Suppose a ∈ S is such that an open interval I ⊂ S contains a . If f ( x ) is differentiable at x = a , then f ( a ) := lim x → a f ( x )- f ( a ) x- a = lim h → f ( a + h )- f ( a ) h . Proof. This proof is left as an exercise. Note . In order for a function to be differentiable at a point x = a , there must be an open interval I containing a on which f ( x ) is defined. Example 5.1. Let f : R → R be defined by f ( x ) = c. Let a ∈ R . Then f ( a ) := lim h → f ( a + h )- f ( a ) h = lim h → c- c h = lim h → h = lim h → 0 = 0 ....
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- Fall '09
- Derivative, Inverse function, open interval, lim φ