Chapter 6

# 3 x 4 to be we 2 6 x0 3 x x equal to y 2 x x let

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Unformatted text preview: l to ∆ y 2 ∆ x = ∆ x Let us try to figure out its value when ∆x is small: 6 x0 ∆ +3( ∆ ) x x ∆ x 2 =6 x0 +3∆ x ∆x → 0 6 x0 Clearly, as ∆x is getting arbitrarily small, the above Clearly, expression converges to just 6 x0 7 Definition of a Derivative The derivative of a function x0 is defined as follows: dy dx x0 ∆y ≡ f ′( x0 ) ≡ lim ∆x → ∆ 0 x x0 f ( x0 + ∆x ) − f ( x0 ) ≡ lim ∆→ x0 ∆x Read as: a derivative of a function at a given x0 is the limit of its Read difference quotient at x0 as ∆x approaches zero. 1) A derivative is a function of just x0 since ∆x is 1) assumed to be infinitely close to zero. The difference quotient is a function of both. quotient 2) Derivatives may not be computed for certain functions. 2) For example, functions with kinks do not have derivatives everywhere in their domain. everywhere 3) When computing a derivative, remember it only makes 3) sense to do so with a particular point in the function’s domain in mind—”loose” derivatives are nonsensical. domain 8 Derivatives and Slopes Consider the following cost function: C ( Q ) =1 +Q 2 Consider C ∆C ∆Q → 0 ∆ Q Q0 f ′( Q0 ) = lim A straight line passing straight through A and B through C1 is crossing the is horizontal axis at angle α at B ∆C ∆Q → ∆ 0 Q = f ′( Q0 ) tan β = lim ∆C whose tangent whose is equal to tan α = C is Q0 − C0 ∆C = Q1 − Q0 ∆Q 1 A C0 β α Q0 As ∆ Q is getting smaller, this straight As line will be approaching the tangent line tangent to C(Q) at Q0 ∆ Q Q1 Q 9 Derivatives and Increasing/Decreasing Functions Since a function’s derivative is representing its slope, we have two Since We shall later study rules of differentiation that will tell us exactly how We handy rules for analyzing the function’s behavior in some set S in its to take derivatives of various functions. to domain: domain: 1) If f ′( x ) > 0 for x ∈ S , function f ( x ) is positively sloped for x ∈ S If df Consider the function f ( x ) = x : its derivative dx = 2 x0 , so it is positive Consider x0 for every x0 > 0 . Hence, function f ( x ) has a positive slope for that range 2 of x. 2) If f ′ ( x ) < 0, x ∈ S , function f 2) ( x) is negatively sloped if x ∈ S For function f ( x ) its derivative f ′( x ) = 2 x is negative for x<0. For Hence, this function has a negative slope for x<0 10 A Note on Kinky Functions The geometric interpretation of the derivative concept makes it clear why The the latter is not defined for functions with the kinks (where the kinks occur). occur). If we only increase x, If However, if we However, this will be the slope. this y decrease x, the decrease the slope line will be quite different quite A At point A this At function has a kink, which makes it problematic to establish the location of the slope line at this point. this x 11 Left­Side and Right­Side Derivatives As the previous example demonstrated, it can sometimes be important As whether we are taking the increments...
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