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Chapter 4 | Applications of Derivatives
This is the Mean Value Theorem. • If f ′( x ) =0 over an interval I , then f is constant over I . • If two differentiable functions f and g satisfy f ′( x ) = g ′( x ) over I , then f ( x ) = g ( x )+ C for some constant C . • If f ′( x ) >0 over an interval I , then f is increasing over I . If f ′( x ) <0 over I , then f is decreasing over I . 4.5 Derivatives and the Shape of a Graph • If c is a critical point of f and f ′( x ) >0 for x < c and f ′( x ) <0 for x > c , then f has a local maximum at c . • If c is a critical point of f and f ′( x ) <0 for x < c and f ′( x ) >0 for x > c , then f has a local minimum at c . • If f ″( x ) >0 over an interval I , then f is concave up over I . • If f ″( x ) <0 over an interval I , then f is concave down over I . • If f ′( c ) =0 and f ″( c ) >0, then f has a local minimum at c . • If f ′( c ) =0 and f ″( c ) <0, then f has a local maximum at c . • If f ′( c ) =0 and f ″( c ) =0, then evaluate f ′( x ) at a test point x to the left of c and a test point x to the right of c , to determine whether f has a local extremum at c . 4.6 Limits at Infinity and Asymptotes • The limit of f ( x ) is L as x →∞ (or as x →−∞) if the values f ( x ) become arbitrarily close to L as x becomes sufficiently large. • The limit of f ( x ) is ∞ as x →∞ if f ( x ) becomes arbitrarily large as x becomes sufficiently large. The limit of f ( x ) is −∞ as x →∞ if f ( x ) <0 and | f ( x ) | becomes arbitrarily large as x becomes sufficiently large. We can define the limit of f ( x ) as x approaches −∞ similarly. • For a polynomial function p ( x ) = a n x n + a n −1 x n −1 +…+ a 1 x + a 0 , where a n ≠0, the end behavior is determined by the leading term a n x n . If n ≠0, p ( x ) approaches ∞ or −∞ at each end. • For a rational function f ( x ) = p ( x ) q ( x ) , the end behavior is determined by the relationship between the degree of p and the degree of q . If the degree of p is less than the degree of q , the line y =0 is a horizontal asymptote for f . If the degree of p is equal to the degree of q , then the line y = a n b n is a horizontal asymptote, where a n and b n are the leading coefficients of p and q , respectively. If the degree of p is greater than the degree of q , then f approaches ∞ or −∞ at each end.
4.7 Applied Optimization Problems • To solve an optimization problem, begin by drawing a picture and introducing variables. • Find an equation relating the variables. • Find a function of one variable to describe the quantity that is to be minimized or maximized.
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