CalculusAdvanced Mathematics D
Advanced Mathematics D
Chapter FourThe Derivatives inGraphing and Application
Chapter Four The Derivatives in Graphing and Application
Increase&DecreaseDefinitionLet f be defined on an interval, and let x1and x, denote points in the interval.O fis increase on the interval if f(x)< f(x2)whenever x < x2O fis decrease on the interval if f (x)> f (x2)whenever x,<x2O fis constant on the interval if f(x)= f(x2) for allpoints X1, X2
Increase & Decrease ⚫Definition Let f be defined on an interval, and let x1 and x2 denote points in the interval. f is increase on the interval if f (x1 )< f (x2 ) whenever x1 < x2 f is decrease on the interval if f (x1 )> f (x2 ) whenever x1 < x2 f is constant on the interval if f (x1 )= f (x2 ) for all points x1 , x2
Increase &Decrease-TheoremLet fbe a function that is continuous on aclosed interval [a, b] and differentiable on theopen interval (a,b)O If f'(x)>0, for all x in (a,b) => fis increase on [a,b]O If f'(x)<0, for all x in (a,b) => fis decrease on[a, b] O If f'(x)=O, for all x in (a,b) => fis constant on [a,b]
Increase & Decrease - ⚫Theorem Let f be a function that is continuous on a closed interval [a,b] and differentiable on the open interval (a,b) If f ’(x)>0, for all x in (a,b) => f is increase on [a,b] If f ’(x)<0, for all x in (a,b) => f is decrease on [a,b] If f ’(x)=0, for all x in (a,b) => f is constant on [a,b]
ConcavityDefinitionO If fis differentiable on an open interval I, thenf is said to be concave up on Iif f' isincreasing on IO fis said to be concave down on Iif f'isdecreasing on I
Concavity ⚫Definition If f is differentiable on an open interval I, then f is said to be concave up on I if f ’ is increasing on I f is said to be concave down on I if f ’ is decreasing on I