Chapter Four The Derivatives in Graphing and Application
Chapter Four The Derivatives in Graphing and Application
Increase Decrease Definition Let fbe defined on an interval,and let x1 and x,denote points in the interval. ofis increase on the interval if f(x1)<f(x2) wheneverx1<x2 O fis decrease on the interval if f(x1)>f(x2) wheneverx1<x2 o fis constant on the interval if f(x)=f(x2)for all points 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 ●Theorem Let fbe a function that is continuous on a closed interval [a,b]and differentiable on the open 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)=0,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]
Concavity ●Definition O If fis differentiable on an open interval I,then f is said to be concave up on Tif f'is increasing on o fis said to be concave down on Iif f'is decreasing 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
Concavity ●Theorem O If f"(x)>0 for all value of x in I,then fis concave up on 2 O If f"(x)<0 for all value of x in I,then fis concave down on 2
Concavity - ⚫Theorem If f ’’(x)>0 for all value of x in I, then f is concave up on I If f ’’(x)<0 for all value of x in I, then f is concave down on I