Infi-1 12

Infi-1 12

Derivative

f′(x0)=limx→x0f(x)−f(x0)x−x0f′(x0)=limh→0f(x0+h)−f(x0)h

Exercise

f(x)=x3f′(x)=?Solution:Let a∈Rf′(a)=limh→0f(a+h)−f(a)h=limh→0(a+h)3−a3h=limh→0a3+3a2h+3ah2+h3−a3h==limh→03a2+3ah+h2=3a2⟹f′(x)=3x2

Exercise

f(x)={x2sin⁡(1x2)x≠00x=0Find f′(0)Solution:f′(0)=limh→0f(0+h)−f(0)h=limh→0f(h)h=⏟h≠0limh→0h2sin⁡(1h2)h=limh→0hsin⁡(1h2)⏟−1≤sin⁡(1h2)≤1=0

Exercise

f(x)={4xx≤0sin⁡(3x)x>0Find f′(0)Solution:f′(0)=limx→0f(x)−f(0)x−0=limx→0f(x)xlimx→0+f(x)x=limx→0+sin⁡(3x)x=limx→0+sin⁡(3x)3x⋅3=3limx→0−f(x)x=limx→0−4xx=4limx→0+f(x)x≠limx→0−f(x)x⟹∄f′(0)
f(x)=|x|f′(0)=limh→0f(0+h)−f(0)h=limh→0|h|hlimh→0+|h|h=1limh→0−|h|h=−1limh→0+|h|h≠limh→0−|h|h⟹∄f′(0)

Exercise

f(x)={1xsin⁡(ax2)x<02x2+bx+1x>00x=0Find a,b∈R such that f is differentiable on RSolution:limx→0+f(x)=limx→0+2x2+bx+1=1⟹limx→0f(x)≠0⟹f is not continuous at 0 for any a,b⟹f is not differentiable at 0 for all a,b

Exercise

f(x)={ax+9x≥bbx−x2x<bFind a,b∈R such that f is differentiable on RSolution:limx→bf(x)=?f(b)limx→b−f(x)=limx→b−bx−x2=x(b−x)=0limx→b+f(x)=limx→b+ax+9=ab+9ab+9=0⟺f is continuous at bLet ab+9=0b=−9af′(b)=limh→0f(b+h)−f(b)h=limh→0f(b+h)hlimh→0+f(b+h)h=limh→0+ab+ah+9h=alimh→0−f(b+h)h=limh→0−b(b+h)−(b+h)2h=limh→0−=b2+bh−b2−2bh−h2h==limh→0−−bh−h2h=−b∃f′(b)⟺a=−b{ab+9=0a=−b⟹{a=3,b=−3a=−3,b=3