Compute the right-hand and left-hand derivatives as limits to show that the following functions are not differentiable at the point P:1.f(x)=x^2 ,when x<=0. f(x)=x,when x>=0 P(0,0) 2.y=2,when x <=1. y=2x,when x>=1 P(1,2)
对于函数f(x):
当x <= 0时,f(x) = x^2;
当x >= 0时,f(x) = x。
要证明函数在P(0,0)处不可导,我们需要计算左导数和右导数。
左导数(Left-hand derivative):
lim(x->0-) (f(x) - f(0)) / (x - 0)
= lim(x->0-) (x^2 - 0) / (x - 0)
= lim(x->0-) x
= 0
右导数(Right-hand derivative):
lim(x->0+) (f(x) - f(0)) / (x - 0)
= lim(x->0+) (x - 0) / (x - 0)
= lim(x->0+) 1
= 1
左导数和右导数不相等,所以函数f(x)在点P(0,0)处不可导。
对于函数y:
当x <= 1时,y = 2;
当x >= 1时,y = 2x。
要证明函数在P(1,2)处不可导,我们需要计算左导数和右导数。
左导数(Left-hand derivative):
lim(x->1-) (y(x) - y(1)) / (x - 1)
= lim(x->1-) (2 - 2) / (x - 1)
= lim(x->1-) 0
= 0
右导数(Right-hand derivative):
lim(x->1+) (y(x) - y(1)) / (x - 1)
= lim(x->1+) (2x - 2) / (x - 1)
= lim(x->1+) 2
= 2
左导数和右导数不相等,所以函数y在点P(1,2)处不可导。