Derivadas 2

Page 1

1. Determine, si existe, la derivada de f en x = 2 de la funci´ on f definida por  2 ,  x +1 f (x) = 5 ,  (x − 1)2 + 4 ,

Soluci´ on

l´ım−

h→0

=

f (2 + h) − f (2) = h

l´ım−

h→0

h→0−

l´ım

h→0+

=

l´ım

l´ım

h→0+

((2 + h − 1)2 + 4) − (5) h

(1 + 2h + h2 + 4) − (5) h

= l´ım+

h(2 + h) = h

como:

l´ım

h→0

l´ım (4 + h) = 4

h→0−

f (2 + h) − f (2) = h

h→0+

h→0

((2 + h)2 + 1) − (5) h

(22 + 4h + h2 + 1) − (5) h h(4 + h) = h

= l´ım

l´ım−

x < 2, x = 2, x > 2.

h→0−

l´ım (2 + h) = 2

h→0+

f (2 + h) − f (2) 6= h

l´ım

h→0+

f (2 + h) − f (2) h

la derivada no existe en x0 = 2


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