Question

Actividad 1.1
Investigue sobre el tema de diferenciabilidad en un punto para encontrar los valores de "a" y "b" tales que
la función
definida a continuación sea diferenciable en t = 2, luego construya su gráfica.
at +b, sit < 2
f(t) = {2t2 – 1, si 2 st
1​

Answer 1

Answer:

a = 8

b = -8

Step-by-step explanation:

You have the following function:

$f(x)\\\\=at+b;\ \ t$

A function is differentiable at a point c, if the derivative of the function in such a point exists. That is, f'(c) exists.

In this case, you need that the function is differentiable for t=2, then, you have:

$f'(t)=a;\ \ \ \ t$

If the derivative exists for t=2, it is necessary that the previous derivatives are equal:

$f'(2)=a=4(2)\\\\a=8$

Furthermore it is necessary that for t=2, both parts of the function are equal:

$8(2)+b=2(2)^2-1\\\\16+b=8-1\\\\b=-8$

Then, a  = 8, b = -8