Question

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

Answer 1

a = 8

b = -8

Explanation:

You have the following function:

`f(x)\\\\=at+b;\ \ t<2\\\\2t^2-1;\ \ 2\leq 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<2 \\\\f'(t)=4t;\ \ \ 2\leq 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