Activity 1.8.2.
Suppose it is known that for a given differentiable function \(y = g(x)\text{,}\) its local linearization at the point where \(a = -1\) is given by \(L(x) = -2 + 3(x+1)\text{.}\)
(a)
(b)
(c)
Do you expect the value of \(g(-1.03)\) to be greater than or less than the value of \(g(-1)\text{?}\) Why?
(d)
Use the local linearization to estimate the value of \(g(-1.03)\text{.}\)
(e)
Suppose that you also know that \(g''(-1) = 2\text{.}\) What does this tell you about the graph of \(y = g(x)\) at \(a = -1\text{?}\)
(f)
For \(x\) near \(-1\text{,}\) sketch the graph of the local linearization \(y = L(x)\) as well as a possible graph of \(y = g(x)\) on the axes provided.
