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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) Compute the values of
\(L(-1)\) and
\(L'(-1)\text{.}\)
(b) What must be the values of
\(g(-1)\) and
\(g'(-1)\text{?}\) Why?
(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.
