§1.8 The tangent line approximation (continued)

The local linearization

Suppose that a function $y = f(x)$ has its tangent line approximation given by $L(x) = 3 - 2(x-1)$ at the point $(1,3)$, but we do not know anything else about the function $f$. How can we estimate the value of $f(x)$ for $x$ near $1$?

A function $g$ has tangent line approximation $L(x) = -2 + 3(x+1)$ at the point $(-1, -2)$. What is the best estimate of $g(-1.03)$?

A function $f$ satisfies $f(2) = -1$ and $f'(2) = 2$. Which is its tangent line approximation at $2$?

Functions $f$ and $g$ satisfy $f(3) = g(3) = 1$ and $f'(3) = g'(3) = -0.5$. What can be said about their tangent line approximations at $3$?

With $L(x) = -1 + 2(x - 2)$, estimate $f(2.07)$ by evaluating $L(2.07)$, rounded to two decimals.

With the shared linearization $L(x) = 1 - 0.5(x - 3)$, compute $L(3.1)$, rounded to two decimals.

If the tangent line at $x = 2$ lies above the graph of $f$ just to the right of $2$, then the estimate $f(2.07) \approx L(2.07)$ is too large.