§1.8 The tangent line approximation (continued)

The local linearization

A function $g$ has tangent line approximation $L(x) = -2 + 3(x+1)$ at the point $(-1, -2)$, and nothing else is known about $g$. How should $g(x)$ be estimated for $x$ near $-1$?

With $L(x) = -2 + 3(x+1)$, what are $L(-1)$ and $L'(-1)$?

The same $L(x) = -2 + 3(x+1)$ is the tangent line approximation of $g$ at $a = -1$. What must $g(-1)$ and $g'(-1)$ be?

With $g'(-1) = 3$, the value $g(-1.03)$ is expected to be less than $g(-1)$.

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

Suppose in addition that $g''(-1) = 2$. What does this say about the graph of $y = g(x)$ at $a = -1$?

Near $x = -1$, a possible graph of $g$ with $g''(-1) = 2$ lies above the line $L(x) = -2 + 3(x+1)$ on both sides of the tangency point, touching it only at $(-1, -2)$.

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

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

The graph of $f'$ rises for $x \lt 2$, peaks at $x = 2$, and falls for $x \gt 2$. What does the graph of $f''$ look like?

With the same $f'$ (rising for $x \lt 2$, peaking at $2$, falling after), is the slope of the tangent line to $y = f(x)$ increasing, decreasing, or neither at $x = 2$?

For the same $f$, the graph of $y = f(x)$ is concave up for $x \lt 2$, concave down for $x \gt 2$, and steepest near $x = 2$, where it crosses its tangent line $L(x) = -1 + 2(x-2)$.

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.

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

Near $x = 3$ the table shows $L(x)$ always below $f(x)$ and always above $g(x)$ for $x \ne 3$, with $f$ deviating from $L$ less than $g$ does. What explains this?