§1.8 The tangent line approximation (continued)

Exercises (continued)

Summary

Near $x = 3$, a shared tangent line $L$ approximates $f$ and $g$, where $f''(3) = 0.2$ and $g''(3) = -1$. Why does $L$ approximate $f$ better?

A function $p$ has local linearization $L(x) = -2x + 5$ at $a = 3$. What are $p(3)$ and $p'(3)$?

With local linearization $L(x) = -2x + 5$ at $a = 3$, estimate $p(2.79)$, rounded to two decimals.

If $p''(x) \lt 0$ for $x \lt 3$ and $L$ is the linearization of $p$ at $3$, then $L(2.79)$ overestimates $p(2.79)$.

A potato's temperature $F(t)$ is sampled every $10$ minutes. Which expression best estimates $F'(30)$ from the table?

A potato satisfies $F(30) = 167.6$ degrees and $F'(30) \approx 2.065$ degrees per minute. Use the tangent line at $30$ to estimate $F(33)$, rounded to three decimals.

The potato's recorded temperatures increase at a decreasing rate near $t = 30$. Is the tangent line estimate of $F(33)$ too large or too small?

An object has position $s(9) = 4$ feet and velocity $s'(9) = -1.2$ feet per second at $t = 9$ seconds. The tangent line estimate of $s(9.34)$ is

The same object has acceleration $s''(9) = 0.08$ feet per second per second. Is the linear estimate $s(9.34) \approx 3.592$ feet too large or too small?

At $t = 9$ the object is moving toward the origin and slowing down, because its velocity is negative while its acceleration is positive.

A function $f$ has $f'(x) = (x-1)e^{-x^2}$. At which input is $f'(x) = 0$?

The graph of $f'(x) = (x-1)e^{-x^2}$ rises on roughly $-0.37 \lt x \lt 1.37$ and falls outside that interval. What does this say about $f$?

For the same $f$ with $f(2) = -3$, use the tangent line at $2$ to estimate $f(1.88)$, given $f'(2) = e^{-4} \approx 0.0183$. Round to four decimals.

Near $x = 2$ the graph of $f'(x) = (x-1)e^{-x^2}$ is decreasing. Is the estimate $f(1.88) \approx -3.0022$ too large or too small?