§1.3 The derivative of a function at a point (continued)

The Derivative of a Function at a Point

Use the limit definition to compute $f'(2)$ for $f(x) = x - x^2$.

Let $f(x) = 3 - 2x$, a linear function. Using any valid method, what are $f'(2)$, $f'(\pi)$, and $f'(-\sqrt{2})$?

For $f(x) = 3 - 2x$, the average rates of change on $[1,4]$, on $[3,7]$, on $[5,5+h]$, and the instantaneous rate $f'(1)$ all equal $-2$.

A ball's height is $s(t) = -16t^2 + 16t + 32$ feet. Which list correctly gives the intercepts and vertex of the graph of $s$?

For $s(t) = -16t^2 + 16t + 32$ feet with $t$ in seconds, compute the average velocity on the interval $[1, 2]$, in feet per second.

A ball's height is $s(t) = -16t^2 + 16t + 32$ feet at time $t$ seconds. Using the limit definition, what is $s'(1)$?

For the ball with $s(t) = -16t^2 + 16t + 32$ and the population $P(t) = 25000e^{t/5}$, two lines are drawn at each of $t = 1$ and $t = 2$ respectively: one through the curve points at the ends of an interval starting there, one just touching the curve at that instant. What do the two kinds of lines represent?

For $s(t) = -16t^2 + 16t + 32$ on the ball's flight from $t = 0$ to $t = 2$, where is $s'(a)$ positive, zero, and negative?

A city's population is $P(t) = 25000e^{t/5}$ people with $t$ in decades from today. Compute the average rate of change of $P$ between $t = 2$ and $t = 4$ (years 2030 to 2050). Round to the nearest person per decade.

For $P(t) = 25000e^{t/5}$, the limit definition gives $P'(2) = \lim_{h \to 0} 25000e^{2/5}\left(\frac{e^{h/5}-1}{h}\right)$. Why is this limit difficult to evaluate exactly?

A city's population is $P(t) = 25000e^{t/5}$ people with $t$ in decades. Numerical work gives average growth rates over $[2-h, 2]$ near $7460$ and over $[2, 2+h]$ near $7457$ people per decade. The best estimate of $P'(2)$ is about

A city's population follows $P(t) = 25000e^{t/5}$ with $t$ in decades. As $t$ increases from $0$ to $5$, what do the graph of $P$ and its tangent slopes $P'(a)$ do?