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

Summary

For a smooth function $f$, measurements give the secant slope on $[-3,-1]$ as about $1.15$, the secant slope on $[0,2]$ as about $-0.4$, the tangent slope at $x = -3$ as about $3$, and the tangent slope at $x = 0$ as about $-0.5$. Which pairing of quantity and slope is correct?

A sketch must show a function $f$ with $f(-3) = 3$ whose average rate of change on $[-3, 0]$ equals $-2$. What value of $f(0)$ produces that average rate?

With $P(t) = 1.15(1.014)^t$ billion people, compute the total change in population between the start of 1993 and the start of 2000. Give your answer in billions of people, rounded to four decimal places.

China's population is modeled by $P(t) = 1.15(1.014)^t$ billion people with $t$ years since the start of 1993. What is the average rate of change of the population over $[0, 7]$?

For the model $P(t) = 1.15(1.014)^t$, computations give $AV_{[0,7]} \approx 0.0168$ and $P'(7) \approx 0.0180$ billion people per year. How do these compare, and why?

For $P(t) = 1.15(1.014)^t$, the instantaneous growth rate at the start of 1993 is $P'(0) = \displaystyle\lim_{h \to 0} \frac{P(h) - P(0)}{h}$.

Use the limit definition to compute $f'(2)$ for $f(x) = x^2 - 3x$.

Use the limit definition to compute $f'(1)$ for $f(x) = \frac{1}{x}$.

Let $f(x) = 2 - |x - 1|$. Difference quotients at $a = 1$ give $1$ for $h \lt 0$ and $-1$ for $h \gt 0$. What does this mean for $f'(1)$?

For $f(x) = \sin(x)$, the point $(\frac{\pi}{2}, 1)$ is the top of a hump in the graph, and the data indicate $f'\left(\frac{\pi}{2}\right) = 0$.