§1.5 Interpreting, estimating, and using the derivative

Introduction

Units of the derivative function

Toward more accurate derivative estimates

With $T(s) = \frac{50}{s}$ hours for a $50$ mile trip at $s$ mph, compute $T(75)$ as a fraction of an hour.

The function $y = P(t)$ measures the population of a city (in thousands) at the start of year $t$, with $t = 0$ corresponding to 2020. The statement $P'(2) = 21.37$ means that

Driving $50$ miles takes $T(s) = \frac{50}{s}$ hours at speed $s$ mph, and $T'(60) \approx -0.0139$. What does this value say?

Let $V(m)$ be the value in dollars of a car after $m$ miles of driving. The statement $V'(10250) = -0.89$ means that

Let $W(h)$ measure the amount of water (in liters) in a tank filled to a depth of $h$ meters. The statement $W'(0.75) = 3.43$ means that

Let $S(t)$ be the temperature in degrees Celsius of a soda in a refrigerator at time $t$ in minutes. The statement $S'(20) = -0.527$ means that after $20$ minutes, one more minute should lower the soda's temperature by about $0.527$ degrees Celsius.

Let $C(s)$ measure the rate at which a cyclist burns calories (in calories per hour) when riding at speed $s$ kilometers per hour. The statement $C'(19) = 52.1$ means that

A function satisfies $f(1) = 2.5$, $f(2) = 3.25$, and $f(3) = 3.625$. Using a central difference at $a = 2$, the estimate of $f'(2)$ is

A potato's temperature in degrees Fahrenheit is sampled at $t = 10, 20, 30$ minutes with values $94.7$, $141.7$, $167.6$. Use a central difference to estimate $F'(20)$ in degrees per minute, rounded to three decimals.

The same potato has temperature readings $F(30) = 167.6$ and $F(50) = 197.9$ degrees Fahrenheit. Use a central difference to estimate $F'(40)$ in degrees per minute, rounded to three decimals.

A baking potato's temperature increases less and less as time goes on: central difference estimates gave $F'(20) \approx 3.645$ and $F'(40) \approx 1.515$ degrees per minute. Without further calculation, which should be greater, $F'(50)$ or $F'(60)$?

If $F(46) = 192.5$ degrees and $F'(46) = 1.39$ degrees per minute, then $F(47)$ should be about $193.89$ degrees.

On the time interval $[10, 40]$ minutes, the potato's temperature readings rise throughout, but the central difference estimates of its rate of change fall from $3.645$ toward $1.515$ degrees per minute while staying positive. Which sentence describes this behavior?

Fuel consumption data gives $f(80) = 0.015$, $f(90) = 0.02$, and $f(100) = 0.027$ liters per kilometer at speeds of $80$, $90$, and $100$ km/h. The central difference estimate of $f'(90)$ is

For the fuel consumption function $f$ (liters per kilometer at speed $s$ in km/h), the statement $f(80) = 0.015$ means that

For the fuel consumption function $f$ with $f'(90) \approx 0.0006$ liters per kilometer per km/h: driving at $91$ km/h instead of $90$ km/h should raise the per-kilometer consumption by about $0.0006$ liters per kilometer.