Integrated Language

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For each of the first three problems, write an equivalent statement using function and/or integral notation.

If $f$ is the population of Hicktown,

(a): The population of Hicktown was $5.4$ million in 2010.
(b): The population of Hicktown was increasing at a rate of $110000$ per year in 2010.
(c): The population of Hicktown grew by $1,000,000$ people between 2010 and 2018.

Joan decides to take a run outside. Let $p(t)$ be the position Joan is from home (east or west) after $t$ minutes.

(a): At $t = 10$ , Joan is running at $100$ meters per minute.
(b): She comes to a hill. At $t=12$ , her velocity is dropping at a rate of $10$ meters per minute each minute.
(c): At $12$ minutes, Joan was $185$ meters east of where she was at $10$ minutes. (Side Note: This may or may not mean the same as Joan traveled $185$ meters over that period of time- why?).

If $T(t) =$ the temperature at OSU at time $t$ hours after noon,

(a): The temperature at $11$ AM is $67$ degrees.
(b): At $2$ PM the temperature was $75$ degrees and rising.
(c): Between $5$ PM and $6$ PM, the temperature rose less than it did between $2$ PM and $5$ PM.

Let $H$ stand for the height of a balloon (in feet above the ground) as a function of time (in minutes). Translate the following into English:

(a): $H(0) = 1000$
(b): $H'(0) = 0$
(c): $H'(5) = -52$
(d): $H(5) - H(0) = -201$ (Is there another way we can write this now?)

Let $R(t)$ be Rachel’s height at $t$ years of age and $J(t)$ be Joe’s height at $t$ years of age. Translate the following into English.

(a): $R(14) > J(14)$
(b): $R'(14) < J'(14)$
(c): $\int _{12}^{15} R'(t) \, dt > \int _{12}^{15} J'(t) \, dt$
(d): $R(3) + \int _3^{16} R'(t)\, dt$ (What does this describe and is there another way to write this?)

Debbie the Dieter goes on every fad diet that she sees advertised on TV, thus wreaking havoc on her system. She is constantly losing and gaining weight. While her TV was being repaired one day, Deb, using her many charts of her weight through the years, figured out that her weight (in pounds) at time $t$ years (her age) is given by the function $W(t)$ . What would $\int _{30.5}^{37} W'(t)\, dt$ mean in terms of her age and weight?

2024-10-10 13:51:05