Free response · Unit 5 Beta

Greenhouse temperature: average rate, Mean Value Theorem and extremes

Modeling with a given function · Part A: graphing calculator allowed · 9 points

The question

Name: ______________________   Date: ____________   Greenhouse temperature: average rate, Mean Value Theorem and extremes (9 points)

Graph

Temperature in a greenhouse

The temperature inside a greenhouse, in degrees Fahrenheit (∘^\circF), is modeled by H(t)H(t)=62= 62+3te−0.15t+ 3te^{-0.15t}+4sin⁡(t2)+ 4\sin\left(\frac{t}{2}\right) for 0≤t≤140 \le t \le 14, where tt is measured in hours after 6 a.m. A graphing calculator is required; use radian mode.
246810121458606264666870727476tHH

  1. (a) Find the average rate of change of H(t)H(t) over the interval 0≤t≤140 \le t \le 14. Indicate units of measure. [2 points]

  2. (b) Find H′(5)H'(5). Using correct units, interpret the meaning of H′(5)H'(5) in the context of the problem. [2 points]

  3. (c) Is there a time tt, for 0<t<140 < t < 14, at which H′(t)H'(t) is equal to the average rate of change found in part (a)? Justify your answer. [2 points]

  4. (d) Find the absolute maximum temperature and the absolute minimum temperature of the greenhouse for 0≤t≤140 \le t \le 14. Justify your answers. [3 points]