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The function f(x)=−11sin(7/3x+1/6)−2 models the average rate of change in temperature of a substance monitored in an experiment. In the function, x represents the number of minutes since the commencement of the experiment and the temperature of the substance is measured in degrees Fahrenheit. Over what range of temperatures does the average rate of change in temperature fall?

The range in the average rate of change in temperature of the substance is from a low temperature of ____°F to a high of ____°F.

May 20, 2020

#1
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The function f(x)=−11sin(7/3x+1/6)−2 models the average rate of change in temperature of a substance monitored in an experiment. In the function, x represents the number of minutes since the commencement of the experiment and the temperature of the substance is measured in degrees Fahrenheit. Over what range of temperatures does the average rate of change in temperature fall?

Hello Guest!

The range in the average rate of change in temperature of the substance is from a low temperature of  -13°F to a high of +9°F.

!

May 20, 2020
edited by asinus  May 20, 2020
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$$f(x)=-11*sin((7/3)*x+1/6)-2$$

$$\frac{df(x)}{dx}=-11\cdot cos(\frac{7}{3}x+\frac{1}{6})\cdot \frac{7}{3}\\ \frac{df(x)}{dx}=-\frac{77}{3}\cdot cos(\frac{7}{3}x+\frac{1}{6})=0$$

$$cos(\frac{7}{3}x+\frac{1}{6})=0\\ (\frac{7}{3}x+\frac{1}{6})\in\{\frac{\pi}{2},\frac{3\pi}{2}\}$$

$$x_1=(\frac{\pi}{2}-\frac{1}{6})\cdot \frac{3}{7}=\color{blue}0.60177\\ x_2=(\frac{3\cdot \pi}{2}-\frac{1}{6})\cdot \frac{3}{7}=\color{blue}1.9482$$

$$f(x)=-11*sin((7/3)*x+1/6)-2$$

$$f(x_1)=-13°F\\ f(x_2)=9°F$$

!

asinus  May 20, 2020