Question

1. (20pts) Given below is the graph of the rate of change of temperature over an eight- hour period of time, i.e. degrees per hour. rate of chango of temp. ime 2 a. List all the times the temperature reaches a local maximum and provide a short explanation b. List an interval where the graph of temperature with respect to time is concave down. c. Is the temperature at time t = 8 hours cooler than the same as, or hotter than the temperature at t = 2? Make your choice clear, and explain your thinking using the provided rate of change graph.

Answer 1

a)

we are given derivative (rate of change of temp) graph and we know that there is relative maxima when derivative changes from positive to negative.

Here derivative changes from positive (above x axis) to negative(below x axis) at t=4. So there is local maxima at t=4

b)

Temp. graph would be concave down when rate of change is decraesing because then second derivative of tem graph would be negative .

And we see that derivative graph is decreasing from t=2 to 2=5

So temp graph is concave down on interval  

$2\leq t\leq 5$

c)

If we need to find temp. from derivative graph then we need to find area under the derivative graph for that particular time.

$T(t)= T_{o}+\int_{0}^{t}f(x) dx$

We see that area under the curve for interval 0<t<2 is less than the area under the curve for interval 0<t<8.

That means temp at time t=8 is hotter than the temp at t=2