Answer
a) The temperature is increasing the fastest on its 101th day.
b) It increases by approximately $0.637$ degrees per day.
Work Step by Step
$$y=37\sin\Big[\frac{2\pi}{365}(x-101)\Big]+25$$
a) Since we are concerned with temperature change, we need to find derivative of $y$: $$y'=37\cos\Big[\frac{2\pi}{365}(x-101)\Big]\Big(\frac{2\pi}{365}(x-101)\Big)'$$ $$y'=37\cos\Big[\frac{2\pi}{365}(x-101)\Big]\Big(\frac{2\pi}{365}(1-0)\Big)$$ $$y'=37\cos\Big[\frac{2\pi}{365}(x-101)\Big]\frac{2\pi}{365}$$ $$y'=\frac{74\pi}{365}\cos\Big[\frac{2\pi}{365}(x-101)\Big]$$
- The temperature increases the fastest when $y'$ reaches its maximum value.
We know that $$\cos\Big[\frac{2\pi}{365}(x-101)\Big]\le1$$ $$\frac{74\pi}{365}\cos\Big[\frac{2\pi}{365}(x-101)\Big]\le\frac{74\pi}{365}$$ $$y'\le\frac{74\pi}{365}$$
So the temperature increases the fastest when $y'=(74\pi)/365$, or when $$\cos\Big[\frac{2\pi}{365}(x-101)\Big]=1$$ $$\frac{2\pi}{365}(x-101)=0$$ $$x-101=0$$ $$x=101$$
which corresponds to the 101th day.
So on the 101th day, the temperature increases the fastest.
b) Since $y'$ represents the temperature change, its maximum value is also the temperature increase when it is increasing at its fastest.
So when the temperature is increasing at its fastest, it is increasing by $$y'_{\max}=\frac{74\pi}{365}\approx0.637\text{(degrees per day)}$$
