Direction Fields — Question 10

Question 10 – Solution

Strategy. Interpret the moving zero-slope curve as a temperature comparison, then verify the response and locate its horizontal tangents.

Step 1: Heating and cooling. Below $y=T(x)$, $y'>0$ and the object warms; above it, $y'<0$ and it cools. At the three listed points, the slopes are $\boxed{1,-1,1}$.

Step 2: Check both curves. Along $y=T(x)$ the right-hand side is zero, but $T'(x)=\cos x$ is not identically zero on any interval. Thus $T$ is not a solution. By contrast, $$r'=\frac{\cos x+\sin x}{2}=2+\sin x-r,$$ so $r$ is a solution for all real $x$.

Step 3: Extrema and delay. Rewrite $$r(x)=2+\frac 1{\sqrt 2}\sin(x-\pi/4).$$ On the stated interval, its extreme values are $$\boxed{r_{\max}=2+\frac 1{\sqrt 2}\text{ at }x=\frac{3\pi}{4},\qquad r_{\min}=2-\frac 1{\sqrt 2}\text{ at }x=\frac{7\pi}{4}.}$$ The endpoint values both equal $3/2$ and lie between these extremes. The ambient maximum occurs at $x=\pi/2$, so the response maximum follows it by $\boxed{\pi/4}$ time units.

Step 4: Field interpretation. At a response extremum, $r'=T-r=0$, so the response meets the zero-slope curve. Elsewhere it need not equal $T$. A locus of zero assigned slopes can move with time without being a solution curve itself.