A teacher asks students to sketch all integral curves on the slope field for that…

MATHEMATICS · Single correct · AP · Calculus AB · Differential Equations · Differential Equations

Problem

A teacher asks students to sketch all integral curves on the slope field for dydx=sin⁡x+cos⁡y\dfrac{dy}{dx}=\sin x + \cos y that pass through the initial points (0,0)(0,0) and (π,0)(\pi,0). Which of the following statements about these two curves is best supported by the slope field? (a) Both curves initially fall because the slope at each given point is negative. (b) The curve through (0,0)(0,0) is horizontal, whereas the curve through (π,0)(\pi,0) rises, because cos⁡0=0\cos 0=0 makes dydx=0\dfrac{dy}{dx}=0 at (0,0)(0,0). (c) Both curves initially rise because the slope at each given point is positive. (d) The curve through (0,0)(0,0) rises, whereas the curve through (π,0)(\pi,0) falls, because sin⁡π=1\sin \pi=1 gives slopes of opposite sign.

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