a. Find equations for the tangents to the curves y = sin 2x and y = -sin (x/2) at the origin. Is there anything special about how the tangents are related? Give reasons for your answer.

b. Can anything be said about the tangents to the curves y = sin mx and y = -sin (x/m) at the origin (m a constant ≠ 0)? Give reasons for your answer.

c. For a given m, what are the largest values the slopes of the curves y = sin mx and y = -sin (x/m) can ever have? Give reasons for your answer.

d. The function y = sin x completes one period on the interval [0, 2p], the function y = sin 2x completes two periods, the function y = sin (x/2) completes half a period, and so on. Is there any relation between the number of periods y = sin mx completes on [0, 2p] and the slope of the curve y = sin mx at the origin? Give reasons for your answer.