In math land, there is a clock. The hour hand of this clock is 1 ft long. However, the clock is weird, in that it takes precisely 2π hours for the hour hand to do a full rotation. Let’s position the clock in the xy-plane so that the center of the clock is at (0, 0), and the tip of the hour hand at "midnight" is at (0, 1). (a) It follows from the definition of sin and cos that at time t, the tip of the hour hand is at position (x, y) = (cos(t),sin(t)). Check that this makes sense by “manually” finding the position of the clock at time t = 0, π/2 , π, 3π/2 , and 2π. (b) The equation describing the circle that the hour hand traces out is x2 +y2 = 1. We’re going to calculate how fast the x-coordinate of the tip of the hour hand changes with time. We’ll do this in two ways. Using the fact that x = cos(t), what is dx/dt ? (c) Next, we’ll compute dx/dt in another way. Using the fact that y = sin(t), what is dy/dt ?
In math land, there is a clock. The hour hand of this clock is 1 ft long. However, the clock is weird, in that it takes precisely 2π hours for the hour hand to do a full rotation. Let’s position the clock in the xy-plane so that the center of the clock is at (0, 0), and the tip of the hour hand at "midnight" is at (0, 1).
(a) It follows from the definition of sin and cos that at time t, the tip of the hour hand is at position (x, y) = (cos(t),sin(t)). Check that this makes sense by “manually” finding the position of the clock at time t = 0, π/2 , π, 3π/2 , and 2π.
(b) The equation describing the circle that the hour hand traces out is x2 +y2 = 1. We’re going to calculate how fast the x-coordinate of the tip of the hour hand changes with time. We’ll do this in two ways. Using the fact that x = cos(t), what is dx/dt ?
(c) Next, we’ll compute dx/dt in another way. Using the fact that y = sin(t), what is dy/dt ?
(d) Next, use implicit differentiation to find dx/dy . Express your answer first in terms of x and y, and then in terms of t.
(e) Now find dx/dy · dy/dt in terms of t.
(f) Does your answer to part (e) match your answer to part (b)? Was this a coincidence? Why or why not?
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