(b) Use the chain rule in the form to find a formula for in terms of .
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(c) Draw the tangent lines to the curve in part (a) at all of the plotted points from to = , and use part (b) to find the slope of each tangent line.
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Exercise 11.2: A spring is causing an object to oscillate up and down sinusoidally between and . At the object is at maximum height , and it makes one full motion (from 3 to -3 and back to 3) twice a second. Meanwhile, the whole apparatus is moving at constant velocity 2 units/sec down the x-axis, starting at 0 at time t = 0.

(a)Find x(t).
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(b) Find y(t). Namely, use the information given to (i) graph y as a sinusoidal function of time in the ty-plane, and (ii) find A, B, C and D in the formula .

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(c) Graph the path of the object in the xy-plane.

Find the formula for in two ways:
(i) Use the graph of the path to find a, b, c and d in (where we're using small letters for the constants, so as to avoid confusion with the sinusoidal function in part (b)).

(ii) Eliminate t, and use part (b) to write y directly in terms of x. Use both methods, and check that your answers agree.
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Exercise 11.3: The same as Exercise 11.2, except suppose that the whole apparatus is moving down the x-axis according to the formula . Notice that the vertical motion y(t) as a function of time is unaffected by this change. However, in the horizontal direction the object is accelerating i.e., its velocity, rather than being constant, is steadily increasing according to the formula v(t) = ; and the acceleration is constant. Here can be thought of as , where the constant acceleration g here is 2 / (and of course, unlike gravity, which pulls in the negative y-direction, here the constant acceleration is pulling in the positive x-direction). Also notice that your graph of the path is no longer sinusoidal: the distance between peaks becomes greater as the object accelerates to the right. The formula for can be obtained by substituting in place of t in the formula in part (b).

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Exercise 11.4: A wheel of radius 2 ft is centered at the point (0 ft,100 ft) and is rotating clockwise at the constant rate of one revolution every 2 seconds. At time t = 0 a pebble on the perimeter of the wheel is located at the point (0,102). At time t = 0 the wheel is dropped. Write down formulas for the x- and y-coordinates of the pebble as functions of time, and graph its trajectory for t between 0 and 2.5 sec (plot points at intervals of 1/4 sec).
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Exercise 11.5: A wheel rolls down the x-axis from left to right without slipping at constant speed 6 m/sec. The diameter of the wheel is 1.5 m. As it passes over the origin at time t = 0 a pebble gets stuck in a spoke right under the center at a distance 25 cm from the rim. Find x(t) and y(t) for the pebble.

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Exercise 11.6: In an episode of the TV program ``Northern Exposure,'' the hero Chris and a mathematician named Amy send his motorcycle careening over a cliff. We want to describe the path of the point P at the top of one of the wheels at the instant t = 0 when the wheel starts over the cliff. Suppose that the motorcycle wheel has a radius 0.25 m, and until time t = 0 the motorcycle is traveling at 10 m/sec. Neglect air resistance, and take g = 9.8 / . Suppose that the wheel is traveling in the xy-plane, the motorcycle is moving horizontally (along the negative x-axis) until time t = 0, and the origin is the point at the edge of the cliff where the wheel is touching at the instant t = 0. Find the coordinates of P at time t (while the motorcycle is falling).

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