y t is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known

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4 A particle is moving along a curve so that its position at time t is x t, y t, where x t t 4t 8 and y t is not explicitly given. Both x and y are measured in meters, and t is measured in seconds. It is known that te t 3 1. (a) Find the speed of the particle at time t 3 seconds. (b) Find the total distance traveled by the particle for t 4 seconds. (c) Find the time t, t 4, when the line tangent to the path of the particle is horizontal. Is the direction of motion of the particle toward the left or toward the right at that time? Give a reason for your answer. (d) There is a point with x-coordinate 5 through which the particle passes twice. Find each of the following. (i) The two values of t when that occurs (ii) The slopes of the lines tangent to the particle s path at that point 1 (iii) The y-coordinate of that point, given y 3 e (a) Speed x 3 y 3.88 meters per second (b) x t t 4 Distance 4 t 3 t 4 te or meters : 1 : integral t 3 (c) when te 1 and t 4 This occurs at t.794. Since x.794, the particle is moving toward the right at time t.7 or.8. 3 : 1 : considers 1 : t.7 or.8 1 : direction of motion with reason (d) x t 5 at t 1 and t 3 At time t 1, the slope is At time t 3, the slope is 3 1 y 1 y e t 1 t 1 t 3 t : 1 : t 1 and t 3 1 : slopes 1 : y-coordinate

5 The velocity vector of a particle moving in the plane has components given by 14cos t sin e t and At time t, the position of the particle is, 3. 1 sin t, for t 1.5. (a) For t 1.5, find all values of t at which the line tangent to the path of the particle is vertical. (b) Write an equation for the line tangent to the path of the particle at t 1. (c) Find the speed of the particle at t 1. (d) Find the acceleration vector of the particle at t 1. (a) The tangent line is vertical when x t and y t. On t 1.5, this happens at t 1.53 and t or : 1 : sets (b) t 1 y 1 x x 1 x t y 1 3 y t : 1 : 1 1 : x 1 1 : y 1 1 : equation t The line tangent to the path of the particle at t 1 has equation y x (c) Speed x 1 y (d) Acceleration vector: x 1, y ,.161 : 1 : x 1 1 : y 1

6 A diver leaps from the edge of a diving platform into a pool below. The figure above shows the initial position of the diver and her position at a later time. At time t seconds after she leaps, the horizontal distance from the front edge of the platform to the diver s shoulders is given by x t, and the vertical distance from the water surface to her shoulders is given by y t, where x t and y t are measured in meters. Suppose that the diver s shoulders are 11.4 meters above the water when she makes her leap and that.8 and t, for t A, where A is the time that the diver s shoulders enter the water. (a) Find the maximum vertical distance from the water surface to the diver s shoulders. (b) Find A, the time that the diver s shoulders enter the water. (c) Find the total distance traveled by the diver s shoulders from the time she leaps from the platform until the time her shoulders enter the water. (d) Find the angle,, between the path of the diver and the water at the instant the diver s shoulders enter the water. (a) only when t Let b The maximum vertical distance from the water surface to the diver s shoulders is 11.4 b y b 1.61 meters. Alternatively, y t t 4.9 t, so y b 1.61 meters. 3 : 1 : considers 1 : integral or y t (b) 11.4 A y A A 4.9A A seconds. when : 1 : equation (c) A meters : 1 : integral (d) At time A, t A The angle between the path of the diver and the water is 1 tan or : 1 : at time A

7 A particle moves along the x-axis so that its velocity at time t, for t 6, is given by a differentiable function v whose graph is shown above. The velocity is at t, t 3, and t 5, and the graph has horizontal tangents at t 1 and t 4. The areas of the regions bounded by the t-axis and the graph of v on the intervals, 3, 3, 5, and 5, 6 are 8, 3, and, respectively. At time t, the particle is at x. (a) For t 6, find both the time and the position of the particle when the particle is farthest to the left. Justify your answer. (b) For how many values of t, where t 6, is the particle at x 8? Explain your reasoning. (c) On the interval t 3, is the speed of the particle increasing or decreasing? Give a reason for your answer. (d) During what time intervals, if any, is the acceleration of the particle negative? Justify your answer. (a) Since v t for t 3 and 5 t 6, and v t for 3 t 5, we consider t 3 and t 6. 3 x 3 v t x 6 v t Therefore, the particle is farthest left at time t 3 when its position is x 3 1. (b) The particle moves continuously and monotonically from x to x 3 1. Similarly, the particle moves continuously and monotonically from x 3 1 to x 5 7 and also from x 5 7 to x : 3 : 1 : identifies t 3 as a candidate 1 : considers 1 : conclusion 6 v t 1 : positions at t 3, t 5, and t 6 1 : description of motion 1 : conclusion By the Intermediate Value Theorem, there are three values of t for which the particle is at x t 8. (c) The speed is decreasing on the interval t 3 since on this interval v and v is increasing. (d) The acceleration is negative on the intervals t 1 and 4 t 6 since velocity is decreasing on these intervals. with reason : 1 : justification

8 A particle moving along a curve in the xy-plane has position x t, y t at time t with The particle is at position 1, 5 at time t 4. (a) Find the acceleration vector at time t 4. 3t and t 3cos. (b) Find the y-coordinate of the position of the particle at time t. (c) On the interval t 4, at what time does the speed of the particle first reach 3.5? (d) Find the total distance traveled by the particle over the time interval t 4. (a) a 4 x 4, y 4.433, (b) 4 t y 5 3cos 1.6 or : 1 : integrand 1 : uses y 4 5 (c) Speed x t y t t 3t 9cos : 1 : expression for speed 1 : equation The particle first reaches this speed when t.5 or.6. (d) 4 t 3t 9cos : 1 : integral

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