Problem 2: Velocity and Speed Directions: Suppose that an object moving along a straight line has position function s(t) = (t^2 - 9t + 19)e^{t-1} in feet, with t > 0 measured in minutes. Use this position function to solve the following problems. Instructor Note: Position, velocity, and speed were some of the motivating influences in the early days of calculus. a) Show that the velocity function, v(t), is equal to v(t) = (t^2 - 7t + 10)e^{t-1}. Since the answer for this part is given, it is especially important to show each step in your work. b) Write a definite integral that gives the displacement of the object during the time interval [1, 4]. Then calculate the displacement. Integral: Displacement: c) Write a definite integral which gives the average velocity of the object over the time interval [1, 4]. Then find its value. Integral:
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The velocity function (in meters per second) is given for a particle moving along a line. v(t) = t^2 - 7t + 10, 1 ≤ t ≤ 6 (a) Write a single integral that will represent the displacement by the particle during the given time interval. (b) Find the displacement during the given time interval. Give an exact value. (c) Write a single integral that will represent the total distance traveled by the particle over the time interval. (d) Find the total distance traveled by the particle during the given time interval. Rewrite your single integral from part (c) so that it no longer contains an absolute value. Then determine the exact value (you may use your calculator to evaluate the integral(s)).
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