a partial moves along the x-axis so that its velocity at time t, for 0< = t = < 6, is given by a differentiable function v whose graph is shown above. The velocity is 0 at t=0, t=5, and the graph has horizontal tangents at t=4. the areas of the regions bounded by the t-axis and the graph of v on the intervals [0,3], [3,5] and [5,6] are 8,3 and 2, respectively. at time t= -2.
a) for 0 < = t < = 6, find both the time and the position of the particle when the particle is farther to the left.
b) for how many values of t where 0 < = t < = 6 is particle at x= -8?
c) on the interval 2<t<3 is the speed of the particle increasing or decreasing?
d) During what time intervals if any is the acceleration of the particle negative?
Without the graph that is supposed to be "above", I don't see how we can help you find the answers.
To answer the questions, we need to analyze the given information and use concepts from calculus, such as velocity, position, area, and acceleration.
a) To find when the particle is farther to the left, we need to consider the position of the particle. Since the velocity function, v(t), gives us information about the velocity of the particle, we can determine the position by integrating the velocity function. The position function, denoted as s(t), can be found by integrating the velocity function:
s(t) = ∫ v(t) dt
Integrating the velocity function gives us the displacement of the particle from the starting point at t=0. By finding the values of t where the position is the farthest to the left, we can answer this part of the question.
b) To determine how many times the particle is at x = -8, we need to find the values of t where the position function, s(t), is equal to -8. By solving the equation s(t) = -8, we can find the values of t for which the particle is at x = -8.
c) To determine if the speed of the particle is increasing or decreasing on the interval 2 < t < 3, we need to consider the velocity function, v(t). The speed is the magnitude of the velocity, so we need to examine the sign of the derivative of the velocity function within the interval 2 < t < 3. If the derivative is positive, the speed is increasing. If the derivative is negative, the speed is decreasing.
d) To find the time intervals during which the acceleration of the particle is negative, we need to analyze the acceleration function, a(t). Since the acceleration is the derivative of the velocity function, we can examine the sign of the derivative of the velocity function within different time intervals. When the derivative of the velocity function is negative, the acceleration is negative.
Overall, using calculus concepts such as integration and differentiation, we can analyze the given information to answer each of the questions.