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?

To answer these questions, we need to analyze the given information and use the concepts of velocity and acceleration.

a) To find the time and position when the particle is farther to the left, we need to look for the points on the graph where the velocity is negative. This indicates the particle is moving to the left.

Looking at the graph, we see that the velocity is positive from t = 0 to t = 4, then becomes zero at t = 5, and finally becomes negative after t = 5.

Therefore, the particle is farther to the left when the velocity is negative, which happens after t = 5. At t = 6, the particle is at its farthest left position.

b) To determine the number of times the particle is at x = -8, we need to find the points on the graph where the position is -8.

We know that position is the integral of velocity. Since the areas under the graph represent displacement, we can calculate the position by finding the definite integrals of the velocity function over the intervals [0,3], [3,5], and [5,6].

Given 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] as 8, 3, and 2 respectively, we can write the following equations:

∫[0,3] v(t) dt = 8
∫[3,5] v(t) dt = 3
∫[5,6] v(t) dt = 2

By solving these equations, we can find the values of t where the position is -8.

c) The speed of the particle is given by the absolute value of the velocity. To determine if the speed is increasing or decreasing on the interval 2 < t < 3, we need to examine the slope of the velocity graph during that interval.

If the slope of the velocity graph is positive, the speed is increasing. If the slope is negative, the speed is decreasing.

d) The acceleration is given by the derivative of the velocity. If the derivative of the velocity is negative, it means the acceleration is negative.

To find the time intervals when the acceleration is negative, we need to look for intervals where the derivative of the velocity function is negative.