Exercises

1.

Consider the function f(x) = x³-6x²+5x+12 on the interval [-1,3]. Use the command leftsum to approximate the definite integral

$$\int_{-1}^{3} f(x) \, dx$$

to two decimal places. Then explain why the leftsum and rightsum commands give the same numerical values for the same number of subintervals. (Hint - use the leftbox and rightbox commands to see what is going on.)

2.

Suppose that the velocity of a particle moving in one dimension is given by

$$v(t) = t\sin(t^2)$$

Find the average value of v(t) on the following time intervals.

(a)

[0,3]

(b)

[0,6]

(c)

[3,6]

3.

Let the continuous function v(t) represent the velocity of a particle moving in one dimension. Let a and b be real numbers in the domain of v and let c be (a+b)/2. That is, c is exactly halfway between a and b. Show that the following equation is always satisfied

$$\bar{v}_{ab} = \frac{\bar{v}_{ac} + \bar{v}_{cb}}{2}$$

where $\bar{v}_{ab}$ is the average value of v on the interval [a,b], $\bar{v}_{ac}$ is the average value of v on the interval [a,c], and $\bar{v}_{cb}$ is the average value of v on the interval [c.b]. Note - this is not a Maple problem. This problem is a thinking problem. I would suggest leaving some space on your worksheet and writing your answer in by hand. Part of using Maple is learning when not to use it.