Calculus Online: Lab 2 Solutions

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Question 1

Part a (2 marks)

You were asked to shade the largest region which lies underneath the graph.

Part b (2 marks)

You were asked to shade the smallest region which lies contained all of the area underneath the graph.

Part c (2 marks)

You were asked to use Parts a and b to estimate the area underneath the graph.

Question 2

Part a (2 marks)

You are shown a sequence of bars whose width denotes the approximate displacement over a given time interval. As the time interval becomes smaller, the approximation becomes better. If you shrink the time intervals as small as possible, you can then estimate the total displacement.

Part b (2 marks)

The average velocity is simple the total displacement divided by the time taken which is, in this example, 5 seconds.

Part c (2 marks)

You were asked to determine approximately where the particle is moving most slowly. This position occurs when the bars are very short since this means that the displacement over a given time interval is small.

Question 3

Part a (2 marks)

A local maximum occurs when the function f(x) is zero and decreasing. This is because, as we increase x, the definite integral begins to add in minus the new area undercovered.

Part b (2 marks)

A local minimum occurs when the function f(x) is zero and increasing. This is because, as we increase x, the definite integral begins to add in the new area undercovered.

Part c (2 marks)

If you zoom in around the point 3, you can measure the change in the area that is produced by a small change in x. This enables you to estimate quite accurately the derivative of the area function which is g(3).

Question 4

Part a (2 marks)

From the graph on the right, you can see that the length of each piece of the chain is 1 and that there are 10 pieces. This means that the total length of the chain is 10 units.

Part b (2 marks)

On the graph on the right, you are shown py where y is the distance that a given piece is lifted. For instance, the first piece is lifted up 1 unit and the height of this segment on the plot on the right is 3. This means that the density is 3.

Part c (2 marks)

If we were to break the chain into more pieces each of shorter length, we would have a better approximation to the total work done. With a very short length for each of the segments, the graph on the right starts to resemble more closely a triangle. The area under this triangle represents the total work done. In this case, this area is