MHB Understand Truncation Order 2: Conditions & Examples

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I do not understand how to do the question in part b.

Suppose I show that the function is twice differentiable (how do i do so), is that sufficient to show that the 'method' (does this refer to the prediction-correction method) has a truncation order of 2?

What is truncation?

Please see attached image.
I have the solutions for this question, however I do not understand them one bit. If anyone would like me to post them, I'd be more then happy too.

thanks
EDIT: I've solved this.

In order to show for a truncation of order 2, there are two conditions which must be satisfied. That is all the question is asking.

Thanks anyway.
 

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So what is meant by "truncation?" My journey into the depths of Google didn't really get me anywhere. I know it has to do with the accuracy of the area calculation, but couldn't get further than that.

-Dan
 

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