The key to these problems relies on the very last thing that I said in class, which goes something like:

If $v(t)$ is a velocity function, then the distance travelled during $t=a$ to $t=b$ is the same as the area under the curve of $v$ over $[a,b]$.

Example 7.1 in Section 7.1 talks explicitly about this and I will say a little more about why this the case at the beginning of class tomorrow (Monday). To attack these problems, you really need to find the area under the curve between the specified $t$ values. For example, in Exercise 7.1.1, if you find the area under the curve over the interval $[1,2]$, then you will know exactly how far the object travelled during this interval. Then you can just add this number to 5 (the given position at $t=1$) to determine where the object is at $t=2$. Finding the area under the curve is easy here (no rectangle approximations required) since the graph of the velocity function is just a straight line (think triangles and rectangles).

I hope that helps.