1.  To take a result to 42 significant figures, you can use DERIVE (enter expression, Simplify, Approximate).  For example, if you want 57/117 to 100 places, enter 57/117, choose Simplify, choose Approximate, and when it asks you for digits of precision change the number (which probably reads 10) to 42.  Click on Approximate, and it'll appear as 0.495726495726495726495726495726495726495726.

2.  Expand a binomial:  (a + b)^2 = a ( a + b) + b ( a + b) = a^2 + ab + ba + b^2 = a^2 + 2 a b + b^2.  You should know how this works.  You should remember that (a + b)^2 = a^2 + 2 a b + b^2, and you should always be able to verify this using the distributive law, etc., as demonstrated above.

3.  Sequences:  There are two fairly obvious things you can do to a sequence:

• you can find the differences of the numbers in the sequence, or
• you can find the ratios of the numbers in the sequence 

One of these operations will often give you a new sequence with a more obvious pattern.  If you can get the pattern of the new sequence and extend it, you can then use your results to extend the original sequence.

You can take this idea a little further.  You can for example find the sequence of differences, then you can find the differences of that new sequence.  Or you could find the sequence of differences, then use this to find the sequence of ratios of the differences.  Other combinations are possible.

4.  For a ball traveling down an 80 cm ramp in 5 seconds, we reason out average velocity as follows:

• Average velocity is average rate of change of position with respect to clock time.
• Average rate of change of A with respect to B is (change in A) / (change in B).
• We apply the definition with A being the position and B being the clock time.
• Thus average velocity is (change in position) / (change in clock time) = 80 cm / (5 sec) = 16 cm / sec.

The question might also ask for the average rate of change of velocity with respect to clock time, in which case you should start with the definition of average rate, identify the A and B quantities, figure out what you can about the change in each of these quantities, and do the required calculation.

5.  If there are 100 demented and 900 sane people, then if 20% of the demented become sane, it means that 20 demented people will become sane.  If 10% of the sane people become demented, then 90 sane people will become demented.  From the standpoint of the sane population, you start with 900, lose 90 and gain 20.  From the standpoint of the demented population, you start with 100, lose 20 and gain 90.

6.  If a tree has 500 bugs and its nearest neighbors have 300 bugs and 600 bugs, respectively, and if 10% of the bugs on every tree travel to each of its nearest neighbors, then the first tree loses 50 bugs to one neighbor and 50 to the other.  It gains 30 from one neighbor and 60 from the other, so it ends up with a net loss of 10 bugs.  Its population after the transition will therefore be 490.

We know each neighbor gains 50 from the first tree, and we know how many each gains from the first tree.  If we knew the populations of each of the neighbors' other neighbor, we could therefore calculate the new populations of the neighbors. 

7.  If you expand the binomial in the expression 6 ( t - 5)^2 you get 6 ( t^2 - 10 t + 25).  Multiplying through by 6 we get 6 t^2 - 60 t + 150.  The result is in the form a b^2 + b t + c, with a = 6, b = -60 and c = 150.