Math H161    Brown    Au04

Homework 5,  due Wednesday, October 27, 2004

1.  Textbook, page 269, #2, 4, 12, 14, 18

2.  (a) Verify that y = e5x  and  y = e-7x  are both solutions to the differential equation

y'' + 2y' – 35y = 0, by substituting each of the two functions into the differential equation.

(b) Factor the polynomial  equation t2 + 2t – 35 = 0.  What connection do you see between the roots of this polynomial and the solutions of  y'' + 2y' – 35y = 0?

3.  Textbook, page 276 - 277,  #2aceh

4.  Prove that if  d > 0, then  (dr)s = d(rs).

5.  (a)  Find the value of the following limit L:     limx ® 0  [ln(1 + x) – ln 1]/x = L  by simply observing that a particular derivative is actually defined to be this limit.  And you can find this derivative by using your standard differentiation formulas, thereby easily evaluating the limit L.

(b)  The number  (1 + x)1/x  has been defined as  e[ln(1 + x)]/x (acetate 98). Then we know that because ln 1 = 0, we must have limx ® 0  (1 + x) 1/x , which is

limx ® 0 e[ln(1 + x)]/x , must equal  limx ® 0 e[ln(1 + x) – ln 1]/x.  Because  ex  is a continuous function, the last limit in this string is eL.  So, what is  limx ® 0  (1 + x) 1/x ?

6,  The following problem has no solution because the various facts stated impose incompatible constraints.  Change the problem as little as possible, state your new version of the problem carefully, and then solve it.

Throw a ball straight up from the top of a tower 25 feet high at a speed of  15  feet/second.  It rises for  2  seconds before turning back toward the ground and falling into a hole  5  feet deep.  When does it hit the bottom of the hole?