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 = e^{5x} and y = e^{-7x} are both solutions to the differential
equation

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

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

3. Textbook, page 276 - 277, #2aceh

4. Prove that
if d > 0, then (d^{r})^{s} = d^{(rs)}.

5. (a) Find the value of the following limit
L: lim_{x }_{¨}_{ 0} [ln(1 + x) Ð 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 lim_{x }_{¨}_{ 0} (1 + x)^{ 1/x}
, which is

lim_{x }_{¨}_{ 0} e^{[ln(1
+ x)]/x} , must equal
lim_{x }_{¨}_{ 0} e^{[ln(1
+ x) Ð ln 1]/x}.
Because e^{x} is a continuous function, the last limit in this string is e^{L}. So, what is lim_{x }_{¨}_{ 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?