6.4 #4
a) P(t) = 4000 - 100t, and reaches 0 after 40 hours.
b) P(t) = 4000(.95)^t and never goes to zero. Doesn't make so much sense,
does it? What that means is that the approximation we're using (which
for example lets us have half a bacterium) isn't very accurate at low
numbers of bacteria. But at t=40 this is ~514, and so one might suspect
that it has a way left to go before going to zero. (If you prefer, you
can figure out when it gets to 1; this is much later than t=40.)
#5
30-20=10 net new people per 1000, so relative growth rate of 10/1000 = 1%.
#7
a) P(t) = 100 + 10t.
b) P(t) = 100 (1.1)^t.
c) They cross each other at t=0 and t=1. Between those times a) beats b),
but then b) catches up and grows much larger.
#16
a) f'(x)/f(x) = (n x^(n-1) ) / x^n = n/x, which decreases as x increases.
b) g'(x)/g(x) = (k e^kx) / e^kx = k, which is constant. So for x large enough
(larger than n/k), the relative growth rate of f is less than that of g.
6.5
Instead of the answer, I'll list which rules are involved in finding it, if it's inside the back cover.
#31
The general antiderivative is 2x + 2x^2 + 5/3 x^3 + c; at x=0 this equals c. So we need c=0.
#32
The general antiderivative is -cos x + c. At x=0 this equals c-1. So we need c=1.
6.6 #5
First break this into two pieces int(y^2) and int(y^4), and then add them. The left one is y^3/3 from 0 to 1, so 1/3; the right is y^5/5 from 0 to 1, so 1/5. Together, 1/3+1/5 or 8/15 if you prefer.
#13
The integral is -cos @, from @ between 0 and 1, so -cos 1 - (-cos 0) which is 1-cos(1). You could find out that cos(1)=.54 approximately, but best not to until you _really_ need a number.
#18
int(6x^2+1) from 0 to 2. Actually doing it: it's 2x^3+x from 0 to 2, so (2(2^3)+2)-(0)=18, like the problem number.
#21
a) int(r) from 0 to 5. The only hard part of this is reading to see what
t means. t is not years from 0 AD; it's very clearly stated that it's
years from beginning of 1990.
b) To do the integral, the 32 comes out (because it doesn't change as
we change t), and the integral is 32/.05 e^.05t from 0 to 5, so
32/.05 (e^.25-1) = 181+ billions of barrels of oil.