x
for a certain function f converges to
14. The Taylor series about
5
f x for all x in its interval of
x
convergence. The nth derivative of f at
5
is given by
n
1
n
!
1
n
and
f
5
f
5
n
2
2
n
2
x
Show that the sixth-degree Taylor polynomial for f about
5
approximates
f
6
with an error less than
1
.
1000
2
dP
P
15. The population
P t of unicorns in a forest satisfies the logistic differential equation
3
P
.
dt
6000
(a) If
P
0
4000
, what is
lim
P t
? Is the solution curve increasing or decreasing? Justify your answer.
t
(b) If
P
0
10, 000
, what is
lim
P t
? Is the solution curve increasing or decreasing? Justify your
t
answer.
(c) If
P
0
20,000
, what is
lim
P t
? Is the solution curve increasing or decreasing? Justify your
t
answer.
(d) If
, what is the population when it is growing the fastest? Where is the solution curve
P
0
4000
concave up? Concave down? Justify your answer.
dx
dy
3
2
t . At time
t , the object at
16. (Calculator Permitted) Given
cos
t
and
3sin
t
for 0
3
2
dt
dt
position
4,5 .
t .
(a) Find the speed of the object at time
2
t .
(b) Find the total distance traveled by the object over the time interval 0
1
t .
(c) Find the position of the object at time
3
2
n
1
1 2
x
3
x
n
17. (Calculator Permitted) Given
f x
x
n
1
3
9
27
3
1
f x
3
(a)
lim
.
x
x
0
1
(b) Write the first three nonzero terms and the general term for the infinite series that represents
.
f x dx
0
(c) Find the sum of the series found in part (b).
18. (Calculator Permitted) The function f has derivatives of all orders for all real numbers x . Assume that
4
f
f
f
,
,
,
, and
x
f
2
5
2
3
2
4
2
1
f
3
for all x in
2, 2.2 .
x .
(a) Write the third-degree Taylor polynomial for f about
2
(b) Use your answer to (a) to approximate
f
2.15
. Give your answer correct to five decimal places.
(c) Use the Lagrange error bound on the approximation of
f
2.15
to explain why
f
2.15
4.7
.