Calculus BC
Section ________
Name _____________________
Date ______________________
~Assignment 9-B
1. Approximate the sum of the alternating series
n 1
1
n 1
n3
with an error less than or equal to 0.001.
2. If the first four terms are used to approximate the series
1
2n
n 1
n 1
1
3
find an upper bound for the
remainder.
3. Approximate e 1 with a sixth degree Maclaurin Polynomial and find an upper limit of the
Alternating Series Remainder.
4. How many terms of a Maclaurin Polynomial are needed to approximate sin 1 with an error of
less than 0.001?
5. How many terms of a Maclaurin Polynomial are needed to approximate sin 2 with an error of
less than 0.001?
6. If a Taylor Polynomial centered at 1 is used to approximate ln 2 with an error of less than
0.001, how many terms are needed?
7. If f 4 x 4 find the Lagrange error bound if a third degree Taylor Polynomial centered at
x 1 is used to approximate f 2 . Assume the series converges for x = 2.
8. If P3 2 5 for the function from problem 7, find the range of possible values for f 2 .
9. If f
6
x 200sin x
and x .5 is in the interval of convergence of the power series for f ,
then find the error when a fifth-degree Taylor polynomial, centered at x 0 is used to
approximate f .5 .
10. If a sixth degree Taylor Polynomial centered at x 0 is used to approximate f 3 , find the
Lagrange error bound for each of the following if the graph shown is a portion of the graph of
f 7 x . Assume the series converges for x = 3.
a.
b.
y
c.
y
y
x
x
x
11. Assuming the function from problem 10 is represented by an alternating series, which of the
three answers would be the same using an alternating series error bound?
12. The function f x e2x is approximated by the polynomial f x 1 2 x 2 x 2 x 3 .
3
4
For what x-values will this approximation have an error of less than 0.001?
13. For f x ln x, c 1 :
a. Write a Taylor Polynomial P4 x .
b. Write a power series for f x using notation.
c. Approximate f 1.3 using P4 1.3 .
d. Find the actual value of f 1.3 .
e. Find the Lagrange error (remainder) bound, R4 1.3 .
f. Find the number of terms from the Taylor Polynomial needed to approximate f 1.3 with an
error (remainder) less than .001.
x3
14. Find an upper limit for the error when the Taylor polynomial T x x
is used to
3!
approximate f x sin x at x 0.5 .
15. Let f x be a function whose Taylor series converges for all x . If f
n
x 1 what is the
minimum number of terms of the Taylor series, centered at x 1 , necessary to approximate
f 1.2 with an error less than 0.00001? Assume the series has no zero terms.
16. (calculator allowed)
x h x h x h x h x h 4 x
2
3
4
5
6
7
Let h be a function having derivatives of all orders for x 0 , selected values of h and its first four
derivatives are indicated in the table above.
(a) Write the first-degree Taylor polynomial for h about x = 2 and use it to approximate
h 1.9 .
(b) Write the third-degree Taylor polynomial for h about x = 2 and use it to approximate
h 1.9 .
(c) Assuming the fourth derivative of h is an increasing function, use the Lagrange error bound
to show that the third-degree Taylor polynomial for h about x = 2 approximates h 1.9
with error less than 3 105 .
3
d2y
2 x , find
.
x
dx 2
17.
If y
18.
Find the point(s) where the line(s) tangent to the graph of f ( x ) x 3 x 2 x 3 is/are parallel
1
3
to the graph of y x 5 .
Integrate:
x
2
x 1
dx
x
19.
3 4t
t 2 9 dt
22.
Find a general solution of the differential equation t
23.
The rate of growth of bacteria in a culture is proportional to the number of bacteria present at any
time t. If there were 2000 bacteria present 3 days after the
introduction of bacteria into the culture, and 5000 present 2 days later, find:
a. the growth rate for the bacteria in the culture.
b. the number of bacteria initially introduced into the culture. (Round to the nearest whole
number.)
c. the estimated number of bacteria in the culture 12 days after the bacteria were initially
introduced. (Round to the nearest hundred.)
24.
A small dog kennel with 8 individual rectangular holding pens of equal size is to be constructed
using 144 ft of chain link fencing material. One side of the kennel is to
be placed against a building and requires no fencing, as shown in the figure below.
20.
arcsin
4x
2
dx
21.
dy
dt
2y
dy
dt
a. Find the dimensions (for each holding
pen) that produce a maximum area for
each pen.
b. What is that maximum area for each
holding pen?
25.
A block of ice is exposed to heat in such a way that the
block maintains a similar shape as it melts. The block
of ice is initially 2 feet wide, 2 feet high, and 3 feet long,
as shown at right. If the change in the width of the ice
is 1 ft/hr, find:
3
hint: let u
Find lim
2
2 cos
e 2 1
. Solve for y.
Building
2 ft
3 ft
a. the rate of change in the volume of the block of ice when the width is 1 ft.
b. the amount of time it will take for the block of ice to completely melt.
26.
x 1
2 ft
Selected Answers:
1. .902
2. R
6. 1000 terms
1
249
3. .368, R
7. R
1
6
1
5040
4. three terms
8. 4 f 2 5
5
6
10a. R .867 or .868
b. R 1.735 or 1.736
12. .197 or .196 x .196 or .197
13a. P4 x 1
x 1
2
x 1
3
2
3
f 1.3 P4 1.3 .261975 ,
c.
x 1
d.
1
6
9. R .0043
c. R 1.301 or 1.302
4
5
3
2
2
20.
1
2
2
R .000029 .00003
1
17. y 6 x 3 x
arcsin
x
2
C
2
18.
n 1
,
n
15. five terms
0,3 ,
3
2,
h 1.9 2.624
19. arctan 2 ln t 2 9 C
t
3
11
3
21. 2 x 1 2arctan x 1 C
22.
y C t 1
23. a. k .458 b. 505 or 506 bacteria c. 123,500 bacteria
24. a. 8 ft. by 9 ft.
n
h 1.9 2.6 ,
(b) h x 3 4 x 2 2 x 2 x 2 ,
(c)
11. c
1 x 1
, b. f x
4
n 1
f 1.3 ln 1.3 .262364 ,
e. R4 1.3 .000486 , f. four terms
14. Alt Series Error .00026 or Lagrange Error .0026
16. (a) h x 3 4 x 2 ,
5. five terms
b. 72 ft 2
25. a.
3 ft 3
2 hr
b. 6 hours
26. 2
2
0
