Math 2414 Activity 18 (74 parts)(2 per person)(Due by August 18)
1. Using first and second Taylor polynomials with remainder, show that for x  0 ,
x x2
x
1   1 x 1 .
2 8
2
2. Using a second Taylor polynomial with remainder, find the best constant C so that for x  0 ,
1

x x2 
3
1  x   1     Cx3 .
 3 9 
3. The nth Derivative Test : Suppose that f has n continuous derivatives and
n1
n
f   c   f   c    f    c   0 , but f    c   0 .
Case I: If n is even and f 
f  x  f c 
f  x  f c 
f
n
 z
f
f
n
z
n!
n
c
n!
n
c  0 ,
 x  c
 x  c
n
n
then from Taylor’s Theorem, we know that
for some z between x and c.
Or equivalently,
n
. Since f   is continuous and f    c   0 , for x close
n
 0 . This means that f  x   f  c   0 , or f  x   f  c  for x
n!
close to c. So f has a local minimum at c.
n
a) Investigate Case II: If n is even and f    c   0 .
to c,
 x  c
n
b) Investigate Case III: If n is odd and f    c   0 .
4. Test x  0 as a local extremum for the following functions:
a) x3  2
b) sin x  x
3
4
x2
x
x
c) sin x  
d)  cos x  1 
2
6 24
4
3
2
5. Let f  x   x  12 x  44 x  2 x  1
n
a) Find the second Maclaurin polynomial for f  x  .
b) Find the fourth Maclaurin polynomial for f  x  .
c) Find the fourth Taylor polynomial centered at x  3
for f  x  .
x2 x4
6. Explain why the polynomial 1  x  
cannot
2 8
be the fourth Maclaurin polynomial for the
function graphed:
(12 parts)
7. Let M 2  x   a0  a1 x  a2 x 2 be the second Maclaurin polynomial generated by the function f
graphed below. Determine the signs of a0 , a1 , and a2 .
8. Given the graph of the differentiable function f,
which has a minimum at x  0 and inflection point at
x  2 , determine the signs of the coefficients in the
following Taylor polynomials for f:
a) P2  x   a0  a1 x  a2 x 2
b) P2  x   a0  a1  x  1  a2  x  1
2
c) P2  x   a0  a1  x  2   a2  x  2 
2
d) P2  x   a0  a1  x  3  a2  x  3
2
9. How accurate are the following Maclaurin polynomial approximations if x  .1 ?
a) e x  1  x  12 x 2  16 x3  241 x 4
1
b) sin x  x  16 x3  120
x5
c) ln 1  x   x  12 x 2  13 x3  14 x 4
d) 1  x  1  12 x  18 x 2
10. For f  x   sin x , the nth Maclaurin polynomial is
Pn  x   0  x  2!0 x2  3!1 x3 

n
f    0
n!
xn .
(9 parts)
 sin c ; n  1  4k
 cos c ; n  1  4k  1
n 1
f    z  n1
 n 1
And the remainder is Rn  x    n1! x , where f
. So for a
 c   

sin
c
;
n

1

4
k

2

 cos c ; n  1  4k  3
fixed value of x , Rn  x  
f
n 1
z
 n1!
x
n 1
f
z
 n1!


a) Perform the ratio test on the series
n 1

n 0
x
n 1
, but f
 n1
n 1
x
.
 c   1 , so Rn  x  
 n  1!
n 1
x
.
 n  1!
n 1
x
b) What does the result of part a) tell you about lim
for any value of x?
n  n  1!
c) What does the result of part b) tell you about lim Rn  x  ?
n
11. Use the first Maclaurin polynomial with remainder for f  x   1  x  , with n  1 and
x  1 to get an inequality between f  x  and 1  nx .
n
12. Find the fourth Taylor polynomial centered at 2 for the function f  x   x 4 , and show that it
represents f exactly.
13. Find the third Taylor polynomial centered at 1 for the function f  x   x3  2 x 2  3x  5 , and
show that it represents f exactly.
14. The fourth Maclaurin polynomial for sin x , P4  x  , is really a third degree polynomial since
x3
the coefficient of x is zero. So sin x  x   R4  x  .
6
a) Show that if 0  x  .5 , then R4  x   .0002605 .
4
.5
b) Approximate
.5
 sin x dx with  P  x  dx , and give an upper bound on the error.
4
0
0
15. Find Maclaurin series for the following functions:
a) cos  x 
b) e
d) sin  x 4 
e) x 2e x
 2x
c) x tan 1 x
f) x cos 2 x
(14 parts)
16. Express the following antiderivatives as infinite series:
sin x
a) x cos  x3  dx
b)
dx
x



c)
ex  1
dx
x
17. Use series to approximate the following definite integrals to within .001 of their exact
values:
1
2
1
a)

x cos  x3  dx
b)
18. Use series to evaluate the following limits:
x  tan 1 x
a) lim
x0
x3
1  cos x
x 0 1  x  e x
b) lim
sin x  x  16 x3
c) lim
x0
x5
x 0
2
0
0
e) lim

x 2e  x dx
tan x  x
x 0
x3
d) lim
x  sin x
x3 cos x
sin x  tan x
x 0 sin 1 x  tan 1 x
f) lim
19. Use multiplication, division , or a trig identity to find at least the first three nonzero terms in
the Maclaurin series of the following functions:
a) e x cos x
b) sec x
c)
x
sin x
d)
2 tan x
tan 2 x  1
20. Use Maclaurin series to find the sum of the following series:


d)

a)
n 0

n 0

x4n

1
 
n!
n
b)

n 0
3n
5n n !
n
e) 3 
 ln 2 
f) 1  ln 2 
2!
2
 ln 2 

3
3!

 1  2 n
6 2 n  2 n !

sin 2 x sin 3 x sin 4 x
h) f  x   1  sin x 



2!
3!
4!
c)

n 0
 1  2n1
42 n1  2n  1!
n
9 27 81 243

 

2! 3! 4! 5!
g) f  x   x 
x 2 x 4 x 6 x8
   
3! 5! 7! 9!

i) f  x  

n 1
3n  1 n
 1 n x
2 n!
n
(24 parts)
21. Find the following derivatives of the given function at x  0 :
a) f 15  0  , f  x   sin  x 3 
b) f 15  0  , f  x   x sin x
x
16 
 0 , f  x   cos  x 2 
19
 0 , f  x   xe
c) f 
e) f
d) f 
17 
x
f) f
 20
 0 , f  x  

e t dt
2
0
 0 , f  x   ln 1  x 2 
22. Given the two Maclaurin series:
1

1  x 1  2 x 


n 0
1  2x
an x and

1  x 1  2 x 1  4 x 
n
an and bn for n  0,1,2,


bn x n , find an equation relating
n 0
.
{Hint: Partial fractions.}
23. What is the coefficient of x100 in the Maclaurin series for e2 x ?

24. Consider the improper integral

xe x
xe x
x
x
6
.
For
large,
, and
x
dx



1  e x
1  e x e x 16 x3 x 2
0
 x  x  12 x 
xe

lim

1 2
x 0 1  e  x
x 0
 x 2x 
x
2

  1 . So the improper integral converges.

3
lim
a) Make the substitution u  1  e x to convert the integral into a different improper integral.

b) Use the series  ln 1  x  

n 0
x n1
and the fact that
n 1


n 1
1 2

to find the value of
n2 6
the improper integral.


3
an x n , where a0  1 and an    an1 for n  1 .
n
n 0
a) Find the first four terms and the general term of the series.
25. Consider the power series f  x  
b) What function is represented by this power series?
c) Find the exact value of f  1
(13 parts)
26. Suppose that f has derivatives of all orders for all numbers and that f  0   3 , f   0   2 ,
f   0   7 , and f   0   5 .
a) Find the third Maclaurin Polynomial for f, and use it to approximate f .2  .
b) Find the fourth Maclaurin Polynomial for the function g if g  x   f  x 2  .
(2 parts)