<< стр. 14(всего 14)СОДЕРЖАНИЕ
nв†’в€ћ nв†’в€ћ

в€ћ
|ak | converges to the ratio test, since
Part (a) If L < 1, then the series
k=1

|an+1 | an+1
= L < 1.
lim = lim
nв†’в€ћ |an | an
nв†’в€ћ

в€ћ
Hence, the series ak converges absolutely.
k=1
в€ћ
|ak | diverges by the ratio test if L > 1,
Part (b) As in Part (a), the series
k=1
since
|an+1 | an+1
= L > 1.
= lim
lim
nв†’в€ћ |an | an
nв†’в€ћ
в€ћ
|ak | converges by the root test, since
Part (c) If M < 1, then the series
k=1
1/n
lim |ak | = M < 1.
nв†’в€ћ
в€ћ
|ak | diverges by the root test as in
Part (d) If M > 1, then the series
k=1
Part (c).
в€ћ в€ћ
в€ћ
1 1
1
and , L = M = 1, but diverges
Part (e) For the series
k2
k k
k=1 k=1
k=1
в€ћ
1
and converges by the p-series test. Thus, L = 1 and M = 1 fail to
k2
k=1
determine convergence or divergence.
346 CHAPTER 8. INFINITE SERIES

This completes the proof of Theorem 8.5.3.

Exercises 8.3 Determine the region of convergence of the following series.
в€ћ в€ћ
(в€’1)n xn (в€’1)n (x + 2)n
71. 72.
3n n2
2n
n=1 n=1

в€ћ в€ћ
(в€’1)n (x в€’ 1)n (в€’1)n n!(x в€’ 1)n
73. 74.
5n
n!
n=1 n=1

в€ћ в€ћ
(x + 2)n
nn
75. (в€’2) x 76.
2n n2
n=0 n=1

в€ћ в€ћ
+ 1)n (в€’1)n (x в€’ 3)n
n (x
77. 78.
(в€’1)
3n n3 n3/2
n=1 n=1

в€ћ в€ћ
(2x)n (в€’1)n xn
79. 80.
n! (2n)!
n=1 n=1

в€ћ в€ћ
(n + 1)!(x в€’ 1)n (в€’1)n (2n)!xn
81. 82.
4n n!
n=1 n=1

в€ћ в€ћ
(в€’1)n n!(x в€’ 1)n
2 n
83. n (x + 1) 84.
1 В· 3 В· В· В· 5 В· В· В· (2n + 1)
n=1 n=1

в€ћ в€ћ
(в€’1)n (n!)2 (x в€’ 1)n (в€’1)n 3n xn
85. 86.
3n (2n)! 23n
n=1 n=1

в€ћ в€ћ
(в€’1)n (x + 1)n ln(n + 1)2n (x + 1)n
87. 88.
(n + 1) ln(n + 1) n+2
n=1 n=1

в€ћ в€ћ
(в€’1)n (ln n)3n xn (в€’1)n 1 В· 3 В· 5 В· В· В· (2n + 1) n
89. 90. x
4n n2 2 В· 4 В· 6 В· В· В· (2n + 2)
n=1 n=1
8.6. POWER SERIES 347

8.6 Power Series
Deп¬Ѓnition 8.6.1 If a0 , a1 , a2 , . . . is a sequence of real numbers, then the
series в€ћ ak xk is called a power series in x. A positive number r is called
k=1
the radius of convergence and the interval (в€’r, r) is called the interval of
convergence of the power series if the power series converges absolutely for
all x in (в€’r, r) and diverges for all x such that |x| > r. The end point x = r is
included in the interval of convergence if в€ћ ak rk converges. The end point
k=1
x = в€’r is included in the interval of convergence if the series в€ћ (в€’1)k ak rk
k=1
converges. If the power series converges only for x = 0, then the radius of
convergence is deп¬Ѓned to be zero. If the power series converges absolutely
for all real x, then the radius of convergence is deп¬Ѓned to be в€ћ.

в€ћ
cn xn converges for x = r = 0, then the
Theorem 8.6.1 If the series
n=1
в€ћ
cn xn converges absolutely for all numbers x such that |x| < |r|.
series
n=0
в€ћ
cn rn converges. Then, by the Divergence Test,
Proof. Suppose that
n=0

lim cn rn = 0.
nв†’в€ћ

For = 1, there exists some natural number m such that for all n в‰Ґ m,
|cn rn | < = 1.
Let
M = max{|cn rn | + 1 : 1 в‰¤ n в‰¤ m}.
Then, for each x such that |x| < |r|, we get |x/r| < 1 and
в€ћ в€ћ
x n
n n
|cn x | = |cn r | В·
r
n=0 n=0
в€ћ
x n
в‰¤ M
r
n=0
M
< в€ћ.
=
1в€’ x
r
348 CHAPTER 8. INFINITE SERIES

в€ћ
|cn xn | converges for x such that |x| <
By the comparison test the series
n=0
|r|. This completes the proof of Theorem 8.6.1.
в€ћ
cn (x в€’ a)n converges for some x в€’ a =
Theorem 8.6.2 If the series
n=0
в€ћ
cn (x в€’ a)n converges absolutely for all x such that
r = 0, then the series
n=0
|x в€’ a| < |r|.
в€ћ
cn un converges for some u = r. Then
Proof. Let xв€’a = u. Suppose that
n=0
в€ћ
cn un converges absolutely for all u such that
by Theorem 8.6.1, the series
n=0
в€ћ
cn (x в€’ a)n converges absolutely for all
|u| < |r|. It follows that the series
n=0
x such that |x в€’ a| < |r|. This completes the proof of the theorem.
в€ћ
cn xn be any power series. Then exactly one of the
Theorem 8.6.3 Let
n=0
following three cases is true.

(i) The series converges only for x = 0.

(ii) The series converges for all x.

(iii) There exists a number R such that the series converges for all x with
|x| < R and diverges for all x with |x| > R.

Proof. Suppose that cases (i) and (ii) are false. Then there exist two
в€ћ в€ћ
n
cn q n diverges.
nonzero numbers p and q such that cn p converges and
n=0 n=0
By Theorem 8.6.1, the series converges absolutely for all x such that |x| < |p|.
Let
в€ћ
cn pn converges}.
A = {p :
n=0
8.6. POWER SERIES 349

The set A is bounded from above by q. Hence A has a least upper bound, say
R. Clearly |p| в‰¤ R < q and hence R is a positive real number. Furthermore,
в€ћ
cn xn converges for all x such that |x| < R and diverges for all x such that
n=0
|x| > R. We deп¬Ѓne R to be 0 for case (i) and R to be в€ћ for case (ii). This
completes the proof of Theorem 8.6.3.
в€ћ
cn (x в€’ a)n be any power series. Then exactly one
Theorem 8.6.4 Let
n=0
of the following three cases is true:

(i) The series converges only for x = a and the radius of convergence is 0.

(ii) The series converges for all x and the radius of convergence is в€ћ.

(iii) There exists a number R such that the series converges for all x such
that |x в€’ a| < R and diverges for all x such that |x в€’ a| > R.
в€ћ
cn un . The
Let u = x в€’ a and use Theorem 8.6.3 on the series
Proof.
n=0
details of the proof are left as an exercise.
в€ћ
cn rn converges for |x| < R, then
Theorem 8.6.5 If R > 0 and the series
n=0
в€ћ в€ћ
ncn xnв€’1 , obtained by term-by-term diп¬Ђerentiation of c n xn ,
the series
n=1 n=0
converges absolutely for |x| < R.
Proof. For each x such that |x| < R, choose a number r such that |x| < r <
в€ћ
cn xn converges, lim cn rn = 0 and hence {cn rn }в€ћ is bounded.
R. Then n=0
nв†’в€ћ
n=0
There exists some M such that |cn rn | в‰¤ M for each natural number n. Then
в€ћ в€ћ
1x nв€’1
nв€’1 n
n|cn r | В· В·
|ncn x |=
rr
n=1 n=1
в€ћ
M x nв€’1
в‰¤ .
n
r r
n=1
350 CHAPTER 8. INFINITE SERIES

в€ћ
x
x nв€’1
converges by the ratio test, since < 1. It follows
The series n
r r
n=1
в€ћ
ncn xnв€’1 converges absolutely for all x such that |x| < R. This
that
n=1
completes the proof of this theorem.

в€ћ
cn (x в€’ a)n converges for all x
Theorem 8.6.6 If R > 0 and the series
n=0
в€ћ
cn (xв€’a)n may be diп¬Ђerentiated with
such that |xв€’a| < R, then the series
n=0
respect to x any number of times and each of the diп¬Ђerential series converges
for all x such that |x в€’ a| < R.
в€ћ
cn un converges for all u such that |u| < R. By
Proof. Let u = xв€’a. Then
n=0
в€ћ
ncn unв€’1 converges for all u such that |u| < R.
Theorem 8.6.5, the series
n=1
This term-by-term diп¬Ђerentiation process may be repeated any number of
times without changing the radius of convergence. This completes the proof
of this theorem.

Theorem 8.6.7 Suppose that R > 0 and f (x) = в€ћ cn xn and R is radius
n=0
в€ћ n
of convergence of the series n=0 cn x . Then f (x) is continuous for all x
such that |x| < R.

Proof. For each number c such that в€’R < c < R, we have

в€ћ
xn в€’ c n
f (x) в€’ f (c)
= cn
xв€’c xв€’c
n=0
в€ћ
cn nanв€’1
= n
n=1
в€ћ
n cn anв€’1
в‰¤ n
n=1
8.6. POWER SERIES 351

for some an between c and x, for each natural number n, by the Mean Value
в€ћ
n |cn an |nв€’1 converges. Hence,
Theorem. By Theorem 8.6.6, the series
n=1

в€ћ
n cn anв€’1
lim |f (x) в€’ f (c)| = lim |x в€’ c| |c0 в€’ c| + n
xв†’c xв†’c
n=1
в€ћ
n cn anв€’1
=0В· |c0 в€’ c| + n
n=1
= 0.

Hence, f (x) is continuous at each number c such that в€’R < c < R. This
completes the proof of this theorem.

в€ћ
cn xn and R is the radius
Theorem 8.6.8 Suppose that R > 0, f (x) =
n=0
в€ћ
cn xn . For each x such that |x| < R, we deп¬Ѓne
of convergence of the series
n=0

x
F (x) = f (t)dt.
0

Then, for each x such that |x| < R, we get

в€ћ
xn+1
.
F (x) = cn
n+1
n=0
352 CHAPTER 8. INFINITE SERIES

Proof. Suppose that |x| < |r| < R. Then
m
m x x
xn+1
tn
f (t)dt в€’
lim F (x) в€’ = lim cn
cn
n+1 nв†’в€ћ
mв†’в€ћ 0 0
n=0
n=0
m
x
cn tn
f (t) в€’ dt
= lim
mв†’в€ћ 0 n=0
в€ћ
x
cn tn dt
= lim
mв†’в€ћ 0 n=m+1
в€ћ
x
|cn tn | dt
в‰¤ lim
mв†’в€ћ 0 n=m+1
в€ћ
x
|cn rn | dt
в‰¤ lim
mв†’в€ћ 0 n=m+1
в€ћ x
n
в‰¤ lim |cn r | 1 dt
mв†’в€ћ 0
n=m+1

= 0 В· |x|
= 0,
в€ћ
|cn rn | converges.
since
n=0
It follows that
в€ћ
x x
cn tn dt
f (t)dt =
0 0 n=0
в€ћ
xn+1
= cn .
n+1
n=0

This completes the proof of the this theorem.
в€ћ
cn xn for all |x| < R, where R > 0
Theorem 8.6.9 Suppose that f (x) =
n=0
в€ћ
cn xn . Then f (x) has continuous
is the radius of convergence of the series
n=0
8.6. POWER SERIES 353

derivatives of all orders for |x| < R that are obtained by successive term-by-
в€ћ
c n xn .
term diп¬Ђerentiations of
n=0

Proof. For each |x| < R, we deп¬Ѓne
в€ћ
ncn xnв€’1 .
g(x) =
n=1

в€ћ
ncn xnв€’1 .
Then, by Theorem 8.6.5, R is the radius of convergence of the series
n=1
By Theorem 8.6.7, g(x) is continuous. Hence,
в€ћ
x
cn xn = f (x).
c0 + g(x)dx = c0 +
0 n=1

By the fundamental theorem of calculus, f (x) = g(x). This completes the
proof of this theorem.

Deп¬Ѓnition 8.6.2 The radius of convergence of the power series
в€ћ
ak (x в€’ a)k
k=1

is
(a) zero, if the series converges only for x = a;

(b) r, if the series converges absolutely for all x such that |x в€’ a| < r and
diverges for all x such that |x в€’ a| > r.

(c) в€ћ, if the series converges absolutely for all real number x.
If the radius of convergence of the power series in (x в€’ a) is r, 0 < r < в€ћ,
then the interval of convergence of the series is (a в€’ r, a + r). The end points
x = a + r or x = a в€’ r are included in the interval of convergence if the
corresponding series в€ћ ak rk or в€ћ (в€’1)k ak rk converges, respectively. If
k=1
k=1
r = в€ћ, then the interval of convergence is (в€’в€ћ, в€ћ).
354 CHAPTER 8. INFINITE SERIES

Exercises 8.4 In problem 1вЂ“12, determine the Taylor series expansion for
each function f about the given value of a.
1. f (x) = eв€’2x , a = 0 2. f (x) = cos(3x), a = 0

4. f (x) = (1 + x)в€’2 , a = 0
3. f (x) = ln(x), a = 1

5. f (x) = (1 + x)в€’3/2 , a = 0 6. f (x) = ex , a = 2

ПЂ ПЂ
7. f (x) = sin x, a = 8. f (x) = cos x, a =
6 4
ПЂ
10. f (x) = x1/3 , a = 8
9. f (x) = sin x, a =
3
1 1
11. f (x) = sin x в€’ 12. f (x) = cos x в€’
,a = 0 ,a = 0
2 2
n
(x в€’ a)k
(k)
In problems 13-20, determine .
f (a)
k!
k=0
2
14. f (x) = x2 eв€’x , a = 0, n = 3
13. f (x) = ex , a = 0, n = 3

1
15. f (x) = , a = 0, n = 2 16. f (x) = arctan x, a = 0, n = 3
1 в€’ x2

17. f (x) = e2x cos 3x, a = 0, n = 4 18. f (x) = arcsin x, a = 0, n = 3

20. f (x) = (1 + x)1/2 , a = 0, n = 5
19. f (x) = tan x, a = 0, n = 3

8.7 Taylor Polynomials and Series
Theorem 8.7.1 (TaylorвЂ™s Theorem) Suppose that f, f , В· В· В· , f (n+1) are all
continuous for all x such that |x в€’ a| < R. Then there exists some c between
a and x such that
f (x) = Pn (x) + Rn (x)
where
n
(x в€’ a)n+1
(x в€’ a)k (n+1)
(k)
Pn (x) = , Rn (x) = f (c) .
f (a)
k! (n + 1)!
k=0
8.7. TAYLOR POLYNOMIALS AND SERIES 355

The polynomial Pn (x) is called the nth degree Taylor polynomial approxima-
tion of f . The term Rn (x) is called the Lagrange form of the remainder.
Proof. We deп¬Ѓne a function g of a variable z such that
f (z)(x в€’ z) f (z)(x в€’ z)2
g(z) = [f (x) в€’ f (z)] в€’ в€’ в€’ В·В·В·
1! 2!
f (n) (z)(x в€’ z)n (x в€’ z)n+1
в€’ в€’ Rn (x) .
(x в€’ a)n+1
n!
Then
n
f (k) (a)
(x в€’ a)k + Rn (x)
g(a) = f (x) в€’ = 0,
k!
k=0
and
g(x) = f (x) в€’ f (x) = 0.
By the Mean Value Theorem for derivatives there exists some c between a
and x such that g (c) = 0. But
f (z)(x в€’ z)2
в€’ В·В·В·
g (z) = в€’f (z) в€’ [в€’f (z) + f (z)(x в€’ z)] в€’ в€’f (z)(x в€’ z) +
2!
f n (z)(x в€’ z)nв€’1 f (n+1) (z)(x в€’ z)n (n + 1)(x в€’ z)n
в€’в€’ + + Rn (x)
(x в€’ a)n+1
n! n!
(x в€’ z)n (n + 1)(x в€’ z)n
(n+1)
= в€’f (z) + Rn (x)
(x в€’ a)n+1
n!
(x в€’ c)n (n + 1)(x в€’ c)n
(n+1)
g (c) = 0 = в€’f (c) + Rn (x) .
(x в€’ a)n+1
n!
Therefore,
(x в€’ a)n+1 f (n+1) (c) (x в€’ a)n+1
(n+1)
В·
Rn (x) = =f (c)
n+1 n! (n + 1)!
as required. This completes the proof of this theorem.

Theorem 8.7.2 (Binomial Series) If m is a real number and |x| < 1, then
в€ћ
m(m в€’ 1) В· В· В· (m в€’ k + 1) k
m
(1 + x) = 1 + x
k!
k=1
m(m в€’ 1) 2 m(m в€’ 1)(m в€’ 2) 3
x + В·В·В· .
= 1 + mx + x+
2! 3!
356 CHAPTER 8. INFINITE SERIES

This series is called the binomial series. If we use the notation
m(m в€’ 1) В· В· В· (m в€’ k + 1)
m
=
k k!
m
then is called the binomial coeп¬ѓcient and
k
в€ћ
mk
(1 + x)m = 1 + x.
k
k=1

If m is a natural number, then we get the binomial expansion
m
mk
m
(1 + x) = 1 + x.
k
k=1

Proof. Let f (x) = (1 + x)m . Then for all natural numbers n,
f (x) = m(1 + x)mв€’1 , f (x) = m(m в€’ 1)(1 + x)mв€’2 , В· В· В· ,
f (n) (x) = m(m в€’ 1) В· В· В· (m в€’ n + 1)(1 + x)mв€’n .

Thus, f (n) (0) = m(m в€’ 1) В· В· В· (m в€’ n + 1), and
в€ћ
m(m в€’ 1)(m в€’ 2) В· В· В· (m в€’ n + 1) n
x
f (x) =
n!
n=0
в€ћ
mn
= x
n
n=0

m m
= m(m в€’ 1) В· В· В· (m в€’ n + 1) is called the nth
where = 1 and
0 n
binomial coeп¬ѓcient. By the ratio test we get
m(m в€’ 1) В· В· В· (m в€’ n)xn+1 n!
В·
lim
m(m в€’ 1) В· В· В· (m в€’ n + 1)xn
(n + 1)!
nв†’в€ћ

mв€’n
= |x| lim
n+1
nв†’в€ћ
m
в€’1
= |x| lim n 1
nв†’в€ћ 1 +
n
= |x|,
8.7. TAYLOR POLYNOMIALS AND SERIES 357

and, hence, the series converges for |x| < 1.
This completes the proof of the theorem.

Theorem 8.7.3 The following power series expansions of functions are valid.
в€ћ в€ћ
1. (1 в€’ x)в€’1 = 1 + xk and (1 + x)в€’1 = 1 + (в€’1)k xk , |x| < 1.
k=1 k=1

в€ћ
в€ћ k
xk kx
в€’x
2. ex = 1 + , |x| < в€ћ.
, e =1+ (в€’1)
k! k!
k=1
k=1

в€ћ
x2k+1
k
, |x| < в€ћ.
3. sin x = (в€’1)
(2k + 1)!
k=0

в€ћ
x2k k
, |x| < в€ћ.
4. cos x = (в€’1)
(2k)!
k=0

в€ћ
x2kв€’1
, |x| < в€ћ.
5. sinh x =
(2k + 1)!
k=0

в€ћ
x2k
, |x| < в€ћ.
6. cosh x =
(2k)!
k=0

в€ћ
xk+1 k
, в€’1 < x в‰¤ 1.
7. ln(1 + x) = (в€’1)
k+1
k=0

в€ћ
x2k+1
1 1+x
, в€’1 < x < 1.
8. ln =
1в€’x
2 2k + 1
k=0

в€ћ
x2k+1 k
, в€’1 в‰¤ x в‰¤ 1.
9. arctan x = (в€’1)
2k + 1
k=0

в€ћ 2k+1
в€’1/2 kx
, |x| в‰¤ 1.
10. arcsin x = (в€’1)
2k + 1
k
k=0
358 CHAPTER 8. INFINITE SERIES

Proof.
Part 1. By the geometric series expansion, for all |x| < 1, we have
в€ћ в€ћ
1 1
1
xk (в€’1)k xk .
=1+ = =1+
and
1в€’x 1 в€’ (в€’x)
1+x
k=1 k=1

Part 2. If f (x) = ex , then f (n) (x) = ex and f (n) (0) = 1 for each n =
0, 1, 2, В· В· В· . Thus
в€ћ
xn
x
.
e=
n!
n=0
By the ratio test the series converges for all x.
xn+1 n! 1
В· n = |x| lim = 0.
lim
nв†’в€ћ (n + 1)! x nв†’в€ћ n + 1

Part 3. Let f (x) = sin x. Then f (x) = cos x, f (x) = в€’ sin x, f (3) (x) =
в€’ cos x and f (4) (x) = sin x. It follows that, for each n = 0, 1, 2, 3, В· В· В· , we
have
f (4n) (0) = 0, f (4n+1) (0) = 1, f (4n+2) (0) = 0 and f (4n+3) (0) = в€’1.
Hence,
x3 x5
в€’ В·В·В·
sin x = x в€’ +
3! 5!
в€ћ
x2n+1
n
= .
(в€’1)
(2n + 1)!
n=0

By the ratio test, the series converges for all |x| < в€ћ:
x2n+3 (2n + 1)!
n+1
lim (в€’1)
(2n + 3)! x2n+1
nв†’в€ћ
1
= x2 lim
nв†’в€ћ (2n + 3)(2n + 2)

= 0.
Part 4. By term-by-term diп¬Ђerentiation we get
в€ћ
x2n
n
, |x| < в€ћ.
cos x = (sin x) = (в€’1)
(2n)!
n=0
8.7. TAYLOR POLYNOMIALS AND SERIES 359

Part 5. For all |x| < в€ћ, we get
1x
(e в€’ eв€’x )
sinh x =
2
в€ћ в€ћ
xn n
1 nx
в€’
= (в€’1)
2 n=0 n! n=0 n!
в€ћ
x2n+1
= .
(2n + 1)!
n=0

Part 6. By diп¬Ђerentiating term-by-term, we get
в€ћ
x2n
, l |x| < в€ћ.
cosh x = (sinh x) =
(2n)!
n=0

Part 7. For each |x| < 1, by performing term by integration, we get
x
1
ln(1 + x) = dx
1+x
0
в€ћ
в€ћ
(в€’1)n xn dx
=
0 n=0
в€ћ
xn+1 n
= (в€’1) .
n+1
n=0

Part 8. By Part 7, for all |x| < 1, we get
1 1+x 1
[ln(1 + x) в€’ ln(1 в€’ x)]
ln =
1в€’x
2 2
в€ћ в€ћ
xn+1 n+1
1 n (в€’x)
n
в€’
= (в€’1) (в€’1)
2 n + 1 n=0 n+1
n=0
в€ћ
(в€’1)n
1
(1 в€’ (в€’1)n+1 )xn+1
=
2 n+1
n=0
в€ћ
x2k+1
= .
2k + 1
k=0

1 1+x
Recall that arctanh x = ln .
1в€’x
2
360 CHAPTER 8. INFINITE SERIES

Part 9. For each |x| в‰¤ 1, we perform term-by-term integration to get
x
1
arctan x = dx
1 + x2
0
в€ћ
x
(в€’1)k x2k dx
=
0 k=0
в€ћ
x2k+1 k
= (в€’1) .
(2k + 1)
k=0

Part 10. By performing term-by-term integration of the binomial series, we
get
x
1
в€љ dx
arcsin x =
1 в€’ x2
0
x
(1 в€’ x2 )в€’1/2 dx
=
0
в€ћ
x
в€’1/2
(в€’x2 )k dx
=
k
0 k=0
в€ћ
x2k+1
в€’1/2 k
= (в€’1) .
k (2k + 1)
k=0

This series converges for all |x| в‰¤ 1.
This completes the proof of this theorem.

8.8 Applications
Chapter 9

Analytic Geometry and Polar
Coordinates

A double right-circular cone is obtained by rotating a line about a п¬Ѓxed axis
such that the line intersects the axis and makes the same angle with the
axis. The intersection point of the line and the axis is called a vertex. A
conic section is the intersection of a plane and the double cone. Some of
the important conic sections are the following: parabola, circle, ellipse and a
hyperbola.

9.1 Parabola
Deп¬Ѓnition 9.1.1 A parabola is the set of all points in the plane that are
equidistant from a given point, called the focus, and a given line called the
directrix. A line that passes through the focus and is perpendicular to the
directrix is called the axis of the parabola. The intersection of the axis with
the parabola is called the vertex.

Theorem 9.1.1 Suppose that v(h, k) is the vertex and the line x = h в€’ p is
the directrix of a parabola. Then the focus is F (h + p, k) and the axis is the
horizontal line with equation y = k. The equation of the parabola is

(y в€’ k)2 = 4p(x в€’ h).

361
362CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

Theorem 9.1.2 Suppose that v(h, k) is the vertex and the line y = k в€’ p is
the directrix of a parabola. Then the focus is F (h, k + p) and the axis is the
vertical line with equation x = h. The equation of the parabola is

(x в€’ h)2 = 4p(y в€’ k).

9.2 Ellipse
Deп¬Ѓnition 9.2.1 An ellipse is the locus of all points, the sum of whose
distances from two п¬Ѓxed points, called foci, is a п¬Ѓxed positive constant that
is greater than the distance between the foci. The midpoint of the line
segment joining the two foci is called the center. The line segment through
the foci and with end points on the ellipse is called the major axis. The
line segment, through the center, that has end points on the ellipse and is
perpendicular to the major axis is called the minor axis. The intersections
of the major and minor axes with the ellipse are called the vertices.

Theorem 9.2.1 Let an ellipse have center at (h, k), foci at (h В± c, k), ends
of the major axis at (h В± a, k) and ends of the minor axis at (h, k В± b), where
a > 0, b > 0, c > 0 and a2 = b2 + c2 . Then the equation of the ellipse is

(x в€’ h)2 (y в€’ k)2
+ = 1.
a2 b2
The length of the major axis is 2a and the length of the minor axis is 2b.

Theorem 9.2.2 Let an ellipse have center at (h, k), foci at (h, k В± c), ends
of the major axis at (h, k В± a), and the ends of the minor axis at (h В± b, k),
where a > 0, b > 0, c > 0 and a2 = b2 + c2 . Then the equation of the ellipse
is
(y в€’ k)2 (x в€’ h)2
+ = 1.
a2 b2
The length of the major axis is 2a and the length of the minor axis is 2b.

Remark 24 If c = 0, then a = b, foci coincide with the center and the
ellipse reduces to a circle.
9.3. HYPERBOLA 363

9.3 Hyperbola
Deп¬Ѓnition 9.3.1 A hyperbola is the locus of all points, the diп¬Ђerence of
whose distances from two п¬Ѓxed points, called foci, is a п¬Ѓxed positive constant
that is less than the distance between the foci. The mid point of the line
segment joining the two foci is called the center. The line segment, through
the foci, and with end points on the hyperbola is called the major axis. The
end points of the major axis are called the vertices.

Theorem 9.3.1 Let a hyperbola have center at (h, k), foci at (h В± c, k),
в€љ
vertices at (h В± a, k), where 0 < a < c, b = c2 в€’ a2 , then the equation of
the hyperbola is
(x в€’ h)2 (y в€’ k)2
в€’ = 1.
a2 b2

Theorem 9.3.2 Let a hyperbola have center at (h, k), foci at (h, k В± c),
в€љ
vertices at (h, k В± a), where 0 < a < c, b = c2 в€’ a2 , then the equation of
the hyperbola is
(y в€’ k)2 (x в€’ h)2
в€’ = 1.
a2 b2

9.4 Second-Degree Equations
Deп¬Ѓnition 9.4.1 The transformations
x = x cos Оё в€’ y sin Оё
y = x sin Оё + y cos Оё

and
x = x cos Оё + y sin Оё
y = в€’x sin Оё + y cos Оё
are called rotations. The point P (x, y) has coordinates (x , y ) in an x y -
coordinate system obtained by rotating the xy-coordinate system by an angle
Оё.

Theorem 9.4.1 Consider the equation ax2 + bxy + cy 2 + dx + ey + f =
0, b = 0. Let cot 2Оё = (a в€’ c)/b and x y -coordinate system be obtained
364CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

through rotating the xy-coordinate system through the angle Оё. Then the
given second degree equation

ax2 + bxy + cy 2 + dx + ey + f = 0

becomes
ax2+cy2+dx+ey+f =0
where

= a cos2 Оё + b cos Оё sin Оё + c sin2 Оё
a
= a sin2 Оё в€’ b sin Оё cos Оё + c cos2 Оё
c
d = d cos Оё + e sin Оё
= в€’d sin Оё + e cos Оё
e
f =f

Furthermore, the given second degree equation represents
(i) an ellipse, a circle, a point or no graph if b2 в€’ 4ac < 0;

(ii) a hyperbolic or a pair of intersecting lines if b2 в€’ 4ac > 0;

(iii) a parabola, a line, a pair of parallel lines, or else no graph if b2 в€’4ac = 0.

9.5 Polar Coordinates
Deп¬Ѓnition 9.5.1 Each point P (x, y) in the xy-coordinate plane is assigned
the polar coordinates (r, Оё) that satisfy the following relations:

x2 + y 2 = r2 , y = r cos Оё, y = r sin Оё.

The origin is called the pole and the positive x-axis is called the polar axis.
The number r is called the radial coordinate and the angle Оё is called the
angular coordinates. The polar coordinates of a point are not unique as the
rectangular coordinates are. In particular,

(r, Оё) в‰Ў (r, Оё + 2nПЂ) в‰Ў (в€’r, Оё + (2m + 1)ПЂ)

where n and m are any integers. There does exist a unique polar represen-
tation (r, Оё) if r в‰Ґ 0 and 0 в‰¤ Оё < 2ПЂ.
9.6. GRAPHS IN POLAR COORDINATES 365

9.6 Graphs in Polar Coordinates
Theorem 9.6.1 A curve in polar coordinates is symmetric about the

(a) x-axis if (r, Оё) and (r, в€’Оё) both lie on the curve;

(b) y-axis if (r, Оё) and (r, ПЂ в€’ Оё) both lie on the curve;

(c) origin if (r, Оё), (r, Оё + ПЂ) and (в€’r, Оё) all lie on the curve.

Theorem 9.6.2 Let e be a positive number. Let a п¬Ѓxed point F be called the
focus and a п¬Ѓxed line, not passing through the focus, be called a directrix. If
P is a point in the plane, let P F stand for the distance between P and the
focus F and let P D stand for the distance between P and the directrix. Then
the locus of all points P such that P F = eP D is a conic section representing

(a) an ellipse if 0 < e < 1;

(b) a parabola if e = 1;

(c) a hyperbola if e > 1;

The number e is called the eccentricity of the conic.
In particular an equation of the form

ek
r=
1 В± e cos Оё
represents a conic with eccentricity e, a focus at the pole (origin), and a
directrix perpendicular to the polar axis and k units to the right of the pole,
in the case of + sign, and k units to the left of the pole, in the case of в€’
sign.
Also, an equation of the form

ek
r=
1 В± e sin Оё
represents a conic with eccentricity e, a focus at the pole, and a directrix
parallel to the polar axis and k units above the pole, in the case of + sign,
and k units below the pole, in the case of в€’ sign.
366CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

9.7 Areas in Polar Coordinates
Theorem 9.7.1 Let r = f (Оё) be a curve in polar coordinates such that f is
continuous and nonnegative for all О± в‰¤ Оё в‰¤ ОІ where О± в‰¤ ОІ в‰¤ 2ПЂ + О±. Then
the area A bounded by the curves r = f (Оё), Оё = О± and Оё = ОІ is given by
ОІ ОІ
12 1
(f (Оё))2 dОё.
A= r dОё =
2 2
О± О±

Theorem 9.7.2 Let r = f (Оё) be a curve in polar coordinates such that f
and f are continuous for О± в‰¤ Оё в‰¤ ОІ, and there is no overlapping, the arc
length L of the curve from Оё = О± to Оё = ОІ is given by
ОІ
(f (Оё))2 + (f (Оё))2 dОё
L=
О±
2
ОІ
dr
r2 +
= dОё
dОё
О±

9.8 Parametric Equations
Deп¬Ѓnition 9.8.1 A parametrized curve C in the xy-plane has the form
C = {(x, y) : x = f (t), y = g(t), t в€€ I}
for some interval I, п¬Ѓnite or inп¬Ѓnite.
The functions f and g are called the coordinate functions and the variable
t is called the parameter.

Theorem 9.8.1 Suppose that x = f (t), y = g(t) are the parametric equa-
tions of a curve C. If f (t) and g (t) both exist and f (t) = 0, then
dy g (t)
= .
dx f (t)
Also, if f (t) and g (t) exist, then
d2 y f (g)g (t) в€’ g (t)f (t)
= .
dx2 (f (t))2
At a point P0 (f (t0 ), g(t0 )), the equation of
9.8. PARAMETRIC EQUATIONS 367

(a) the tangent line is

g (t0 )
y в€’ g(t0 ) = (x в€’ f (t0 ))
f (t0 )

(b) the normal line is

f (t0 )
(x в€’ f (t0 ))
y в€’ g(t0 ) = в€’
g (t0 )

provided g (t0 ) = 0 and f (t0 ) = 0.

Theorem 9.8.2 Let C = {(x, y) : x = f (t), y = g(t), a в‰¤ t в‰¤ b} where f (t)
and g (t) are continuous on [a, b]. Then the arc length L of C is given by
b
[(f (t))2 + (g (t))2 ]1/2 dt
L=
a
2 1/2
2
b
dy
dx
+
= dt.
dt dt
a

Theorem 9.8.3 Let C = {(x, y) : x = f (t), y = g(t), a в‰¤ t в‰¤ b}, where f (t)
and g (t) are continuous on [a, b].
(a) If C lies in the upper half plane or the lower half plane and there is no
overlapping, then the surface area generated by revolving C around the
x-axis is given by
b
2ПЂg(t) (f (t))2 + (g (t))2 dt.
a

(b) If 0 в‰¤ f (t) on [a, b], (or f (t) в‰¤ 0 on [a, b]) and there is no overlapping,
then the surface area generated by revolving C around the y-axis is
b
2ПЂf (t) (f (t))2 + (g (t))2 dt.
a
368CHAPTER 9. ANALYTIC GEOMETRY AND POLAR COORDINATES

Deп¬Ѓnition 9.8.2 Let C = {(x(t), y(t)) : a в‰¤ t в‰¤ b} for some interval I.
Suppose that x (t), y (t), x (t) and y (t) are continuous on I.

(a) The arc length s(t) is deп¬Ѓned by
t
[(x (t))2 + (y (t))2 ]1/2 dt.
s(t) =
a

(b) The angle of inclination, П†, of the tangent line to the curve C is deп¬Ѓned
by
y (t) dy
П†(t) = arctan = arctan .
x (t) dx

(c) The curvature Оє(t), read kappa of t, is deп¬Ѓned by

|x (t)y (t) в€’ y (t)x (t)|
dП†
= .
[(x (t))2 + (y (t))2 ]3/2
ds

(d) The radius of curvature, R, is deп¬Ѓned by
1
R(t) = .
Оє(t)

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