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Part (i) y = 3 cos(3x), y = в€’9 sin(3x)

Part (ii) y = в€’8x sin(4x2 ), y = в€’8[sin(4x2 ) + x В· (8x) В· cos(4x2 )]

Part(iii) y = 3 sec2 (3x), y = 3[2 sec(3x) В· sec(3x) tan(3x) В· 3]

y = 18 sec2 (3x) tan(3x)

Part(iv) y = в€’5 csc2 (5x), y = в€’10 csc(5x)[(в€’ csc 5x cot 5x) В· 5]

y = 50 csc2 (5x) cot(5x)

Part(v) y = 2 sec(2x) tan(2x)

y = 2[(2 sec(2x) tan(2x)) В· tan(2x) + sec(2x) В· (2 sec2 (2x))]

y = 4 sec(2x) tan2 (2x) + 4 sec3 (2x)

Part(vi) y = в€’2x csc(x2 ) cot(x2 )

y = в€’2[1 В· csc(x2 ) cot(x2 ) + x(в€’2x csc(x2 ) cot(x2 )) В· cot(x2 )
3.5. HIGHER ORDER DERIVATIVES 139

+ x csc(x2 ) В· (в€’2x csc2 (x2 ))]

= в€’2 csc(x2 ) cot(x2 ) + 4x2 csc(x2 ) cot2 (x2 ) + 4x2 csc3 (x2 )

Example 3.5.2 Compute the second order derivative of each of the follow-
ing functions:

(ii) y = cosh(x2 )
(i) y = sinh(3x) (iii) y = tanh(2x)

(iv) y = coth(4x) (v) y = sech(5x) (vi) y = csch(10x)

Part (i) y = 3 cosh(3x), y = 9 sinh(3x)

Part (ii) y = 2x sinh(x2 ), y = 2 sinh(x2 ) + 2x(2x cosh x2 ) or

y = 2 sinh(x2 ) + 4x2 cosh(x2 )

Part (iii) y = 2 sech2 (2x), y = 2 В· (2 sech(2x) В· (в€’sech(2x) tanh(2x) В· 2)),

y = в€’8 sech2 (2x) tanh(2x)

Part (iv) y = в€’4 csch2 (4x), y = в€’4(2(csch(4x)) В· (в€’csch(4x) coth(4x) В· 4))

y = 32 csch2 (4x) coth(4x)

Part (v) y = в€’5 sech (5x) tanh(5x)

y = в€’5[в€’5 sech(5x) tanh(5x) В· tanh(5x) + sech(5x) В· sech2 (5x) В· 5]

y = 25 sech(5x) tanh2 (5x) в€’ 25 sech3 (5x).

Part (vi) y = в€’10 csch(10x) coth(10x)

y = в€’10[в€’10 csch(10x) coth(10x) В· coth(10x)
140 CHAPTER 3. DIFFERENTIATION

+ csch(10x)(в€’10 csch2 (10x))]

y = 100 csch(10x) coth2 (10x) + 100 csch3 (10x)

Example 3.5.3 Compute the second order derivatives for the following func-
tions:

2
(i) y = ln(x2 ) (ii) y = ex (iii) log10 (x2 + 1)
2
(iv) y = 10x (v) y = arcsin x (vi) y = arctan x

2x 2
= = 2xв€’1
Part (i) y =
x2 x
в€’2
y = в€’2xв€’2 = .
x2
2 2 2 2
Part (ii) y = 2xex , y = 2ex + 4x2 ex = (2 + 4x2 )ex .

(x2 + 1) В· 1 в€’ x В· 2x
1 2x 2
В·2
Part (iii) y = ,y = ,
(x2 + 1)2
ln 10 x + 1 ln 10

1 в€’ x2
2
В·
y=
ln 10 (x2 + 1)2
2
Part (iv) y = 10x В· (ln 10) В· 2x
2 2
y = 2 ln 10[10x + x В· 10x ln 10 В· 2x]
2
y = 10x [2 ln 10 + (2 ln 10)2 x2 ]

1
= (1 в€’ x2 )в€’1/2
Part (v) y = в€љ
1 в€’ x2
3.5. HIGHER ORDER DERIVATIVES 141

в€’1
(1 в€’ x2 )в€’3/2 (в€’2x)
y=
2
x
y= .
(1 в€’ x2 )3/2

1
= (1 + x2 )в€’1
Part (vi) y = 2
1+x
в€’2x
y = в€’1(1 + x2 )в€’2 В· 2x =
(1 + x2 )2

Example 3.5.4 Compute the second derivatives of the following functions:

(i) y = arcsinh x (ii) y = arccosh x (iii) y = arctanh x

From Section 1.4, we recall that
в€љ
arcsinh x = ln(x + 1 + x2 )
в€љ
arccosh x = ln(x + x2 в€’ 1) , x в‰Ґ 1
1 1+x 1
= [ln(1 + x) в€’ ln(1 в€’ x)], |x| < 1.
arctanh x = ln
1в€’x
2 2

Then

Part (i)

1
y =в€љ
1 + x2
d2 d
(1 + x2 )в€’1/2
(arcsinh x) =
2
dx dx
в€’1
(2x)(1 + x2 )в€’3/2
=
2
x
=в€’ .
(1 + x2 )3/2
142 CHAPTER 3. DIFFERENTIATION

Part (ii)
1
y =в€љ , x>1
x2 в€’ 1
d2 d
(x2 в€’ 1)в€’1/2
(arccosh x) =
2
dx dx
в€’1
(2x)(x2 в€’ 1)в€’3/2
=
2
x
=в€’ 2 , x>1
(x в€’ 1)3/2
Part (iii)
1
, |x| < 1.
y=
1 в€’ x2
d2 d
(1 в€’ x2 )в€’1
(arctanh x) =
dx dx
= (в€’1)(1 в€’ x2 )в€’2 (в€’2x)
x
, |x| < 1.
=
(1 в€’ x2 )2

Example 3.5.5 Find y for the equation x2 + y 2 = 4.
First, we п¬Ѓnd y by implicit diп¬Ђerentiation.
x
2x + 2yy = 0 в†’ y , .
y
Now, we diп¬Ђerentiate again with respect to x.
y В· 1 в€’ xy
y=
y2
y в€’ x(в€’x/y)
(replace y by в€’x/y)
=в€’
y2
y 2 + x2
=в€’ (Why?)
y3
4
(since x2 + y 2 = 4)
=в€’ 3
y
3.5. HIGHER ORDER DERIVATIVES 143

Example 3.5.6 Compute y for x3 + y 3 = 4xy.
From Example 25 in the last section we found that

4y в€’ 3x2
if 3y 2 в€’ 4x = 0.
y= 2
3y в€’ 4x

To п¬Ѓnd y , we diп¬Ђerentiate y with respect to x to get

(3y 2 в€’ 4x)(4y в€’ 3x2 ) в€’ (4y в€’ 3x2 )(6yy в€’ 4)
, 3y 2 в€’ 4x = 0.
y=
3y 2 в€’ 4x

In order to simplify any further, we must п¬Ѓrst replace y by its computed
value. We leave this as an exercise.

Example 3.5.7 Compute f (n) (c) for the given f and c and all natural num-
bers n:

(i) f (x) = sin x, c = 0 (ii) f (x) = cos x, x = 0 (iii) f (x) = ln(x), c = 1

(iv) f (x) = ex , c = 0 (v) f (x) = sinh x, x = 0 (vi) f (x) = cosh x, x = 0

To compute the general nth derivative formula we must discover a pattern
and then generalize the pattern.

Part (i) f (x) = sin x, f (x) = cos x, f (x) = в€’ sin x, f (x) = cos x, f 4 (x) =
sin x. Then the next four derivatives are repeated and so on. We get

f (4n) (n) = sin x, f (4n+1) (x) = cos x, f (4n+2) (x) = в€’ sin x, f (4n+3) (x) = в€’ cos x.

By evaluating these at c = 0, we get

f (4n) (0) = 0, f (4n+2) (0); f (4n+1) (0) = 1 and f (4n+3) (0) = в€’1,

for n = 0, 1, 2, В· В· В·

Part (ii) This part is similar to Part (i) and is left as an exercise.

Part (iii) f (x) = ln x, f (x) = xв€’1 , f (x) = (в€’1)xв€’2 , f (3) (x) = (в€’1)(в€’2)xв€’3 , . . . .,
f (n) (x) = (в€’1)(в€’2) . . . (в€’(n в€’ 1))xв€’n = (в€’1)nв€’1 (n в€’ 1)!xв€’n , f (n) (1) =
(в€’1)nв€’1 (n в€’ 1)!, n = 1, 2, . . .
144 CHAPTER 3. DIFFERENTIATION

Part (iv) f (x) = ex , f (x) = ex , f (x) = ex , . . . , f (n) (x) = ex , f (n) (0) =
1, n = 0, 1, 2, . . .

Part (v) f (x) = sinh x, f (x) = cosh x, f (x) = sinh x, . . . f (2n) (x) = sinh x,
f (2n+1) (x) = cosh x, f (2n) (0) = 0, f (2n+1) (0) = 1, n = 0, 1, 2, . . .

Part (vi) f (x) = cosh x, f (x) = sinh x, f (x) = cosh x, . . . , f (2n) (x) =
cosh x, f (2n+1) (x) = sinh x, f (2n) (0) = 1, f (2n+1) (0) = 0, n = 0, 1, 2, . . .

Exercises 3.5 Find the п¬Ѓrst two derivatives of each of the following func-
tions f .

1. f (t) = 4t3 в€’ 3t2 + 10 2. f (x) = 4 sin(3x) + 3 cos(4x)

3. f (x) = (x2 + 1)3 4. f (x) = x2 sin(3x)

5. f (x) = e3x sin 4x 6. f (x) = e2x cos 4x

x2
8. f (x) = (x2 + 1)10
7. f (x) =
2x + 1

9. f (x) = ln(x2 + 1) 10. f (x) = log10 (x4 + 1)

11. f (x) = 3 sinh(4x) + 5 cosh(4x) 12. f (x) = tanh(3x)

14. f (x) = x2 ex
13. f (x) = x tan x

15. f (x) = arctan(3x) 16. f (x) = arcsinh (2x)

18. f (x) = (x2 + 1)100
17. f (x) = cos(nx)

Show that the given y(x) satisп¬Ѓes the given equation:

19. y = A sin(4x) + B cos(4x) satisп¬Ѓes y + 16y = 0

20. y = A sinh(4x) + B cosh(4x) satisп¬Ѓes y в€’ 16y = 0
3.5. HIGHER ORDER DERIVATIVES 145

21. y = eв€’x (a sin(2x) + b cos(2x)) satisп¬Ѓes y в€’ 2y + 2y = 0

22. y = ex (a sin(3x) + b cos(3x)) satisп¬Ѓes y в€’ 2y + 10y = 0

Compute the general nth derivative for each of the following:

23. f (x) = e2x 24. f (x) = sin 3x

25. f (x) = cos 4x 26. f (x) = ln(x + 1)

27. f (x) = sinh(2x) 28. f (x) = cosh(3x)

1+x
29. f (x) = (x + 1)100 30. f (x) = ln 1в€’x

Find y and y for the following equations:

31. x4 + y 4 = 20 32. x2 + xy + y 2 = 16
Chapter 4

Applications of Diп¬Ђerentiation

One of the important problems in the real world is optimization. This is the
problem of maximizing or minimizing a given function. Diп¬Ђerentiation plays
a key role in solving such real world problems.

4.1 Mathematical Applications
Deп¬Ѓnition 4.1.1 A function f with domain D is said to have an absolute
maximum at c if f (x) в‰¤ f (c) for all x в€€ D. The number f (c) is called the
absolute maximum of f on D. The function f is said to have a local maximum
(or relative maximum) at c if there is some open interval (a, b) containing c
and f (c) is the absolute maximum of f on (a, b).

Deп¬Ѓnition 4.1.2 A function f with domain D is said to have an absolute
minimum at c if f (c) в‰¤ f (x) for all x in D. The number f (c) is called the
absolute minimum of f on D. The number f (c) is called a local minimum
(or relative minimum) of f if there is some open interval (a, b) containing c
and f (c) is the absolute minimum of f on (a, b).

Deп¬Ѓnition 4.1.3 An absolute maximum or absolute minimum of f is called
an absolute extremum of f . A local maximum or minimum of f is called a
local extremum of f .

146
4.1. MATHEMATICAL APPLICATIONS 147

Theorem 4.1.1 (Extreme Value Theorem) If a function f is continuous
on a closed and bounded interval [a, b], then there exist two points, c1 and c2 ,
in [a, b] such that f (c1 ) is the absolute minimum of f on [a, b] and f (c2 ) is
the absolute maximum of f on [a, b].

Proof. Since [a, b] is a closed and bounded set and f is continuous on [a, b],
Theorem 4.1.1 follows from Theorem 2.3.14.

Deп¬Ѓnition 4.1.4 A function f is said to be increasing on an open interval
(a, b) if f (x1 ) < f (x2 ) for all x1 and x2 in (a, b) such that x1 < x2 . The
function f is said to be decreasing on (a, b) if f (x1 ) > f (x2 ) for all x1 and
x2 in (a, b) such that x1 < x2 . The function f is said to be non-decreasing
on (a, b) if f (x1 ) в‰¤ f (x2 ) for all x1 and x2 in (a, b) such that x1 < x2 . The
function f is said to be non-increasing on (a, b) if f (x1 ) в‰Ґ f (x2 ) for all x1
and x2 in (a, b) such that x1 < x2 .

Theorem 4.1.2 Suppose that a function f is deп¬Ѓned on some open interval
(a, b) containing a number c such that f (c) exists and f (c) = 0. Then f (c)
is not a local extremum of f .

1
Proof. Suppose that f (c) = 0. Let = |f (c)|. Then > 0.
2
Since > 0 and
f (x) в€’ f (c)
f (c) = lim ,
xв€’c
xв†’c

there exists some Оґ > 0 such that if 0 < |x в€’ c| < Оґ, then

f (x) в€’ f (c) 1
в€’ f (c) < |f (c)|
xв€’c 2
f (x) в€’ f (c)
1 1
в€’ |f (c)| < в€’ f (c) < |f (c)|
xв€’c
2 2
f (x) в€’ f (c)
1 1
f (c) в€’ |f (c)| < < f (c) + |f (c)|.
xв€’c
2 2
The following three numbers have the same sign, namely,
1
1
|f (c)| and f (c) + |f (c)|.
f (c), f (c) в€’
2 2
148 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Since f (c) > 0 or f (c) < 0, we conclude that

f (x) в€’ f (c) f (x) в€’ f (c)
0< or <0
xв€’c xв€’c
for all x such that 0 < |x в€’ c| < Оґ. Thus, if c в€’ Оґ < x1 < c < x2 < c + Оґ, then
either f (x1 ) < f (c) < f (x2 ) or f (x1 ) > f (c) > f (x2 ). It follows that f (c) is
not a local extremum.

Theorem 4.1.3 If f is deп¬Ѓned on an open interval (a, b) containing c, f (c)
is a local extremum of f and f (c) exists, then f (c) = 0.

Proof. This theorem follows immediately from Theorem 4.1.2.

Theorem 4.1.4 (RolleвЂ™s Theorem) Suppose that a function f is continuous
on a closed and bounded interval [a, b], diп¬Ђerentiable on the open interval
(a, b) and f (a) = f (b). Then there exists some c such that a < c < b and
f (c) = 0.

Proof. Since f is continuous on [a, b], there exist two numbers c1 and c2
on [a, b] such that f (c1 ) в‰¤ f (x) в‰¤ f (c2 ) for all x in [a, b]. (Extreme Value
Theorem.) If f (c1 ) = f (c2 ), then the function f has a constant value on [a, b]
1
and f (c) = 0 for c = 2 (a + b). If f (c1 ) = f (c2 ), then either f (c1 ) = f (a)
or f (c2 ) = f (a). But f (c1 ) = 0 and f (c2 ) = 0. It follows that f (c1 ) = 0 or
f (c2 ) = 0 and either c1 or c2 is between a and b. This completes the proof
of RolleвЂ™s Theorem.

Theorem 4.1.5 (The Mean Value Theorem) Suppose that a function f is
continuous on a closed and bounded interval [a, b] and f is diп¬Ђerentiable on
the open interval (a, b). Then there exists some number c such that a < c < b
and
f (b) в€’ f (a)
= f (c).
bв€’a

Proof. We deп¬Ѓne a function g(x) that is obtained by subtracting the line
joining (a, f, (a)) and (b, f (b)) from the function f :

f (b) в€’ f (a)
g(x) = f (x) в€’ (x в€’ a) + f (a) .
bв€’a
4.1. MATHEMATICAL APPLICATIONS 149

The g is continuous on [a, b] and diп¬Ђerentiable on (a, b). Furthermore, g(a) =
g(b) = 0. By RolleвЂ™s Theorem, there exists some number c such that a < c < b
and

0 = g (c)
f (b) в€’ f (a)
= f (c) в€’ .
bв€’a
Hence,
f (b) в€’ f (a)
= f (c)
bв€’a
as required.

Theorem 4.1.6 (Cauchy-Mean Value Theorem) Suppose that two functions
f and g are continuous on a closed and bounded interval [a, b], diп¬Ђerentiable
on the open interval (a, b) and g (x) = 0 for all x in (a, b). Then there exists
some number c in (a, b) such that

f (b) в€’ f (a) f (c)
= .
g(b) в€’ g(a) g (c)

Proof. We deп¬Ѓne a new function h on [a, b] as follows:

f (b) в€’ f (a)
h(x) = f (x) в€’ f (a) в€’ (g(x) в€’ g(a)).
g(b) в€’ g(a)

Then h is continuous on [a, b] and diп¬Ђerentiable on (a, b). Furthermore,

h(a) = 0 and h(b) = 0.

By RolleвЂ™s Theorem, there exist some c in (a, b) such that h (c) = 0. Then

f (b) в€’ f (a)
0 = h (c) = f (c) в€’ g (c)
g(b) в€’ g(a)
and, hence,
f (b) в€’ f (a) f (c)
=
g(b) в€’ g(c) g (c)
as required. This completes the proof of Theorem 4.1.6.
150 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Theorem 4.1.7 (LвЂ™HospitalвЂ™s Rule, 0 Form) Suppose f and g are diп¬Ђeren-
0
tiable and g (x) = 0 on an open interval (a, b) containing c (except possibly
at c). Suppose that
f (x)
= L,
lim f (x) = 0 , lim g(x) = 0 and lim
g (x)
xв†’c xв†’c xв†’c

where L is a real number, в€ћ, or в€’в€ћ. Then
f (x) f (x)
lim = lim = L.
g(x) xв†’c g (x)
xв†’c

Proof. We deп¬Ѓne f (c) = 0 and g(c) = 0. Let x в€€ (c, b). Then f and g are
continuous on [c, x], diп¬Ђerentiable on (c, x) and g (y) = 0 on (c, x). By the
Cauchy Mean Value Theorem, there exists some point y в€€ (c, x) such that
f (x) в€’ f (c)
f (x) f (y)
= = .
g(x) в€’ g(c)
g(x) g (y)
Then
f (x) f (y)
lim = lim = L.
g(x) yв†’c+ g (y)
+
xв†’c

Similarly, we can prove that
f (x)
lim = L.
g(x)
в€’
xв†’c

Therefore,
f (x) f (x)
lim = lim = L.
g(x) xв†’c g (x)
xв†’c

Remark 12 Theorem 4.1.7 is valid for one-sided limits as well as the two-
sided limit. This theorem is also true if c = в€ћ or c = в€’в€ћ.

Theorem 4.1.8 Theorem 4.1.7 is valid for the case when

lim f (x) = в€ћ or в€’ в€ћ and lim g(x) = в€ћ or в€’ в€ћ.
xв†’c xв†’c

Proof of Theorem 4.1.8 is omitted.
4.1. MATHEMATICAL APPLICATIONS 151

Example 4.1.1 Find each of the following limits using LвЂ™HospitalвЂ™s Rule.

sin 3x tan 2x sin x
(i) lim (ii) lim (iii) lim
sin 5x tan 3x x
xв†’0 xв†’0 xв†’0

1 в€’ cos x
x
(iv) lim (v) lim (vi) lim x ln x
sin x x
xв†’0 xв†’0 xв†’0

We compute these limits as follows:

sin 3x 3 cos 3x 3
(i) lim = lim =
sin 5x xв†’0 5 cos 5x 5
xв†’0

2 sec2 x
tan 2x 2
(ii) lim = lim =
xв†’0 3 sec2 3x
xв†’0 tan 3x 3
sin x cos x
(iii) lim = lim =1
x 1
xв†’0 xв†’0

x 1
(iv) lim = lim =1
sin x xв†’0 cos x
xв†’0

1 в€’ cos x sin x
(v) lim = lim =0
x 1
xв†’0 xв†’0

1
ln x x
(vi) lim x ln x = lim = lim = lim (в€’x) = 0.
в€’1
1
xв†’0 xв†’0 xв†’0 xв†’0
x2
x

Theorem 4.1.9 Suppose that two functions f and g are continuous on a
closed and bounded interval [a, b] and are diп¬Ђerentiable on the open interval
(a, b). Then the following statements are true:

(i) If f (x) > 0 for each x in (a, b), then f is increasing on (a, b).

(ii) If f (x) < 0 for each x in (a, b), then f is decreasing on (a, b).

(iii) If f (x) в‰Ґ 0 for each x in (a, b), then f is non-decreasing on (a, b).

(iv) If f (x) в‰¤ 0 for each x in (a, b), then f is non-increasing on (a, b).

(v) If f (x) = 0 for each x in (a, b), then f is constant on (a, b).
152 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

(vi) If f (x) = g (x) on (a, b), then f (x) = g(x)+C, for constant C, on (a, b).

Proof.
Part (i) Suppose a < x1 < x2 < b. Then f is continuous on [x1 , x2 ] and
diп¬Ђerentiable on (x1 , x2 ). By the Mean Value Theorem, there exists some c
such that a < x1 < c < x2 < b and

f (x2 ) в€’ f (x1 )
= f (c) > 0.
x2 в€’ x1

Since x2 в€’ x1 > 0, it follows that f (x2 ) в€’ f (x1 ) > 0 and f (x2 ) > f (x1 ). By
deп¬Ѓnition, f is increasing on (a, b). The proof of Parts (ii)вЂ“(v) are similar
and are left as an exercise.
Part (vi) Let F (x) = f (x) в€’ g(x) for all x in [a, b]. Then F is continuous on
[a, b] and diп¬Ђerentiable on (a, b). Furthermore, F (x) = 0 on (a, b). Hence,
by Part (v), there exists some constant C such that for each x in (a, b),

F (x) = C, f (x) в€’ g(x) = c, f (x) = g(x) + C.

This completes the proof of the theorem.

Theorem 4.1.10 (First Derivative Test for Extremum) Let f be continuous
on an open interval (a, b) and a < c < b.

(i) If f (x) > 0 on (a, c) and f (x) < 0 on (c, b), then f (c) is a local maxi-
mum of f on (a, b).

(ii) If f (x) < 0 on (a, c) and f (x) > 0 on (c, b), then f (c) is a local minimum
of f on (a, b).

Proof. This theorem follows immediately from Theorem 4.1.9 and its proof
is left as an exercise.

Theorem 4.1.11 (Second Derivative Test for Extremum) Suppose that f, f
and f exist on an open interval (a, b) and a < c < b. Then the following
statements are true:

(i) If f (c) = 0 and f (c) > 0, then f (c) is a local minimum of f .

(ii) If f (c) = 0 and f (c) < 0, then f (c) is a local maximum of f .
4.1. MATHEMATICAL APPLICATIONS 153

(iii) If f (c) = 0 and f (c) = 0, then f (c) may or may not be a local extremum.

Proof.
Part (i) If f (c) > 0, then by Theorem 4.1.2, there exists some Оґ > 0 such
that for all x in (c в€’ Оґ, c + Оґ),

f (x) в€’ f (c)
f (c)
= > 0.
xв€’c xв€’c
Hence, f (x) > 0 on (c, c + Оґ) and f (x) < 0 on (c в€’ Оґ, c). By the п¬Ѓrst
derivative test, f (c) is a local minimum of f .
Part (ii) The proof of Part (ii) is similar to Part (i) and is left as an exercise.

Part (iii) Let f (x) = x3 and g(x) = x4 . Then

f (0) = g (0) = f (0) = g (0).

However, f has no local extremum at 0 but g has a local maximum at 0.
This completes the proof of this theorem.

Deп¬Ѓnition 4.1.5 (Concavity) Suppose that f is deп¬Ѓned in some open inter-
val (a, b) containing c and f (c) exists. Let

y = g(x) = f (c)(x в€’ c) + f (c)

be the equation of the line tangent to the graph of f at c.

(i) If there exists Оґ > 0 such that f (x) > g(x) for all x in (cв€’Оґ, c+Оґ), x = c,
then the graph of f is said to be concave upward at c. If the graph of f is
concave upward at every c in (a, b), then it is said to be concave upward
on (a, b).

(ii) If there exists Оґ > 0 such that f (x) < g(x) for all x in (cв€’Оґ, c+Оґ), x = c,
then the graph of f is said to be concave downward at c. If the graph of
f is concave downward at every c in (a, b), then it is said to be concave
downward on (a, b).

(iii) The point (c, f (c)) is said to be a point of inп¬‚ection if there exists some
Оґ > 0 such that either
154 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

(i) the graph of f is concave upward on (c в€’ Оґ, c) and concave downward
on (c, c + Оґ), or
(ii) the graph of f is concave downward on (c в€’ Оґ, c) and concave upward
on (c, c + Оґ).

Remark 13 The п¬Ѓrst derivative test, second derivative test and concavity
test are very useful in graphing functions.

Example 4.1.2 Let f (x) = x4 в€’ 4x2 , в€’3 в‰¤ x в‰¤ 3

(a) Locate the local extrema, and point extrema and points of inп¬‚ections.

(b) Locate the intervals where the graph of f is increasing, decreasing, con-
cave up and concave down.

(c) Sketch the graph of f . Determine the absolute maximum and the abso-
lute minimum of the graph of f on [в€’3, 3].

Part (a)

(i) f (x) = x4 в€’ 4x2 = x2 (x2 в€’ 4) = 0 в†’ x = 0, x = в€’2, x = 2 are zeros of f .

в€љ
в€љ
(ii) f (x) = 4x3 в€’ 8x = 0 = 4x(x2 в€’ 2) = 0 в†’ x = 0, x = в€’ 2 and x = 2
are the critical points of f .
1 1
1
(iii) f (x) = 12x2 в€’ 4 = 12 x2 в€’ = 0 в†’ x = в€’ в€љ and x = в€љ are
3 3 3
the x-coordinates of the points of inп¬‚ections of the graph of f , since f
changes sign at these points.

(iv) f (0) = 0, f (0) = в€’4 в†’ f (0) = 0 is a local minimum of f .
в€љ в€љ в€љ
f (в€’ 2) = 0, f (в€’ 2) > 0 в†’ f (в€’ 2) = в€’8 is a local minimum of f .
в€љ в€љ в€љ
f ( 2) = 0, f ( 2) > 0 в†’ f ( 2) = в€’8 is a local minimum of f .
1 в€’11
1
(v) f (x) changes sign at x = В± в€љ and hence В± в€љ , are the points
39
3
of inп¬‚ection of the graph of f .
4.1. MATHEMATICAL APPLICATIONS 155
в€љ в€љ
Part (b) The function f is decreasing on (в€’в€ћ, в€’ 2) в€Є (0, 2) and is increasing
в€љ в€љ в€’1
on (в€’ 2, 0) в€Є ( 2, в€ћ). The graph of f is concave up on в€’в€ћ, в€љ в€Є
3
в€’1 1
1
в€љ , в€ћ and is concave down on в€љ , в€љ .
3 33
(c) f (в€’3) = f (3) = 45 is the absolute maximum of f and is obtained at the
end points of the interval.
в€љ в€љ
Also, f (в€’ 2) = f ( 2) = в€’8 is the absolute minimum of f on [в€’3, 3].
We note that f (0) = 0 is a local maximum of f . The graph is sketched
with the above information.

graph

Example 4.1.3 Consider g(x) = x2 в€’x2/3 , в€’2 в‰¤ x в‰¤ 3. Sketch the graph of
g, locating extrema, zeros, points of inп¬‚ection, intervals where f is increasing
or decreasing, and intervals where the graph of f is concave up or concave
down.
Let us compute the zeros and critical points of g.
(i) g(x) = x2/3 (x4/3 в€’ 1) = 0 в†’ x = 0, в€’1, 1.
3/4
1 1
2
g (x) = 2x в€’ xв€’1/3 = 2xв€’1/3 x4/3 в€’ =0в†’x=В± .
3 3 3
3/4
1
We note that g (0) is undeп¬Ѓned. The critical points are, 0, В± .
3
2
(ii) g (x) = 2 + xв€’4/3 > 0 for all x, except x = 0, where g (x) does not
9
exist.
3/4 3/4
1 1
в€’в€ћ, в€’
The function g is decreasing on and 0, .
3 3
3/4 3/4
1 1
в€’ в€Є ,в€ћ .
The function g is increasing on ,0
3 3
156 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

(iii) The point (0, 0) is not an inп¬‚ection point, since the graph is concave up
everywhere on (в€’в€ћ, 0) в€Є (0, в€ћ).

Exercises 4.1 Verify that each of the following Exercises 1вЂ“2 satisп¬Ѓes the
hypotheses and the conclusion of the Mean Value Theorem. Determine the
1. f (x) = x2 в€’ 4x, в€’2 в‰¤ x в‰¤ 2
2. g(x) = x3 в€’ x2 on [в€’2, 2]
3. Does the Mean Value Theorem apply to y = x2/3 on [в€’8, 8]? If not, why
not?
4. Show that f (x) = x2 в€’ x3 cannot have more than two zeros by using
RolleвЂ™s Theorem.
5. Show that f (x) = ln x is an increasing function. (Use Mean Value The-
orem.)
6. Show that f (x) = eв€’x is a decreasing function.
7. How many real roots does f (x) = 12x4 в€’ 14x2 + 2 have?
8. Show that if a polynomial has four zeros, then there exists some c such
that f (c) = 0.
A function f is said to satisfy a Lipschitz condition with constant M if
|f (x) в€’ f (y)| в‰¤ M |x в€’ y|
for all x and y. The number M is called a Lipschitz constant for f .
9. Show that f (x) = sin x satisп¬Ѓes a Lipschitz condition. Find a Lipschitz
constant.
10. Show that g(x) = cos x satisп¬Ѓes a Lipschitz condition. Find a Lipschitz
constant for g.
In each of the following exercises, sketch the graph of the given function over
the given interval. Locate local extrema, absolute extrema, intervals where
the function is increasing, decreasing, concave up or concave down. Locate
the points of inп¬‚ection and determine whether the points of inп¬‚ection are
oblique or not.
4.2. ANTIDIFFERENTIATION 157

x2
12. f (x) = x2 (1 в€’ x)2 , [в€’2, 2]
11. f (x) = 2 , [в€’1, 1]
2x + 1
1
14. f (x) = 2x2 +
13. f (x) = |x в€’ 1| + 2|x + 2|, [в€’4, 4] , [в€’1, 1]
x2

15. f (x) = sin x в€’ cos x, [0, 2ПЂ] 16. f (x) = x в€’ cos x, [0, 2ПЂ]

2x
18. f (x) = 2x3/5 в€’ x6/5 , [в€’2, 2]
17. f (x) = , [в€’4, 4]
x2 в€’ 9
2
19. f (x) = (x2 в€’ 1)eв€’x , [в€’2, 2] 20. f (x) = 3 sin 2x + 4 cos 2x, [0, 2ПЂ]

Evaluate each of the following limits by using the LвЂ™HospitalвЂ™s Rule.
x + sin ПЂx
sin 3x
22. lim
21. lim
x в€’ sin ПЂx
tan 5x xв†’0
xв†’0

ex в€’ 1
x ln x
23. lim 24. lim
xв†’1 1 в€’ x xв†’0 ln(x + 1)

ex в€’ 1 10x в€’ 1
25. lim 26. lim
x x
xв†’0 xв†’0

sin 3x 1
в€’ csc x
27. lim 28. lim
sinh(5x) x
xв†’0 xв†’0

(1 в€’ x2 )
x + tan x
29. lim 30. lim
xв†’1 (1 в€’ x3 )
xв†’0 x + sin x

4.2 Antidiп¬Ђerentiation
The process of п¬Ѓnding a function g(x) such that g(x) = f (x), for a given
f (x), is called antidiп¬Ђerentiation.

Deп¬Ѓnition 4.2.1 Let f and g be two continuous functions deп¬Ѓned on an
open interval (a, b). If g (x) = f (x) for each x in (a, b), then g is called an
antiderivative of f on (a, b).
158 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Theorem 4.2.1 If g1 (x) and g2 (x) are any two antiderivatives of f (x) on
(a, b), then there exists some constant C such that

g1 (x) = g2 (x) + C.

Proof. If h(x) = g1 (x) в€’ g2 (x), then

h (x) = g1 (x) в€’ g2 (x)
= f (x) в€’ f (x)
=0

for all x in (a, b). By Theorem 4.1.9, Part (iv), there exists some constant c
such that for all x in (a, b),

C = h(x) = g1 (x) в€’ g2 (x)
g2 (x) = g1 (x) + C.

Deп¬Ѓnition 4.2.2 If g(x) is an antiderivative of f on (a, b), then the set
{g(x)+C : C is a constant} is called a one-parameter family of antiderivatives
of f . We called this one-parameter family of antiderivatives the indeп¬Ѓnite
integral of f (x) on (a, b) and write

f (x)dx = g(x) + C.

The expression вЂњ f (x)dxвЂќ is read as вЂњthe indeп¬Ѓnite integral of f (x) with
respect to x.вЂќ The function вЂњf (x)вЂќ is called the integrand, вЂњ вЂќ is called the
integral sign and вЂњxвЂќ is called the variable of integration. When dealing with
indeп¬Ѓnite integrals, we often use the terms antidiп¬Ђerentiation and integration
interchangeably. By deп¬Ѓnition, we observe that
d
f (x)dx = g (x) = f (x).
dx

Example 4.2.1 The following statements are true:
xn+1
1
3
x dx = x4 + c n
+ c, n = в€’1
1. 2. x dx =
4 n+1
4.2. ANTIDIFFERENTIATION 159

1
dx = ln |x| + c sin x dx = в€’ cos x + c
3. 4.
x

в€’1
5. sin(ax)dx = cos(ax) + c 6. cos x dx = sin x + c
a

1
tan x dx = ln | sec x| + c
7. cos(ax)dx = sin(ax) + c 8.
a

1
ln | sec(ax)| + c cot x dx = ln | sin x| + c
9. tan(ax)dx = 10.
a

1
ex dx = ex + c
ln | sin(ax)| + c
11. cot(ax)dx = 12.
a

1 ax
eв€’x dx = в€’eв€’x + c eax dx =
13. 14. e +c
a

15. sinh xdx = cosh x + c 16. cosh x dx = sinh x + c

tanh x dx = ln | cosh x| + c
17.

coth x dx = ln | sinh x| + c
18.

1
19. sinh(ax) = cosh(ax) + c
a

1
20. cosh(ax)dx = sinh(ax) + c
a

1
ln | cosh ax| + c
21. tanh(ax)dx =
a

1
ln | sinh(ax)| + c
22. coth (ax)dx =
a
160 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

sec x dx = ln | sec x + tan x| + c
23.

csc x dx = в€’ ln | csc x + cot x| + c
24.

1
ln | sec(ax) + tan(ax)| + c
25. sec(ax)dx =
a

в€’1
ln | csc(ax) + cot(ax)| + c
26. csc(ax)dx =
a

sec2 xdx = tan x + c
27.

1
sec2 (ax)dx =
28. tan(ax) + c
a

csc2 x dx = в€’ cot x + c
29.

в€’1
csc2 (ax)dx =
30. cot(ax) + c
a

tan2 x dx = tan x в€’ x + c
31.

cot2 x dx = в€’ cot x в€’ x + c
32.

1 1 sin 2x
sin2 x dx = (x в€’ sin x cos x) + c = xв€’
33. +c
2 2 2

1 1 sin 2x
cos3 xdx =
34. (x + sin x cos x) + c = x+ +c
2 2 2

35. sec x tan x dx = sec x + c
4.2. ANTIDIFFERENTIATION 161

csc x cot x dx = в€’ csc x + c
36.

Each of these indeп¬Ѓnite integral formulas can be proved by diп¬Ђerentiating
the right sides of the equation. We show some details in selected cases.
Part 3. Recall that
|x|
x
d
(|x|) = = , x = 0.
|x|
dx x
Hence,
|x|
1 1
d
(ln |x| + c) = В· +0 = .
|x|
dx x x
The absolute values are necessary because ln(x) is deп¬Ѓned for positive num-
bers only.

1
d
В· (sec x tan x + sec2 x)
(ln | sec x + tan x|) =
Part 23.
dx sec x + tan x

sec x(tan x + sec x)
=
(sec x + tan x)
= sec x.

d
(tan x в€’ x + c) = sec2 x в€’ 1 = tan2 x.
Part 31.
dx

d 1
(x в€’ sin x cos x) + c
Part 33.
dx 2

d x sin 2x
в€’
= (Trigonometric Identity)
dx 2 4

1 2 cos 2x
в€’
=
2 4
1
(1 в€’ cos x)
=
2

= sin2 x (Trigonometric Identity)
162 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

d 1
Part 34. (x + sin x cos x) + c
dx 2

d x sin 2x
= +
dx 2 4

11
= + cos 2x
22
1
= (1 + cos 2x)
2

= cos2 x (Trigonometric Identity)

Example 4.2.2 The following statements are true:
в€љ
1 x
в€љ в€љ dx = в€’ 1 в€’ x2 + c
1. dx = arcsin x + c 2.
1 в€’ x2 2
1в€’x
в€љ
1 1
в€љ в€љ dx = 1 + x2 + c
3. dx = arcsinh x + c 4.
1 + x2 1 + x2
в€љ
= ln(x + 1 + x2 ) + c

в€љ
x
1
в€љ в€љ dx = x2 в€’ 1 + c
dx = arccosh x + c 6.
5.
2в€’1 x2 в€’ 1
x в€љ
= ln |x + x2 в€’ 1| + c

1 1
7. dx = arctan x + c 8. dx = arctanh x + c
1 + x2 1 в€’ x2
1+x
1
+c
= ln
1в€’x
2

bx
1 x
в€љ
9. dx = arcsec x + c 10. b dx = + c, b > 0, b = 1
ln b
2в€’1
|x| x

All of these integration formulas can be veriп¬Ѓed by diп¬Ђerentiating the right
sides of the equations.
4.2. ANTIDIFFERENTIATION 163

Remark 14 In the following exercises, use the substitution to reduce the
integral to a familiar form and then use the integral tables if necessary.

Exercises 4.2 In each of the following, evaluate the indeп¬Ѓnite integral by
using the given substitution. Use the formula:

f (g(t))g (t)dt = f (u)du, where u = g(t), du = g (t)dt.

1 1
в€љ в€љ
1. dx, x = 2 sin t 2. dx, x = 2 cosh t
4 в€’ x2 4 + x2

1 1
в€љ в€љ dx, x = 3 sec t
3. dx, x = 3 tan t 4.
x x2 в€’ 9
9 + x2

2
xeв€’x dx, u = в€’x2
5. 6. sin(7x + 1)dx; u = 7x + 1

sec2 (3x + 1)dx, u = 3x + 1 cos2 (2x + 1)dx, u = 2x + 1
7. 8.

x sin2 (x2 )dx, u = x2 tan2 (5x + 7)dx, u = 5x + 7
9. 10.

sec(2x в€’ 3) tan(2x в€’ 3)dx, u = 2x в€’ 3
11. 12. cot(5x + 2)dx, u = 5x + 2

x
x(x2 + 1)10 dx, u = x2 + 1 dx, u = x2 + 1
13. 14. 2 + 1)1/3
(x

e2x в€’ eв€’2x
1
dx, u = e2x + eв€’2x
dx, u = ex
15. 16.
x + eв€’x 2x + eв€’2x
e e

sin3 (2x) cos 2x dx, u = sin 2x esin 3x cos 3x dx, u = sin 3x
17. 18.

sec2 x tan x dx, u = sec x tan10 x sec2 x dx, u = tan x
19. 20.
164 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

x ln(x2 + 1) x
dx, u = ln(x2 + 1) dx, u = 4 + x2
в€љ
21. 22.
2+1
x 4 + x2

x dx x
, u = 4 в€’ x2 dx, u = 9 + x2
в€љ
23. 24.
9 + x2
4 в€’ x2

1 1
в€љ в€љ
25. dx, u = 2 cosh x
dx, u = 2 sinh x 26.
x2 в€’ 4
4 + x2

4.3 Linear First Order Diп¬Ђerential Equations
Deп¬Ѓnition 4.3.1 If p(x) and q(x) are deп¬Ѓned on some open interval, then
an equation of the form
dy
+ p(x)y = q(x)
dx
is called a linear п¬Ѓrst order diп¬Ђerential equation in the variable y.

Example 4.3.1 (Exponential Growth). A model for exponential growth is
the п¬Ѓrst order diп¬Ђerential equation

dy
= ky, k > 0, y(0) = y0 .
dx
To solve this equation we divide by y, integrate both sides with respect to x,
dy
replacing dx by dy as follows:
dx

1 dy
dx = k dx
y dx
1
dy = kx + c
y
ln |y| = kx + c
|y| = ekx+c = ec ekx
y = В±ec ekx .
4.3. LINEAR FIRST ORDER DIFFERENTIAL EQUATIONS 165

Next, we impose the condition y(0) = y0 to get

y(0) = В±ec = y0
y = y0 ekx .

The number y0 is the value of y at x = 0. If the variable x is replaced by the
time variable t, we get
y(t) = y(0)ekt .
If k > 0, this is an exponential growth model. If k < 0, this is an example of
an exponential decay model.

Theorem 4.3.1 (Linear First Order Diп¬Ђerential Equations) If p(x) and q(x)
are continuous, then the diп¬Ђerential equation

dy
+ p(x)y = q(x) (1)
dx
has the one-parameter family of solutions

y(x) = eв€’ p(x)dx
p(x)dx
q(x)e dx + c .

p(x)dx
Proof. We multiply the given diп¬Ђerential equation (1) by e , which is
called the integrating factor.

p(x)dx dy p(x)dx p(x)dx
e + p(x)e y = q(x)e . (2)
dx
Since the integrating factor is never zero, the equation (2) has exactly the
same solutions as equation (1). Next, we observe that the left side of the
equation is the derivative of the product the integrating factor and y:

d p(x)dx p(x)dx
e y = q(x)e . (3)
dx
By the deп¬Ѓnition of the indeп¬Ѓnite integral, we express equation (3) as follows:

p(x)dx p(x)dx
e y= q(x)e dx + c. (4)
166 CHAPTER 4. APPLICATIONS OF DIFFERENTIATION

Next, we multiply both sides of equation (4) by eв€’ p(x)dx
:

y = eв€’ p(x)dx
p(x)dx
q(x)e dx + c . (5)

Equation (5) gives a one-parameter family of solutions to the equation. To
pick a particular member of the family, we specify either a point on the curve,
or the slope at a point of the curve. That is,

or y (0) = y0 .
y(0) = y0

Then c is uniquely determined. This completes the proof.

Example 4.3.2 Solve the diп¬Ђerential equation

y + 4y = 10 , y(0) = 200.

Step 1. We multiply both sides by the integrating factor
4dx
= e4x
e
dy
e4x + 4e4x y = 10e4x . (6)
dx
Step 2. We observe that the left side is the derivative of the integrating factor
and y.
d
(e4x y) = 10e4x . (7)
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