2617 words - 11 pages

XEQ 201: Calculus II

Contents

Course description

References

iv

iv

Chapter 1. Applications of Diﬀerentiation

1.1. Mean value theorems of diﬀerential calculus

1.2. Using diﬀerentials and derivatives

1.3. Extreme Values

iii

1

1

5

7

Course description

Application of diﬀerentiation. Taylor theorem. Mean Value theorem of

diﬀerential calculus. Methods of integration. Applications of integration.

References

1. Calculus: A complete course by Robert A. Adams and Christopher

Essex.

2. Fundamental methods of mathematical economics by Alpha C. Chiang.

3. Schaum’s outline series: Introduction to mathematical economics

by Edward T. Dowling

iv

CHAPTER 1

Example 1.1.2.

Show that sin x < x for all x > 0.

Solution. If x > 2π, then sin x ≤ 1 < 2π < x. If 0 < x ≤ 2π, then by

MVT, ∃ c ∈ (0, 2π) such that

sin x

sin x − sin 0

d

=

= [MVT on [0, x]] =

sin x

x

x−0

dx

= cos c < 1

x=c

which implies that sin x < x in this case too.

increasing

decreasing

functions

Definition 1.1.1 (Increasing and decreasing functions). Suppose that

the function f is deﬁned on an interval I and that x1 and x2 are two points

in I.

(a) If f (x2 ) > f (x1 ) whenever x2 > x1 , we say that f is increasing on

I.

(b) If f (x2 ) < f (x1 ) whenever x2 > x1 , we say that f is decreasing on

I.

(c) If f (x2 ) ≥ f (x1 ) whenever x2 > x1 , we say that f is non-decreasing

on I.

(d) If f (x2 ) ≤ f (x1 ) whenever x2 > x1 , we say that f is non-increasing

on I.

Diagram Fig 2.31

2

XEQ 201

Theorem 1.1.2.

Let J be an open interval and let I be an interval consisting of all points

in J and possibly one or both of the end points of J. Suppose that f is

continuous on I and diﬀerentiable on J.

(a)

(b)

(c)

(d)

If

If

If

If

f

f

f

f

(x) > 0

(x) < 0

(x) ≥ 0

(x) ≤ 0

for

for

for

for

all

all

all

all

x ∈ J,

x ∈ J,

x ∈ J,

x ∈ J,

then

then

then

then

f

f

f

f

is

is

is

is

increasing on I.

decreasing on I.

non-decreasing on I.

non-increasing on I.

derivative of

increasing and

decreasing

functions

Example 1.1.3.

On what intervals is the function f (x) = x3 − 12x + 1 increasing? On what

intervals is it decreasing?

Solution. f (x) = 3x2 − 12 = 3 (x − 2) (x + 2). It follows that f (x) >

0 when x < −2 or x > 2 and f (x) < 0 when −2 < x < 2. Therefore f

is increasing on the intervals (−∞, −2) and (2, ∞) and is decreasing on the

interval (−2, 2).

Diagram Fig 2.32

Example 1.1.4.

Show that f (x) = x3 is increasing on any interval.

Solution. Let x1 , x2 be any two real numbers satisfying x1 < x2 . Since

f (x) = 3x2 > 0 for all x = 0, we have that f (x1 ) < f (x2 ) if either

x1 < x2 ≤ 0 or 0 ≤ x1 < x2 . If x1 < 0 < x2 , then f (x1 ) < 0 < f (x2 ). Thus

f is increasing on every interval.

Theorem 1.1.3.

If f is continuous on an interval I and f (x) = 0 at every interior point of

I, then f (x) = C, a constant on I.

3

derivetive of

constant

function

XEQ 201

derivative at

interior extreme

point

Theorem 1.1.4.

If f is deﬁned on an open interval (a, b) and achieves a maximum (or a

minimum) at the point c ∈ (a, b), and if f (c) exists, then f (c) = 0.

Values of x where f (x) = 0 are called critical points of the function f .

Theorem 1.1.5 (Rolle’s Theorem).

Suppose that the function g is continuous on the closed ﬁnite interval [a, b]

and if it is diﬀerentiable on the open interval (a, b). If g (a) = g (b), ∃ a

point c ∈ (a, b) such that g (c) = 0.

Theorem 1.1.6 (The Generalized Mean Value Theorem).

If functions f and g are both continuous on [a, b] and diﬀerentiable on (a, b),

and if g (x) = 0 for every x ∈ (a, b), then...

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