Consumer Theory Essay

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

...view middle of the document...

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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