Consumer Theory Essay

2617 words - 11 pages

XEQ 201: Calculus II

Course description


Chapter 1. Applications of Differentiation
1.1. Mean value theorems of differential calculus
1.2. Using differentials and derivatives
1.3. Extreme Values



Course description
Application of differentiation. Taylor theorem. Mean Value theorem of
differential calculus. Methods of integration. Applications of integration.
1. Calculus: A complete course by Robert A. Adams and Christopher
2. Fundamental methods of mathematical economics by Alpha C. Chiang.
3. Schaum’s outline series: Introduction to mathematical economics
by Edward T. Dowling



...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
= [MVT on [0, x]] =
sin x

= cos c < 1

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


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

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.



(x) > 0
(x) < 0
(x) ≥ 0
(x) ≤ 0



x ∈ J,
x ∈ J,
x ∈ J,
x ∈ J,




increasing on I.
decreasing on I.
non-decreasing on I.
non-increasing on I.

derivative of
increasing and

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.

derivetive of

XEQ 201

derivative at
interior extreme

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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