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Economics

ECON 2P30

Lester Kwong

Winter

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Univariate Functions and Optimization
Lester M.K. Kwong
Created: August 12, 2004
This Version: October 23, 2008
1 Limits and Derivatives
With the de▯nition of a function, we are often concerned with the function
value as the variable concerned approaches arbitrarily close to a certain point.
Hence, we use the notion of limits to address this issue. For example, when
we write:
x!z f(x) (1)
this is to denote the function value of the function f(x) as x approaches
z. Similarly, we can write this limit in alternative ways. For example,
limx!z f(x) = lim ▯!0 f(z + ▯). When the limit of a function f(x) as x
approaches some value z is say y, we write limx!zf(x) = y. This is to say
that the value of f(x) gets closer and closer to y as x gets closer and closer to
z. However, also note that in discussing about limits, we are never concerned
with the case when x = z. In fact, we are always talking about the value
of f(x) when x approaches z. In this sense, even if the function f(x) is not
de▯ned when x = z the limit as x approaches z may still exist.
There are often times when we are only concerned with the limit of a
function when it approaches a point in the domain from a particular side.
This may occur if the limit from the opposing side does not belong in the
domain of the function. Therefore, we write lim x!z +f(x) to denote the
Copyright
2004 Lester M.K. Kwong. Department of Economics, Brock Univer-
sity, 500 Glenridge Ave., SCatharines, Ontario, L2S 3A1, Canada.Email: lk-
[email protected] Tel: +1 (905) 688-5550, Ext. 5137.
1 limit of f(x) as x approaches z from the right, or for x > z and write
limx!z ▯f(x) to denote the limit of f(x) as x approaches z from the left, or
+
for x < z. It then follows that lx!z f(x) = y if and only if lix!z f(x) =
limx!z ▯f(x) = y.
There are many laws that one may rely on when computing limits. We
state four here with no particular order. Let f(x) and h(x) be two functions
and that their limits exist at z. Then:
1. limx!z(f(x) ▯ h(x)) = lim x!z f(x) ▯ limx!z h(x)
2. lim (c(f(x)) = clim f(x), where c 2 R is a constant
x!z x!z
3. limx!z(f(x)h(x)) = lim x!z f(x)lim x!z h(x)
f(x) lix!zf(x)
4. limx!z h(x)= lix!zh(x)if lix!z h(x) 6= 0
In addition, there are two special limits that one should be aware of.
Namely, if c 2 R is a constant, then limx!zc = c and lim x!z x = z. It then
n n
naturally follows from the third law that lx!z x = z where n is a positive
integer.
x ▯2x +5
Example 1 Let f(x) = x +5x . Determine lim x!1f(x).
To solve for this limit we apply the above laws and special limits. Hence:
x ▯ 2x + 5 lim x!1(x ▯ 2x + 5)
lim f(x) = lim =
x!1 x!1 x + 5x 3 lim x!1(x + 5x )
3 2
limx!1 x ▯ 2lim x!1x + lim x!15
= 4 3
lim x!1x + 5lim x!1 x
1 ▯ 2 + 5 2
= =
1 + 5 3
Equipped with the knowledge of limits, one may now approach the ques-
tion regarding the derivative of a function. Heuristically, the derivative pro-
vides us with the rate of change of the function value with respect to a
variable at a speci▯c point. In other words, it is the slope of the function at
a point.
2 Consider the point x, the slope of a function at a point x may be approx-
imated by:
f(x + ▯) ▯ f(x)
(2)
x + ▯ ▯ x
for some ▯ 2 R so long as x + ▯ is in the domain of f. It turns out that
the slope, or the derivative, then is the limit of the above approximation as
▯ ! 0. More speci▯cally, we write and de▯ne:
df(x) = f (x) = lim f(x + ▯) ▯ f(x) (3)
dx ▯!0 x + ▯ ▯ x
From our discussion on limits, it is then clear that the derivative at a
point x exists so long as the limit at that point exists. This further implies
0
that f (x) exists at x if and only if:
f(x + ▯) ▯ f(x) f(x + ▯) ▯ f(x)
lim = lim (4)
▯!0 + x + ▯ ▯ x ▯!0 ▯ x + ▯ ▯ x
Some general rules for derivatives are provided below:
1. If f(x) = ax then f (x) = anx n▯1 ,
0
2. If f(x) = lnx then f (x) = 1=x.
However, we often deal with more complex functions than those stated
above, and we provide some more general rules below. Suppose f(x) and
g(x) are functions and that their derivatives exist. Then:
0 0 0
1. (f(x) ▯ g(x)) = f (x) ▯ g (x),
2. (f(x)g(x)) = f (x)g(x) + f(x)g (x), 0
3. (f(x)=g(x)) = (f (x)g(x) ▯ f(x)g (x))=(g(x)) , 2
0 0 0
4. (f(g(x))) = f (g(x))g (x).
...........................................................................
3 1.1 Review Exercises
...........................................................................
Exercise 1 Evaluate the following limits as x ! 1. (i) f(x) = x ▯ 1, (ii)
x +2▯5x 3 3 2
f(x) = x ▯2 , (iii) f(x) = 3x ▯ 3x .
Exercise 2 Prove that the following limit does not exist:
jxj
lim (5)
x!0 x
Exercise 3 Prove that if f(x) = ax n then f (x) = anx n▯1 where a is a
constant and n a positive integer.
Exercise 4 Determine the derivatives for the following functions:
2 6 x ▯7x4
(i) f(x) = x + x ▯ 3 (ii) f(x) = 2x +1
(iii) f(x) = (x + 8x )3 2 (iv) f(x) = ln(x ▯ 5x + 2)
(v) f(x) = x ▯ 4lnx (vi) f(x) = ln(ln(lnx))
x e2x
(vii) f(x) = x (viii) f(x) = x2
...........................................................................
2 Properties of Functions on R
In this section, we provide some very useful theorems and de▯nitions. Of
particular interest is often the notion of continuity of functions.
0
De▯nition 1 A function f : R ! R is continuous at a point x if 8▯ > 0,
9▯ > 0 such that d(x;x ) < ▯ implies that d(f(x);f(x )) < ▯.
Note that d(x;y) = jx ▯ yj is the distance operator between x and y.
Naturally, a function is said to be continuous if it is continuous for all
0
x in its domain. Such classes of functions are referred to as the C space
k
of functions. It is clear then, that the space C refers to the class of func-
tions which are k-times continuously di▯erentiable. 1 Aside from De▯nition
0
1, an alternative way to de▯ne continuity of a function at some point x is if
lim 0 f(x) = f(x ). Thus, this immediately follows.
x!x
1
Sometimes, one may refer to a function as being smooth to imply the existence of
su▯cient continuous derivatives. The exact measure of smoothness is somewhat arbitrary
and is simply a re
ection of the requirements of the problem at hand. Hence, the degree
0 1
of smoothness may range from C to C functions.
4 0
Proposition 1 If a function, f : R ! R is di▯erentiable at x then it is
continuous at x .0
If f and g are continuous at x then from the de▯nition of continuity, as
well as properties of limits, the following holds:
1. f ▯ g is continuous at x . 0
0
2. f ▯ g is continuous at x .
3. f=g is continuous at x provided that g(x ) 6= 0.
Note that we have only de▯ne a general notion of continuity. There
are, other forms of continuity such as pointwise continuity and piecewise
2
continuity for which is beyond the scope of this course. However, we should
note the existence of such alternative and generally weaker de▯nitions.
Aside from continuity, we may sometimes be concerned with the behavior
of a function with respect to its argument. We begin by de▯ning monotonic-
ity.
De▯nition 2 A function f is said to be strictly (weakly) monotone increas-
ing if for all x;z 2 R, if x > z then f(x) > (▯)f(z). Similarly, a function
f is said to be strictly (weakly) monotone decreasing if for all x;z 2 R, if
x > z then f(x) < (▯)f(z).
It follows then that strictly monotone functions are \1 ▯ 1" relations. 3
Hence, a general observation is that strictly monotonic functions are invert-
ible. The inverse of a function, as discussed before, is denoted f ▯1(x) with
▯1 ▯1
the property that f (f(x)) = f(f (x)) = x. Hence the inverse function is
simply a re
ection across the 45 line in a two dimensional geometric inter-
pretation. It also follows by the rules of di▯erentiation that:
df ▯1(x) 1
=
dx f (f▯1 (x))
2Interested readers should refer to any book on real analysis for a detailed treatment
of continuity.
3For classes of functions f 2 C , strictly (weakly) monotone increasing functions imply
f > (▯)0 and strictly (weakly) monotone decreasing functions imply f < (▯)0.
4
This, of course, hinges on the fact that the functions are \Onto" relations. But this
may be recti▯ed with no cost simply by restricting the codomain to the range in de▯ning
the relevant function.
5 Note that in the above de▯nition of the derivative of the inverse of f, the
mere existence of f ▯1 implies that f is strictly monotonic and hence f > 0 0
0
for all x 2 D or f < 0 for all x 2 D where f : D ! R. Hence, the existence
of the derivative of f ▯1 is solely dependent of the existence of the derivative
of f.
In addressing the curvature of a function f, we de▯ne concave and convex
functions below.
De▯nition 3 A function f : D ! R is said to be concave over the domain
D if for all x;z 2 D, f(▯x+(1▯▯)z) ▯ ▯f(x)+(1▯▯)f(z) for all ▯ 2 (0;1).
De▯nition 4 A function f : D ! R is said to be convex over the domain D
if for all x;y 2 D, f(▯x+(1▯▯)z) ▯ ▯f(x)+(1▯▯)f(z) for all ▯ 2 (0;1).
The geometric interpretation of a concave function may be obtained as
follows. For a function de▯ned over D ▯ R, if you take any two points in
D and connect them with a line the line should lie below (or equal to) the
function. Conversely, for convex functions, this line should lie above (or equal
to ) the function over this region.
2
As it turns out, the curvature of a function f 2 C is quite easily testable
since if f : D ! R with D ▯ R, then f is concave over D if f (x) ▯ 0 for a

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