Lecture One: Single Variable Calculus ∗

Hongjun Li †

July 5&6 2014

Contents

1 Introduction 2

2 Functions and Limits 22.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.3 Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

3 Derivatives 43.1 First-order Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43.2 Chain Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Applications 64.1 Graphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74.2 Maximum and Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 Taylor Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5 Integration 10

6 Homework 13

∗This is the lecture note for the class of Mathematical Foundations for Economists. Comments are welcome.†Email: econlee@163.com. Phone: 83952487. Address: Chengming Build. 331, Capital University of Economics

and Business, Beijing, China.

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1 Introduction

Economics is the social science that studies economic activity to gain an understanding of theprocesses that govern the production, distribution and consumption of goods and services in anexchange economy. It does not just describe what goes on in the economy, but also attempts toexplain how the economy operates and to make predictions about what may happen to specifiedeconomic variables if certain changes take place, e.g. what effect a crop failure will have on cropprices, what effect a given increase in sales tax will have on the price of finished goods, what willhappen to unemployment if government expenditure is increased. It also suggests some guidelinesthat firms, governments or other economic agents might follow if they wished to allocate resourcesefficiently. Mathematics is fundamental to any serious application of economics to these areas.What will be used mostly in economic analyses are Calculus, Linear Algebra, and ProbabilityTheory.

For example, let us denote the stock price of some stock ABC as yt. We use yt = 0.9yt−1 + εt todescribe the dynamic behavior of yt. To determine the equilibrium price and output, we constructthe demand function and supply function. Suppose we come to the economic world without math.Can you describe the dynamic behavior of stock ABC’s price accurately? Can you find a way toobtain the equilibrium price and output for the economic system? Your answer may be ”Maybe”.But we know it is hard to do so most of the time. We need mathematical tools to help us describeand analyze the economic world.

2 Functions and Limits

2.1 Sets

A set is a collection of distinct objects. The objects in a set are called the elements or membersof the set. We will denote sets by capital letters A,B,C, · · · and use lowercase letters a, b, c, · · · todenote the elements.

x ∈ A the object x is in the set A

x /∈ A the object x is not in the set A

{x : P} the set of all x which satisfy property P

A ⊂ B A is a subset of B, A is contained in B

B ⊃ B B contains A

A ∪B the union of A and B

A ∩B the intersection of A and B

∅ the empty set

Table 1: Notions and Notation

2.2 Functions

A central goal of economic theory is to express and understand relationships between economicvariables. These relationships are described mathematically by functions.

Definition 2.1 A function f is a rule that assigns to each element x in a set D exactly oneelement, called f(x), in a set D. We may denote it as f : D → E.

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We usually consider functions for which the sets D and E are sets of real numbers. The setD is called the domain of the function. The number f(x) is the value of f at x and is read“f of x.” The range of f is the set of all possible values of as varies throughout the domain,denoted as f(D). A symbol that represents an arbitrary number in the domain of a function f iscalled an independent variable. A symbol that represents a number in the range of f is calleda dependent variable.

Image, Range, and Preimage Let D and E be two sets and let f be a function from D toE. For a set C ⊂ A, f(C) is the set of all elements f(x) for x ∈ C, i.e. f(C) = {b ∈ E : b =f(x) for some x ∈ C}. We call f(C) the image of C under f . Obviously the range f(D) ⊂ B, butit is not always the case that f(D) = E. Let V ⊂ E, then f−1(V ) denotes the set of all elementsin D such that f(D) ∈ V , i.e.f−1(V ) = {x ∈ D : f(x) ∈ V }. We call f−1(V ) the preimage of Vunder f .

Monotonicity Let x1 and x2 be any two numbers in X such that x1 < x2. f is monotonicallyincreasing if f(x1) ≤ f(x2) and monotonically decreasing if f(x1) ≥ f(x2). f is said to be strictlymonotonically increasing (decreasing) if f(x1) < f(x2)(f(x1) > f(x2)).

Boundedness The function f is bounded if ∃K ∈ R such that |f(x)| < K, for ∀x ∈ X.

Definition 2.2 Given two functions f and g, the composite function f ◦ g (also called the compo-sition of f and g ) is defined by

(f ◦ g)(x) = f(g(x))

Example 2.1 Let f and g are the functions y = f(x) = x2 : A→ B and z = g(y) = ey : B → C.Then we have a composite function z = (f ◦ g)(x) = ex

2: A→ C.

2.3 Limits

We could begin by saying that limits are important in calculus, but that would be a majorunderstatement. Without limits, calculus would not exist. Every single notion of calculus is a limitin one sense or another.

Definition 2.3 Let f be a function defined at least on an open interval (c−p, c+p) except possiblyat c itself. We say that limx→c f(x) = L if for each ε > 0, there exists a δ > 0 such that if0 < |x− c| < δ, then |f(x)− L| < ε.

Example 2.2

limx→2

x3 − 8

x− 2= 12

Example 2.3 Suppose the function f is defined as

f(x) =

{3x− 4, x 6= 0

10, x = 0.

Then limx→0 f(x) = −4.

Theorem 2.1 (Uniqueness of Limit) If limx→c = M and limx→c = L, then L = M .

Theorem 2.2 If limx→c f(x) = L and limx→c g(x) = M , then

• limx→c[f(x) + g(x)] = L+M ,

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• limx→c[αf(x)] = αL, with α a real number,

• limx→c[f(x)g(x)] = LM ,

• limx→c1

f(x) = 1L if L 6= 0.

Theorem 2.3 (Pinching Theorem) Let p > 0. Suppose that, for all x such that 0 < |x−c| < p,h(x) ≤ f(x) ≤ g(x). If limx→c h(x) = L and limx→c g(x) = L, then

limx→c

f(x) = L.

Definition 2.4 (Continuity) The function f is called continuous at the point p ∈ X if

limx→p

f(x) = f(p).

This is equivalent to saying that for every ε > 0 we can find δ > 0 such that |f(x) − f(p)| < ε forall x for which |x− p| < δ. f is called continuous if it is continuous at every point of X.

3 Derivatives

3.1 First-order Derivative

We begin with a function f . On the graph of f we choose a point (x, f(x)) and a nearby point(x+ h, f(x+ h)). Through these two points we draw a secant line (see Figure 3.1).

Figure 1: Secant Lines.

The slope of a secant line is given by the difference quotient

f(x+ h)− f(x)

h

Definition 3.1 A function f is said to be differentiable at x if limh→0f(x+h)−f(x)

h exists. If thislimit exists, it is called the derivative of f at x and is denoted by f ′(x).

Example 3.1 Find the first order derivative of f(x) = x2.

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To find f(x), we form the difference quotient

f(x+ h)− f(x)

h=

(x+ h)2 − x2

h= 2x+ h.

Thus, we get

f ′(x) = limh→0

f(x+ h)− f(x)

h= lim

h→0(2x+ h) = 2x.

Figure 2: Derivative of x2.

Example 3.2 Find the first order derivative of f(x) =√x, x ≥ 0.

The difference quotient is given by

√x+ h−

√x

h=

x+ h− x(√x+ h+

√x)h

=1√

x+ h+√x.

It is now easy to check that f ′(x) = 12√x.

Figure 3: Derivative of√x.

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Theorem 3.1 (Derivatives of Sums and Scalar Multiples) Let α be a real number. If f andg are differentiable at x, then f + g and αf are differentiable at x. Moreover,

(f + g)′(x) = f ′(x) + g′(x) and (αf)′(x) = αf ′(x).

Theorem 3.2 (The Product Rule) If f and g are differentiable at x, then so is their product,and

(f · g)′(x) = f(x)g′(x) + g(x)f ′(x).

Theorem 3.3 (The Reciprocal Rule) If g is differentiable at x and g(x) 6= 0, then 1/g is dif-ferentiable at x and (

1

g

)′= − g′(x)

[g(x)]2.

Theorem 3.4 (The Quotient Rule) If f and g are differentiable at x and g(x) 6= 0, then thequotient f/g is differentiable at x and(

f

g

)′= −g

′(x)f(x)− f(x)g′(x)

[g(x)]2.

When we differentiate a function f we get a new function f ′, the derivative of f . Now supposethat f ′ can be differentiated. If we calculate (f ′)′, we get the second derivative of f . This isdenoted f ′′. So long as we have differentiability, we can continue in this manner, forming the thirdderivative of f , written f ′′′, and so on.

Definition 3.2 A function F is called an antiderivative of f on an interval I if F ′(x) = f(x) forall x in I.

Example 3.3 sinx is an antiderivative of cosx.

3.2 Chain Rule

Suppose that y is a differentiable function of u and u in turn is a differentiable function of x.Then y is a composite function of x. The chain rule suggests that

dy

dx=dy

du

du

dx(3.1)

Example 3.4 Find the first order derivative of y = ex2.

We can introduce two functions: y = f(u) = eu and u = g(x) = x2. We know that f ′(u) = dydu = eu

and g′(x) = dudx = 2x. Thus, the first order derivative of y, dy

dx = ex22x = 2xex

2.

4 Applications

The derivative contains much information about the important properties of the function. Usingthem, we can solve kinds of economic problems.

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4.1 Graphing

There are two results concerning the graph shape of a function that can be captured by deriva-tives.

Theorem 4.1 Suppose that the function f is continuously differentiable at x0. Then,

1. if f ′(x0) > 0, there is an open interval containing x0 on which f is increasing, and

2. if f ′(x0) < 0, there is an open interval containing x0 on which f is decreasing.

Sketch of the Proof.We consider the first case with f ′(x0) > 0. Since f ′(x0) > 0, we get

limh→0

f(x0 + h)− f(x0)

h= f ′(x0) > 0.

This inequality implies that if h is small and positive, f(x0+h)−f(x0) > 0. Denote x1 = x0+h,we obtain that for any x1 near x0,

x1 > x0 ⇒ f(x1) > f(x0).

So we can conclude that f is increasing near x0.

Theorem 4.2 Let f be differentiable on (a, b). Then the following holds:

1. If f ′(x) ≥ 0 for all x ∈ (a, b), then f is monotonically increasing.

2. If f ′(x) = 0 for all x ∈ (a, b), then f is constant.

3. If f ′(x) ≤ 0 for all x ∈ (a, b), then f is monotonically decreasing.

Figure 4: Convex Function.

Definition 4.1 (Convex and Concave) A function f is called convex on an interval I if andonly if

f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y)

for all x, y ∈ I and all t ∈ [0, 1]. Similarly, a function f is called concave on an interval I if andonly if

f(tx+ (1− t)y)) ≥ tf(x) + (1− t)f(y)

for all x, y ∈ I and all t ∈ [0, 1].

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It is easy to check that if function f has second order derivative f ′′(x) ≥ 0 (≤ 0) on interval I, thenit is convex (concave) on I.

Theorem 4.3 (Mean Value Theorem) Let f be a continuous function on the interval [a, b] andlet f be differentiable on (a, b). Then there is a point x ∈ (a, b) at which

f(b)− f(a) = (b− a)f ′(x).

Figure 5: Mean Value Theorem.

4.2 Maximum and Minimum

One of the major uses of calculus in mathematical models is to find and characterize maximaand minima of functions.

Definition 4.2 Let x0 be a point in the domain D of a function f . Then f(x0) is the

• global maximum value of f on D if f(x0) ≥ f(x) for all x in D;

• global maximum value of f on D if f(x0) ≥ f(x) for all x in D.

Definition 4.3 A function f has a local maximum at a point p if there exists δ > 0 such thatf(x) ≤ f(p) for all x such that |p − x| < δ. Similarly, a function f has a local minimum if thereexists δ > 0 such that f(x) ≥ f(p) for all x such that |p− x| < δ.

Example 4.1 The graph of the function

f(x) = 3x4 − 16x3 + 18x2 − 1 ≤ x ≤ 4

is shown in the above figure. You can see that f(1) = 5 is a local maximum, whereas the absolutemaximum is f(−1) = 37. Also, f(0) = 0 is a local minimum and f(3) = −27 is both a local andan absolute minimum.

Before we introduce the way of find global maximum (minimum), we show how to find the localmaximum (minimum) firstly.

Definition 4.4 A critical point of a function f is a point c in the domain of f such that eitherf ′(c) = 0 or f ′(c) does not exist.

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Figure 6: Maximum and Minimum.

Example 4.2 Find the critical points of f(x) = x35 (4− x).

You can check that the critical points are 32 (f ′

(32

)= 0) and 0 (f ′(0) does not exist).

Theorem 4.4 Let f be a function defined on the interval [a, b]. If f has a local maximum at apoint x ∈ (a, b) and if f ′(x) = 0 exists, then f ′(x) = 0. The same is true for local minimum.

This theorem says that if a function has a local maximum at a point and if its derivative existsat that point then the derivative has to be equal to zero at that point. But how can we determinewhether a critical point x0 is local max, local min, or neither? The answer to this question lies inthe second derivative of f at x0.

Theorem 4.5 Suppose f is a function defined on I and f ′(x0) = 0 at point x0 ∈ I.

• If f ′′(x0) < 0, then x0 is a local max of f ;

• If f ′′(x0) > 0, then x0 is a local min of f ; and

• If f ′′(x0) = 0, then x0 can be a local max, a local min, or neither.

Given the approach of finding local max (min), we can establish the method of finding theglobal max (min). There are three cases to consider:

• f has only one critical point in its domain;

• f ′′ > 0 or f ′′ < 0 throughout the domain of f;

• the domain of f is a closed finite interval.

Theorem 4.6 (The Extreme Value Theorem) If f is continuous on a closed interval [a, b],then f attains a global maximum value f(x) and a global minimum value f(y) at some points xand y in [a, b].

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Theorem 4.7 Suppose that f is defined on an interval I ⊂ R with x0 the only critical point on I.If x0 is a local maximum of f , then x0 is the global maximum of f on I.

Theorem 4.8 If f is a C2 function whose domain is an interval fand if f ′′ is never zero on I,then f has at most one critical point in I. This critical point is a global maximum (minimum) iff ′′ > 0 (< 0).

4.3 Taylor Expansion

As discussed above, f ′(x) is the slope of the line that is tangent to the graph of f at point x.Approximating the function with its tangent, we can write

f(x+ h) ≈ f(x) + f ′(x)h

Let us define difference between the left hand side and the right hand side as the remainderterm

R(h, x) = f(x+ h)− f(x)− f ′(x)h.

Then by the definition of the derivative,

R(h, x)

h→ 0 as h→ 0.

So not only does the remainder tend to 0 as h gets smaller, it also tends to 0 faster than h.We can approximate a differntiable function with a polynomial if we use higher order derivatives,

which is summarized as

Theorem 4.9 Let f : R → R be k + 1 times continuously differentiable on some open set U, andlet a, a+ h ∈ U. Then

f(a+ h) = f(a) + f ′(a)h+f (2)(a)

2!h2 + · · ·+ f (k)(a)

k!hk +

f (k+1)(a)

(k + 1)!hk+1

where a is between a and a+ h.

5 Integration

Differentiation was concerned with the slope of the graph of a function. Integration is concernedwith the area under the graph of the function. The Fundamental Theorem of Calculus demonstratethat integration and differentiation basically inverse operations.

Consider a function f(x). The area under the graph of the function between points x = a and

x = b is denoted by∫ ba f(x)dx, and is called the definite integral of f(x) between a and b.

Definition 5.1 (Definite Integral) If f is a function defined for a ≤ x ≤ b , we divide theinterval [a, b] into n subintervals of equal width 4x = (b − a)/n. We let x0(= a), x1, x2, · · · , xn(=b) be the endpoints of these subintervals and we let x∗1, x

∗2, · · · , x∗n be any sample points in these

subintervals, so x∗i lies in the ith subinterval [xi−1, xi] . Then the definite integral of f from a to bis ∫ b

af(x)dx = lim

n→∞

n∑i=1

f(x∗i )4x (5.1)

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Figure 7: Integration.

provided that this limit exists and gives the same value for all possible choices of sample points. Ifit does exist, we say that is integrable on [a, b].

Theorem 5.1 If is f continuous on [a, b], or if f has only a finite number of jump discontinuities,

then f is integrable on [a, b]; that is, the definite integral∫ ba f(x)dx exists.

Theorem 5.2 If f is integrable on [a, b], then∫ b

af(x)dx = lim

n→∞

n∑i=1

f(xi)4x

where 4x = b−an and xi = a+ i4x.

Theorem 5.3 (Properties of the Integral) Given that f(x) and g(x) are integrable on [a, b],[a, c], and [c, b]:

1.∫ ba cdx = c(b− a), where c is any constant.

2.∫ ba [f(x) + g(x)]dx =

∫ ba f(x)dx+

∫ ba g(x)dx.

3.∫ ba cf(x)dx = c

∫ ba f(x)dx, where c is any constant.

4.∫ ba f(x)dx =

∫ ca f(x)dx+

∫ bc f(x)dx.

5. If f(x) ≤ g(x) for all x ∈ [a, b], then∫ ba f(x)dx ≤

∫ ba g(x)dx.

Theorem 5.4 (Fundamental Theorem of Calculus I) If f is continuous on [a,b], then thefunction g defined by

g(x) =

∫ x

af(t)dt a ≤ x ≤ b

is continuous on [a,b] and differentiable on (a, b), and g′(x) = f(x).

Example 5.1 Find ddx

∫ x4

1 sin tdt.

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Theorem 5.5 (Fundamental Theorem of Calculus II) If f is continuous on [a, b], then∫ b

af(x)dx = F (b)− F (a) (5.2)

where F is any antiderivative of f , that is, a function such that F ′ = f .

Because of the relation given by the Fundamental Theorem between antiderivatives and inte-grals, the notation

∫f(x)dx is traditionally used for an antiderivative of and is called an indefinite

integral. Thus ∫f(x)dx = F (x) means F ′(x) = f(x). (5.3)

Theorem 5.6 (Substitution Rule) If u = g(x) is a differentiable function whose range is aninterval I and f is continuous on I, then∫

f(g(x))g′(x)dx =

∫f(u)du. (5.4)

Example 5.2 ∫xex

2dx =

∫1

2ex

2d(x2)

=1

2ex

2.

Theorem 5.7 (Integration by Parts) If u = u(x), v = v(x), and the differentials du = u′(x)dxand dv = v′(x)dx, then integration by parts states that∫

u(x)v′(x) dx = u(x)v(x)−∫u′(x)v(x) dx.

Example 5.3 ∫xexdx = xex −

∫exdx

= xex − ex.

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6 Homework

Question 1. Using the definition of derivative to find the first order derivative of the followingfunctions:

• f(x) = x3

• f(x) = 1x

• f(x) = sinx

Question 2. Differentiate the following functions:

• f(x) = x−23

• f(x) = x2+3x+2√x

• f(x) = x2ex

• f(x) = xx+ c

xwhere c ∈ R is a constant.

• V (x) = (3x+ 2)(x4 − 3x)

• f(x) = e√x

• y = eax2+bx+c where a, b, c ∈ R are constants.

• f(x) = ln(

x1+x

).

Question 3. Show that | sinx− cosx| ≤√

2 for all x.

Question 4. Find the absolute maximum value of the function

f(x) =1

1 + |x|+

1

1 + |x− 2|.

Question 5. Find (a)∫x2 ln(x)dx; (b)

∫ 10 x

3e4xdx.

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