Lecture
THE DELIGHTFUL DERIVATIVE

1. Functions

• Variable: If x is a variable, it can take on a range of values.
• Interval: Set of points between two numbers a, b, with b > a
• Closed interval: Both endpoints are included
• Open interval: Neither endpoint is included; -1 < x < 5
• Function: A rule by which a variable x is transformed into some unique (single-valued) number, y.
  Note: For each value of x, there exists only one corresponding y value.

1.1 Examples of functions (provide the graphs)

• Constant function: y = b (Horizontal line)
• Linear function: y = mx + b (Straight line)
• Quadratic function: y = x² - 3x + 2 (Parabola)
• Rational function (ratio of two polynomials): y = a/x (Rectangular hyperbola)

1.2 Domain and Range

• x: the argument of the function, or the independent variable
• y: the value of the function, or the dependent variable
• Domain: Set of all permissible x values
• Range: Set of all images (Image of an x value is the corresponding y value)

1.3 Example

2. Limit

• Left-side limit: Value of f(x) as x approaches x₀ from the left
• Right-side limit: Value of f(x) as x approaches x₀ from the right

The limit exists only if the left-side limit and right-side limit are equal and finite.

2.1 Example: f(x) = 2x + 1

• What are left-side and right-side limits as x approaches 1?
• The limit exists: lim f(x) = 3.

3. Continuity

The function f(x) is said to be continuous at x₀ if:

The limit of the function as x --> x₀ in fact equals the value of the function at x₀:
lim f(x) = f(x₀) as x --> x₀

Note 1: The function described in the above example (2.1) is continuous at x = 1.

Note 2: A function does not have to be "smooth" at a point to be continuous at that point.

4. Differentiability

4.1 Difference quotient

• Consider the function y = f(x)
• Now consider a change in x, say from x₀ to x₁: Δx = x₁ - x₀
• Obtain the corresponding change in the value of the function: Δy = f(x₁) - f(x₀).
• Obtain the difference quotient: Δy/Δx = [f(x₁) - f(x₀)] / (x₁ - x₀)

4.2 Derivative

• A function y = f(x) is differentiable at x = x₀ if and only if the limit of the difference quotient at x₀ exists.
• This limit is called the derivative of f(x) at x₀.
• Notation: dy/dx, f'(x), f₁(x)

4.3 The derivative and slope

• Consider the curve for the function y = f(x)
• Start at Point A with co-ordinates (x₀,y₀)
• Consider a small change in x. Now you are at point B on the curve (x₁,y₁).
• Join A and B by a straight line: Slope of AB = Δy/Δx = (y₁ - y₀) / (x₁ - x₀)
• As x₁ approaches x₀, note that the difference quotient, Δy/Δx, measures the slope of the tangent at point A.
• Thus, the derivative of f(x) at x₀ = Slope of curve at point A

4.4 Interpretation of the derivative

• Consider the function z = f(x). What does dz/dx represent?
• The effect of a "small" change in x on z
• If dz/dx = -5, it means that an infinitesimal increase in x will cause z to decrease by 5

4.5 Examples

(a) Consider the function y = f(x) = 3x² + 2

1. Obtain the difference quotient
2. Obtain the limit of the difference quotient as x --> 0.
3. Sketch the curve corresponding to the f(x). What is the slope of the curve at x = 0? At x = 1? At x = -2?

(b) Consider the function y = | x |.

1. Is the function continuous at x = 0?
2. Is the function differentiable at x = 0?

Note: A function may be continuous at a point, yet not be differentiable at that point.

5. Economics applications

5.1 Marginal cost (MC)

Consider a firm's cost function: C(Q) = Q² + 6, where C is total cost and Q is the amount of output.

1. What does marginal cost mean?
2. Sketch the cost function and the marginal cost function.
3. What does the shape of the MC curve indicate?

5.2 Marginal revenue (MR)

Consider a firm's demand function: P(Q) = a - bQ, where Q is quantity demanded by consumers and P is the price of a unit of the good. a and b are parameters.

1. Obtain the firm's revenue function R(Q). [Note: R(Q) = P(Q)Q.]
2. Obtain the marginal revenue function. What does it represent?
3. Sketch the demand curve and the MR curve.

5.3 Elasticity

• Definition of elasticity of demand
  • Roughly: [Percentage change in Q] / [Percentage change in P]

  • More precisely: e = |dQ/dP| (P/Q)

    Note: e is expressed as a positive number.

• Consider the demand function: Q = g - hP
  1. Obtain e as a function of g, h and P:
     e = P / [(g/h) - P].

  2. Identify the regions on the demand curve where:

     (i) e > 1, (ii) e = 1, (iii) e < 1.