Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 2: Math Review Functions A function is a rule that describes a relationship between numbers. » Ex. 𝑦 = 𝑥 2 or 𝑦 = 2𝑥 or 𝑦 = 𝑓(𝑥) or 𝑦 = 𝑓(𝑥1, 𝑥2) » The variable x is often called the independent variable. » The set of values from which x can be drawn is the domain. »

Intermediate Macroeconomics ECN 312 – Summer 2015 Lecture 2: Math Review Functions A function is a rule that describes a relationship between numbers. » Ex. 𝑦 = 𝑥 2 or 𝑦 = 2𝑥 or 𝑦 = 𝑓(𝑥) or 𝑦 = 𝑓(𝑥1, 𝑥2) » The variable x is often called the independent variable. » The set of values from which x can be drawn is the domain. » The variable y is often called the dependent variable. » The set of all corresponding y values is the range. Example: If 𝑓(𝑥) = 𝑥 3 − 4𝑥 + 2, then » 𝑓(1) = (1) 3 − 4(1) + 2 = 1 − 4 + 2 = −1 » 𝑓(−2) = (−2) 3 − 4(−2) + 2 = −8 + 8 + 2 = 2 » 𝑓(𝑎) = (𝑎) 3 − 4(𝑎) + 2 Graphs A graph of a function depicts the behavior of a function pictorially. 𝑦 = 2𝑥 𝑓(0) = 0 𝑓(2) = 4 𝑓(4) = 8 𝑓(6) = 12 𝑓(8) = 14 𝑓(10) = 20 𝑦 = 𝑥 2 𝑓(0) = 0 𝑓(2) = 4 𝑓(4) = 16 𝑓(6) = 36 𝑓(8) = 64 𝑓(10) = 100 Remember that shifts in a curve represents changes to the equation, not to our input value for x. 0 5 10 15 20 25 0 2 4 6 8 10 0 20 40 60 80 100 120 0 2 4 6 8 10 Properties of Functions A continuous function is one that can be drawn without lifting a pencil from the paper: there are no jumps in a continuous function. A smooth function is one that has no “kinks” or corners. » Example of a non-smooth function: 𝑦 = |𝑥| A monotonic function is one that always increases or always decreases. » A positive monotonic function always increases as x increases. » A negative monotonic function always decreases and x increases. Both 𝑦 = 2𝑥 is positive monotonic, but 𝑦 = 𝑥 2 is not, because it is decreasing when x is negative. Equations and Identities An equation asks when a function is equal to some particular number. » Ex. 2𝑥 = 8, 𝑥 2 = 9, 𝑓(𝑥) = 0. The solution to an equation is a value of x that satisfies the equation. » Ex. 4, 3 and -3, x* (we say that x* satisfies the equation𝑓(𝑥) = 0. An identity is a relationship between variables that holds for all values of the variable. » (𝑥 + 𝑦) 2 ≡ 𝑥 2 + 2𝑥𝑦 + 𝑦 2 » 2(𝑥 + 1) ≡ 2𝑥 + 2 Changes and Rates of Change The notation Δ𝑥 is read as “the change in x.” If x changes from x* to x**, then the change in x is just Δ𝑥 = 𝑥 ∗∗ − 𝑥 ∗ , which can also be written as 𝑥 ∗∗ = 𝑥 ∗ + Δ𝑥. Typically, when we write Δ𝑥, we mean a very small change in x, or a marginal change. A rate of change is the ratio of two changes. If y is a function of x, and is given by 𝑦 = 𝑓(𝑥), then the rate of change of y with respect to x is denoted by: Δ𝑦 Δ𝑥 = 𝑓(𝑥 + Δ𝑥) − 𝑓(𝑥) Δ𝑥 Example: 𝑓(𝑥) = 𝑎 + 𝑏𝑥 Δ𝑦 Δ𝑥 = 𝑎 + 𝑏(𝑥 + Δ𝑥) − 𝑎 − 𝑏𝑥 Δ𝑥 = 𝑏Δ𝑥 Δ𝑥 = 𝑏 Example: 𝑓(𝑥) = 𝑥 2 Δ𝑦 Δ𝑥 = (𝑥 + Δ𝑥) 2 − 𝑥 2 Δ𝑥 = 𝑥 2 + 2𝑥Δ𝑥 + (Δ𝑥) 2 − 𝑥 2 Δ𝑥 = 2𝑥 + Δ𝑥 If Δ𝑥 is a marginal change, then it is close to zero, and the rate of change of y with respect to x will be approximately 2x. Slopes and Intercepts The rate of change of a function can be interpreted graphically as the slope of the function. The slope of a linear equation is the “rise” over the “run”. In the case of 𝑦 = 2𝑥, a “run” of 1 (x increase by 1) leads to a “rise” of 2 (y increases by 2). This gives us the slope of 2. Note that the slope of this equation is independent of any additional terms in the equation which do not include x. That is, the slope of 𝑦 = 2𝑥 + 10 is also 2. For a non-linear equation, the rate of change depends of the value of the independent variable, but it will be equal to the slope of the tangent line at that particular point. The tangent line is one that touches the same point as our function at the given x-value, and shares the same rate of change at that point. The vertical intercept of a function is the value of y when x = 0. The horizontal intercept of a function is the value of x when y = 0. Often we will be interested in these intercepts, because they will represent the limits to what is possible in our model. For example, the vertical intercept of the demand curve gives the highest willingness to pay, and marks the maximum value an individual’s consumer surplus could take on. y = 2x 0 5 10 15 20 25 0 2 4 6 8 10 -40 -20 0 20 40 60 80 100 120 0 2 4 6 8 10 Derivatives The derivative of a function 𝑦 = 𝑓(𝑥) is defined to be: 𝑑𝑓(𝑥) 𝑑𝑥 = lim Δ𝑥 →0 𝑓(𝑥 + Δ𝑥) − 𝑓(𝑥) Δ𝑥 Differentiation rules: In the following formulas, u, v, and w are all functions of x, and c and m are constants. 𝑑 𝑑𝑥 (𝑐) = 0 𝑑 𝑑𝑥 (𝑥) = 1 𝑑 𝑑𝑥 (𝑢 + 𝑣 + ⋯ ) = 𝑑 𝑑𝑥 (𝑢) + 𝑑 𝑑𝑥 (𝑣) + ⋯ 𝑑 𝑑𝑥 (𝑐𝑢) = 𝑐 𝑑 𝑑𝑥 (𝑢) 𝑑 𝑑𝑥 (𝑢𝑣) = 𝑢 𝑑 𝑑𝑥 (𝑣) + 𝑣 𝑑 𝑑𝑥 (𝑢) 𝑑 𝑑𝑥 (𝑢𝑣𝑤) = 𝑢𝑣 𝑑 𝑑𝑥 (𝑤) + 𝑢𝑤 𝑑 𝑑𝑥 (𝑣) + 𝑣𝑤 𝑑 𝑑𝑥 (𝑢) 𝑑 𝑑𝑥 ( 𝑢 𝑣 ) = 𝑣 𝑑 𝑑𝑥 (𝑢) − 𝑢 𝑑 𝑑𝑥 (𝑣) 𝑣 2 𝑑 𝑑𝑥 (𝑥 𝑚) = 𝑚𝑥 𝑚−1 𝑑 𝑑𝑥 (𝑢 𝑚) = 𝑚𝑢 𝑚−1 𝑑 𝑑𝑥 (𝑢) 𝑑 𝑑𝑥 (ln 𝑥) = 1 𝑥 𝑑 𝑑𝑥 𝑚𝑥 = 𝑚𝑥 ln 𝑚 𝑑 𝑑𝑥 (𝑒 𝑥 ) = 𝑒 𝑥 Differentiation Examples 𝑓(𝑥) = 4 + 2𝑥 − 3𝑥 2 − 5𝑥 3 − 8𝑥 4 + 9𝑥 5 𝑑 𝑑𝑥 = 0 + 2 − 3(2𝑥) − 5(3𝑥 2 ) − 8(4𝑥 3 ) + 9(5𝑥 4 ) 𝑓(𝑥) = 3 − 2𝑥 3 + 2𝑥 𝑑 𝑑𝑥 = (3 + 2𝑥) 𝑑 𝑑𝑥 (3 − 2𝑥) − (3 − 2𝑥) 𝑑 𝑑𝑥 (3 + 2𝑥) (3 + 2𝑥) 2 = (3 + 2𝑥)(−2) − (3 − 2𝑥)(2) (3 + 2𝑥) 2 = −12 (3 + 2𝑥) 2 𝑓(𝑧) = 𝑧 2 + 3 𝑎𝑛𝑑 𝑔(𝑥) = 2𝑥 + 1 𝑦 = 𝑓(𝑔(𝑥)) = (2𝑥 + 1) 2 + 3 = 4𝑥 2 + 4𝑥 + 4 𝑑𝑦 𝑑𝑥 = 8𝑥 + 4 Chain Rule Given 𝑓(𝑧) and 𝑔(𝑥), for 𝑦 = 𝑓(𝑔(𝑥)): 𝑑𝑦 𝑑𝑥 = 𝑑𝑓(𝑔(𝑥)) 𝑑𝑔(𝑥) 𝑑𝑔(𝑥) 𝑑𝑥 So 𝑑 𝑑𝑥 = 2(2𝑥 + 1) 1 ∗ 2 = 4(2𝑥 + 1) = 8𝑥 + 4 Second Derivatives The second derivative of a function is the derivative of the derivative of that function. » The second derivative measures curvature. A negative second derivative has a slope which is decreasing (concave). A positive second derivative has an increasing slope (convex). First derivatives are often denoted as 𝑓 ′ (𝑥) while second derivatives are noted as 𝑓 ′′(𝑥). Partial Derivatives When there are more than one independent variables, we take the partial derivative with respect to a single independent variable. In other words, you treat the other independent variables as constants. Example: 𝑓(𝑥1, 𝑥2 ) = 𝑥1 0.3𝑥2 0.7 𝑑 𝑑𝑥1 = 0.3𝑥1 −0.7𝑥2 0.7 = 0.3 𝑥2 0.7 𝑥1 0.7 = 0.3 ( 𝑥2 𝑥1 ) 0.7 𝑑 𝑑𝑥2 = 0.7𝑥1 0.3𝑥2 −0.3 = 0.7 𝑥1 0.3 𝑥2 0.3 = 0.7 ( 𝑥1 𝑥2 ) 0.3 Optimization The maximum of a function is found under the following conditions: 𝑑𝑓(𝑥) 𝑑𝑥 = 0 𝑑 2𝑓(𝑥) 𝑑𝑥 2 ≤ 0 The minimum of a function is found under the following conditions: 𝑑𝑓(𝑥) 𝑑𝑥 = 0 𝑑 2𝑓(𝑥) 𝑑𝑥 2 ≥ 0 The condition 𝑑𝑓(𝑥) 𝑑𝑥 = 0 is known as the first-order condition. The condition 𝑑 2𝑓(𝑥) 𝑑𝑥 2 ≤≥ 0 is known as the second-order condition. Example: 𝑓(𝑥) = (10 − 𝑥) 2 𝑑𝑓(𝑥) 𝑑𝑥 = 2(10 − 𝑥)(−𝑥) = 2𝑥 − 20 First Order Condition 2𝑥 − 20 = 0 → 𝑥 = 10 Second Order Condition 𝑑 2𝑓(𝑥) 𝑑𝑥 2 = 𝑑(2𝑥 − 20) 𝑑𝑥 = 2 ≥ 0 → 𝑚𝑖𝑛𝑖𝑚𝑢𝑚 So the function 𝑓(𝑥) = (10 − 𝑥) 2 is minimized when x = 10.

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