Lecture2 - ECON 301 LECTURE #2 Slopes A slope is the...

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1 ECON 301 – LECTURE #2 Slopes A slope is the measure of steepness of a line. That is, the bigger the slope, the steeper the line. Let’s consider the following linear equation of a straight line: Y = aX + b In this case the slope is determined by the coefficient “a” of the independent variable X. The vertical intercept is determined by the constant “b” and the horizontal intercept is determined by the ratio “-b/a”. Since we are investigating a straight line we know that the slope is constant and as such it does not matter which point on the line we evaluate the slope (we should always get the same answer). In the special case where b = 0, we have both the vertical intercept and horizontal intercept at the origin. X 0 b -b/a Slope = a
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2 Slopes of Curves A tangent is a straight line that touches a curve at a single point. The line is referred to as a tangent line and the point on the curve that is touching the line is called the point of tangency. At this point of tangency, both the line and the curve will have the same value of the independent variable. For example, if the straight line Y line = f (X) and the curve Y curve = g (X) are tangent to each other at the point of tangency (X*, Y*), then we have… Y* line = Y* curve = Y* f (X*) = g (X*) So, at each point on a curve, the slope of the curve at that point is defined as the slope of the tangent line at the point of tangency. Y X 0 Curve Tangent line X* Y*
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Why do we care so much about slopes and tangencies in economics? We care because they are very closely related to various marginal measures that we find interesting and useful in economic analysis. Curves / Functions Slopes / Tangents Utility function Marginal utility Production function Marginal product Total cost Marginal cost Total revenue Marginal revenue Total profit Marginal profit Indifference curve Marginal rate of Substitution Isoquant Marginal rate of Technical Substitution Differentiation Rules To calculate the slope of a curve we take the derivative of the function with respect to the independent variable. For example, Curves / Functions Slopes / Tangents Y = a ! Y / ! X = 0 (Constant) Y = aX + b ! Y / ! X = a (Linear) Y = aX 2 + bX + c ! Y / ! X = 2aX + b (Quadratic) Y = aX 3 + bX 2 + cX + d ! Y / ! X = 3aX 2 + 2bX + c (Cubic) Y = X n ! Y / ! X = nX n-1 (Exponential) In addition, we can calculate the slope of a combination of two curves, f (X) and g (X). Curves / Functions Slopes / Tangents Y = f (X) + g (X) ! Y / ! X = [ ! f / ! X] + [ ! g / ! X] (Addition) Y = f (X) - g (X) ! Y / ! X = [ ! f / ! X] - [ ! g / ! X] (Subtraction) Y = f (X) · g (X) ! Y / ! X = [ ! f / ! X] g (X) + [ ! g / ! X] f (X) (Multiplication) Y = f (X) / g (X) ! Y / ! X = [ ! f / ! X] g (X) - [ ! g / ! X] f (X) (Division) [ g (X)] 2 Let’s use each of these rules for the example of f (X) = 2X 3 and g (X) = 7X 2 . Y =
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Lecture2 - ECON 301 LECTURE #2 Slopes A slope is the...

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