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Calculus Review Sheets III. First Derivatives. (This section borrows from Baldani, et. al.) See lists in Appendix A-11 in our P131-132 text. A. Functions. A function, y = f(x), is a rule that specifies the relationship between two variables, where y is the dependent variable and x is the independent variable. The set of values from which the independent variable is drawn id called the domain of the function, while the set of values assigned to the dependent variable is called the range. The function y = f(x) specifies a unique value in the range of y for each value in the domain of x. In business applications the domain the range are usually real-values as opposed to imaginary numbers. For example, variables such as prices, quantities, etc. are non-negative. Thus, the domain and range may be limited to nonnegative real numbers. The functional relationship may also include parameters which may be designated as a, b, c, etc. or a1, a2, a3, etc. or (, (, (, (, etc. Among the types of functions that exist, the following are most common in business applications: y = a constant function. y = ax linear function. y = ax2 + bx + c quadratic function. y = a0 + a1x + a2x2 + a3x3 + ((( + anxn polynomial function of degree n. Y = aex exponential function. Y = alnx logarithmic function. The slope of a function at a point y0 = f(x0) is the slope of the line that is tangent to the function y = f(x) at the point (x0, y0). If (y and (x denote the changes in variables y and x respectfully, then the approximate value of the slope of a function at a point (x0, y0) is given by: This slope is approximate because a tangent line slope is defined at a single point (x0, y0), where x and y are fixed and do not change, where as the formula allows x and y to vary away from point (x0, y0). The degree of discrepancy between a tangent line slope and the slope formula will depend on the size of (x. As (x becomes smaller, the slope formula and the tangent line