微积分CALCULUS-3.pptVIP

Section 2.5 Marginal Analysis The Second Derivative Section 2.4 The Chain Rule When f(x) = (x2+x)3 df/dx = d(x2+x)3/dx = d(x2+x)3/d(x2+x) d(x2+x)/dx = 3(x2+x)2 d(x2+x)/dx d(x2+x)/dx = dx2/dx + dx/dx = 2x+x When f(x) = [1+(x2+1)1/2]-1 df/dx = d[1+(x2+1)1/2]-1/dx = d[1+(x2+1)1/2]-1/d[1+(x2+1)1/2] ? d[1+(x2+1)1/2]/dx = -1[1+(x2+1)1/2]-2 ? d[1+(x2+1)1/2]/dx d[1+(x2+1)1/2]/dx = d1/dx+d[(x2+1)1/2]/dx = d[(x2+1)1/2]/dx d[(x2+1)1/2]/dx = d[(x2+1)1/2]/d(x2+1) ? = d(x2+1)/dx = (?)(x2+1)-1/2 d(x2+1)/dx d(x2+1)/dx = dx2/dx + d1/dx = 2x df/dx = -1[1+(x2+1)1/2]-2 ? (?)(x2+1)-1/2 ? 2x = (-x) [1+(x2+1)1/2]-2(x2+1)-1/2 The ratio between errors (output to input) Suppose x is the correct (or precise) input And x+?x is the incorrect (or approximate) input. Then ?x is called the error of the input. and ?y = f(x +?x) - f(x) is called the error of the output If ?x is small, then y’ ? ?y/ ?x. Or ?y = y’ ? ?x. Marginal Cost: If C(x) is the total cost of producing x units of a commodity. Then the marginal cost of producing x0 units is the derivative C’(x0) , which approximates the additional cost incurred when the level of production is increased by one unit, from x0 to x0+1. Marginal Revenue and Marginal Profit: Suppose R(x) is the revenue function [ generated when x units of a particular commodity are produced], and P(x) is the corresponding profit function. When x=x0 units are being produced, then The marginal revenue is , it approximates , the additional revenue generated by producing one more unit. The marginal profit is , it approximates , the additional profit obtained by producing one more unit Example 22 A manufacturer estimates that when x units of a particular commodity are produced, the total cost will be