\fDEFINITION Average and Marginal Cost The cost function C(x) gives the cost to produce the first x items in a manufacturing process. The average cost to produce x items is C(x) = C(x)/x. The marginal cost C' (x) is the approximate cost to produce one additional item after producing x items. Suppose the cost of producing a items is given by the function 301030- C(x) - -0.02x2 + 50x + 100, for O S x = 1000. Cost (dollars) Cix)- -0.02r' + 50x + 100 Determine the average and marginal cost for the first 900 items and interpret these values. 300 1003 Number of items The average cost of producing 900 items is per item. The marginal cost (the cost of producing the 901st item) isLet f(x) - 283 - 15r- + 24r. = 3 - 15- + 24x Tangent line For what values of x does the line tangent to the graph of f have a slope of 6? has slope & 10- SOLUTION f' (x) = 6r' - 30x + 24 = 6(x - 5 + 3) - 0. Using the quadratic formula, the roots are Tangent line 0- VO -4 0.697 and X 0+ vo 4.303. 2 2 has slope 6. Therefore, the slope of the curve at these points is 6 The value in the red box is The value in the blue box is The value in the green box is