Visual proof Let F 1x2 = 1 x 02a2 - t2 dt. The figure shows that F 1x2 = area of sector OAB + area of triangle OBC. a. Use the figure to prove that F 1x2 = a2 sin-11x>a2 2 + x2a2 - x2 2 . b. Conclude that L 2a2 - x2 dx = a2 sin-11x>a2 2 + x2a2 - x2 2 + C. (Source: The College Mathematics Journal 34, 3, May 2003)
Calculus notes for week of 9/19/16 3.6 Derivatives as Rates of Change Velocity is measured as: V ave(t+∆t) or s(b) – s(a) ∆t b – a (Change in position over change in time.) S’’(t) = V’(t) = A(t) (From left to right: S=Position, V= Velocity, and A=Acceleration) Average and Marginal Cost Suppose C(x) gives the total cost to produce x units of a good cost. Sometimes, C(x) = FC + VC * x FC = Fixed cost which does not change with units produced. VC = Variable cost which is the cost to produce each unit. C(x) = Average cost. C’(x) = Marginal cost, which is approximately the extra cost to produce one more unit beyond x units. C’(x) = lim C(x+∆x) – C(x) ∆x>0 ∆x 3.7 Chain Rule How do we differentiate a composi
