A circular curve of radius R on a highway is banked at an angle so that a car can safely traverse the curve without skidding when there is no friction between the road and the tires. The loss of friction could occur, for example, if the road is covered with a film of water or ice. The rated speed vR of the curve is the maximum speed that a car can attain without skidding. Suppose a car of mass m is traversing the curve at the rated speed vR. Two forces are acting on the car: the vertical force, mt, due to the weight of the car, and a force F exerted by, and normal to, the road (see the figure). The vertical component of F balances the weight of the car, so that | F | cos mt. The horizontal component of F produces a centripetal force on the car so that, by Newtons Second Law and part (d) of 1, | F | sin mv 2 R R (a) Show that v 2 R Rt tan . (b) Find the rated speed of a circular curve with radius 400 ft that is banked at an angle of 128. (c) Suppose the design engineers want to keep the banking at 128, but wish to increase the rated speed by 50%. What should the radius of the curve be?
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