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The following are some brief thoughts on the subject of automobile camshaft dynamics: The subject of this chapter can be applied to any internal combustion engine with an Otto or diesel cycle, two or four stroke, such as racing cars, street cars, motorcycles, and diesel engines The automotive camshaft directs the engine to breathe Manufacturing quality is critical to the best performance Proper alignment of the assembled parts must be achieved For the ultimate manufacturing level use the newest machine tools, the highest quality materials (cost), and the most skilled and talented hands-on artisans

Calculate the slope of the tangent line to the graph of y = f ( x) = x 3 3x at x = 2 Write the equation of the tangent line Draw a figure illustrating these ideas

For high-level demands improved control of heat treatment and foundry production is suggested Tolerances of fabrication should be 00001 to 00005 inch Lubricant and additive plus the lubrication system must be chosen properly with experience Frequent communication with the driver of the automobile is suggested

The fust fraction has limit zero and the second limit 1 The product is zero This problem illustrates a manipulative rule for limits lim A - B = lim A][ lim B ]

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[( 2 + h) 3 3( 2 + h) ] [( 2) 3 3( 2) ] h h 0 [( 8 + 12h 6h2 + h3 ) + ( 6 3h) ] + [2] h h 0 h3 6h2 + 9h h h 0

Hubbard, D, Camshaft Reference Handbook, Don Hubbard, Publisher, Ft Myers, Fla, 2000; dhubbard@peganetcom Dour, J, Megacycle, San Rapael, Calif Personal communication, 2002 Crane, H, formerly of Crane Corporation, Daytona Beach, California Personal communication, 2000

Appendix E CAD/CAM FOR A MEDICAL INSTRUMENT CAM MECHANISM 565 Appendix F OPTIMAL CONTROL THEORY DERIVATION FOR TWO-POINT BOUNDARY VALUE PROBLEM

= 9 We conclude that the slope of the tangent line to the graph of y = x 3 3x at x = 2 is 9 The tangent line passes through ( 2, f ( 2) ) = ( 2, 2) and has slope 9 Thus it has equation y ( 2) = 9( x ( 2) ) The graph of the function and the tangent line are exhibited in Figure 29

A1 ELLIPSE 545 A2 PARABOLA 545 A3 HYPERBOLA 546 A4 LOGARITHMIC SPIRAL 548 A5 INVOLUTE OF A CIRCLE 549

A rubber balloon is losing air steadily At time t minutes the balloon contains 75 10t2 + t cubic inches of air What is the rate of loss of air in the balloon at time t = 1

Continuous functions are defined mathematically, usually over specific intervals The requirements of a continuous function are: 1) it exists at every point in the defrned interval, and 2) the iimit exists at every point and is equal to the value of the function at that point Operationally, continuous functions are ones that can be drawn without lifting your pencil

An ellipse is a curve generated by a point moving so that the sum of the distances from the two xed points (F1 and F2) called foci is a constant The basic equation for the ellipse is X2 Y2 + =1 a2 b2 where X = value of the curve in one direction Y = value of the curve in other direction a = 1/2 major axis b = 1/2 minor axis We note that the foci of the ellipse F1 and F2 are a distance a from the ends of the minor axis Also the major axis equals 2a in length For construction of the ellipse the reader is referred to Sec 23 (Eq A1)

A parabola is a curve generated by a point moving so that its distance from a xed point F, called the focus, is always equal to its distance from a xed straight line called the directrix In Fig A2, the distance X, Y, and e are shown to any point P on the curve AB is the directrix Since from the de nition X = e, we have the basic equation Y 2 = CX

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