Machinery's Handbook, 31st Edition
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Calculus CALCULUS
Problems in engineering and other sciences are often modeled by functions other than simple first- and second-degree polynomials. The features of higher order polynomi - als, as well as trigonometric and other non-algebraic functions, are found using tech- niques of calculus. The essential operations of calculus are differentiation and antidif- ferentiation (or integration ). Brief explanations of these processes, as well as detailed formulas, are given here. Derivatives Between any two points of a linear function y = mx + b , the change in y with respect to x is the constant m . Most functions of interest, however, are nonlinear. For example, the ve- locity of a projectile is not modeled by a line, since objects are subject to gravity, among other outside forces, and therefore accelerate as they fall. The instantaneous rate of change of a function y = f ( x ) at a single point of a curve is a critical feature for any model. This quantity is called the derivative of f with respect to x (or other independent variable). A derivative is notated in one of three ways: y ′ , f ′ ( x ), or dy / dx . As an example, the instantaneous rate of change of displacement s of an object at any instant in time t is its derivative function, velocity ; that is, s ′ ( t ) or ds / dt = v ( t ). Graphically, the derivative function gives the slope of the line tangent to a point of the graph of f . That is, ´ = y m tan . Any constant function y = c has a slope of zero, since its graph is a horizontal line. Hence, y ′ = 0 at every point of a line. Fig. 1a shows a portion of a nonlinear function. The tangent line drawn to the point of the curve at x has a slope equal to the derivative ´ f x ( ). Any group of curves that represent a family of functions, such as the parabolas f 1 , f 2 , and f 3 in Fig. 1b, have the same derivative function, since the slope of the tangent lines at any given value of x is the same for each curve. The slope of the tangent line drawn to each of the curves at x = 0 is f ´(0) = 2(0) + 2 = 2.
(a) (b) Fig. 1. (a) ´ f x ( ) gives the slope of the tangent line to a curve f ( x ) for any x in the domain; (b) A family of parabola functions, all with the same derivative function, ´ = + f x x ( ) 2 2 . Any continuous, smooth function is differentiable (that is, it has a derivative) at each point on its domain. (Roughly speaking, a “continuous” function has no breaks, and a “smooth” function has no sharp corners.) Polynomial, trigonometric, exponential, and logarithmic functions are differentiable everywhere on their domains. To “differenti- ate” a function means to find (or “take”) its derivative. Derivative Formulas.—The formulas used most often in derivative applications are: Constant: If then y c y = ´ = , 0. Coefficient: If then y cx y c = ´ = , . Power: If then for any in the set of real numbe y x y nx n n n = ´ = − , , 1 rs.
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