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Calculus Cheat Sheet Part 3

# Calculus Cheat Sheet Part 3 - 30 Definition of Arc Length...

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30. Definition of Arc Length If the function given by ) ( x f y = represents a smooth curve on the interval [ ] b a , , then the arc length of f between a and b is given by [ ] dx b a x f s 2 ) ( 1 + = . 31. Work and Hooke’s Law 1. If an object is moved a distance D in the direction of an applied constant force F , then the work W done by the force is defined as W = FD . 2. If an object is moved along a straight line by a continuously varying force ) ( x F then the work W done by the force as the object is moved from to b x a x = = is given by W = b a dx x F ) ( . 3. Hooke’s law says that the amount of force F it takes to stretch or compress a spring x units from its natural length is proportional to x . That is, F = kx , where k is the spring constant measured in force units per unit length. 32. Improper Integral b a dx x f ) ( is an improper integral if 1. f becomes infinite at one or more points of the interval of integration, or 2. one or both of the limits of integration is infinite, or 3. both (1) and (2) hold. 33. Parametric Form of the Derivative If a smooth curve C is given by the parametric equations ) ( and ) ( t g y x f x = = , then the slope of the curve C at 0 , is ) , ( ÷ = dt dx dt dx dt dy dx dy y x . Note : The second derivative, dt dx dx dy dt d dx dy dx d dx y d ÷ = = 2 2 . 34. Arc Length in Parametric Form If a smooth curve C is given by ) ( and ) ( t g y t f x = = and these functions have continuous first derivatives with respect to t for b t a

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