Formulas - Calculus III Formulas y6.

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Calculus III Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . θ (cos θ, sin θ ) x y 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x y (1 , 0) π 4 ( 1 2 , 1 2 ) π 6 ( 3 2 , 1 2 ) π 3 ( 1 2 , 3 2 ) π 2 (0 , 1) π ( - 1 , 0) 3 π 2 (0 , - 1) 0 2 π Trigonometric Identities cos 2 θ + sin 2 θ = 1 1 + tan 2 θ = sec 2 θ cot 2 θ + 1 = csc 2 θ cos 2 θ = cos 2 θ - sin 2 θ = 2 cos 2 θ - 1 = 1 - 2 sin 2 θ sin 2 θ = 2 sin θ cos θ cos 2 θ = 1 + cos 2 θ 2 sin 2 θ = 1 - cos 2 θ 2 Differentiation ( fg ) = f g + fg f g = f g - fg g 2 d dx cos u = - sin u du dx d dx sin u = cos u du dx d dx tan u = sec 2 u du dx d dx csc u = - csc u cot u du dx d dx sec u = sec u tan u du dx d dx cot u = - csc 2 u du dx d dx ln | u | = 1 u du dx d dx e u = e u du dx d dx tan - 1 u = 1 1 + u 2 du dx Clarkson University Dr. Kevin Dempsey
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Integration by Parts u dv = uv - v du 13.1 Three-Dimensional Coordinate Systems ( x - h ) 2 + ( y - k ) 2 + ( z - ) 2 = r 2 Equation of a sphere with center C ( h, k, ) and radius r 13.3 The Dot Product a · b = | a | | b | cos θ cos θ = a · b | a | | b | 0 θ π is the angle between the vectors a and b a and b are orthogonal if and only if a · b = 0 13.4 The Cross Product | a × b | = | a | | b | sin θ Two nonzero vectors a and b are parallel if and only if a × b = 0 The length of the cross product a × b is equal to the area of the parallelogram determined by a and b The volume of the parallelepiped determined by the vectors a , b and c is the magnitude of their triple scalar product V = | a · ( b × c ) | 13.5 Equations of Lines and Planes Vector equation of a line r = r 0 + t v Vector equation of a plane n · r = n · r 0 13.6 Cylinders and Quadric Surfaces Ellipsoid x 2 a 2 + y 2 b 2 + z 2 c 2 = 1 Clarkson University Dr. Kevin Dempsey
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Elliptic Paraboloid z c = x 2
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