2.14 Differentials Recall: ( ) ( )' 'dy yf x y xdx x∆= = =∆ all are examples of ways to express derivative and that2 12 1y yyslopex x x−∆= =− ∆ Y2 Y1 ()()( )0limxf x x f xx x x∆ →+ ∆ −+ ∆ − X2 X1 Back to Differentials: Definition: Let ()y f x=represent a function that is diff’able on an open interval containing X. The differential of X (denoted by dx) is any non-zero real number. The differential of Y (denoted by dy) is ()'dy f x dx= ⋅ Y=f(x) ()(),x x f x x+ ∆ + ∆ ∆Y ()'f x x⋅ ∆ ∆x f(x) m= rise m(run) = rise run f’(x)(∆x) = rise X X+∆X *Note that ∆Y ≈f(x+∆x)-f(x) = IF ∆x is “small” ()()()'f x x f xf x x+ − ≈⋅ ∆∆ Or ()()()'f x x f xf x dx+ −⋅∆ ≈ Or ()()()'f x x f xf x dx∆ ≈ ++ Or ()()f x x f x dy+ ∆ ≈ +****Calculating Differentials: Function Derivative Differential 2y x= 2dyxdx= 2dy x dx= ⋅ sin 3y x= 3cos 3dyxdx= 3cos3dy x dx= ⋅ ( )1221y x= + ( )122211 221dyx xdxdy xdxx−= + ⋅=+ 21xdy dxx= ⋅+ **Example: If ∆x is small: Use differentials to approximate 16.5 ()()()'f x x f xf x dx+ ∆ ≈ + ()f x x= ( )121 1'22f x xx−= = ( ).516.5 16 11.562 16= + ≈⋅+ 1 18 24 4.0625≈ + ≈⋅ *Note: With a calculator you get 16.5
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