131notesSection 3 9 Differentials and Linear Approximations Recall that the tangent line to a differentiable function f at a f a is a good approximation to f x if x is close enough to a y f a x a f a The tangent line is y f a f a x a If we write variable for the exact change in the variable then x a f x f a f x and Since f x is close to y the value on the tangent line at x we can write Definition y f a f a x f df f a x f a dx where x dx df is the differential of f near a Since f is a function and we can do this at any a we also write df f x dx 1 Example f x x Approximate 3 df 1 2 x 3 dx 3 3 9 f 9 We know f 8 2 so let a 8 Then dx 9 8 1 and f 8 Then 3 9 2 1 12 so df 1 12 1 and 3 9 2 12 1 12 Example Approximate tan x near 0 f x tan x a 0 f x sec 2 f 0 1 x tan0 0 and x x 0 x y x is the tangent line to f at 0 tan x x if x is near 0 Example A snowball melts and maintains its spherical shape Approximate the change in the volume if the radius decreases by 2 cm For r the radius in cm V 4 r 3 V r 4 r 2 dr 2 3 2 dV 4 r dr 8 r 2 cubiccm Example Approximate the change in the volume of a cube if the sides increase by 1 Let x be the side of the cube 1 of x 01x dx V x dV V 3 03 2 2 dV 3 x dx 3 x 01 x 03 x a change of 3 3
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