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# 10.24 - Hand back homework collect section notes Section...

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Hand back homework, collect section notes Section 2.8: Linear approximation and differentials Main idea? ..?.. Derivatives are good for finding approximate values of functions If f is a differentiable function in the vicinity of x = a , then: for x a , the (usually nonlinear) function y = f ( x ) is well-approximated by the linear function y = L ( x ) = f ( a )+ f ( a )( x a ), aka the tangent line to the graph of y = f ( x ) at the point ( a , f ( a )). We call L ( x ) the linearization of f at a . Equivalently, f ( a + h ) is well-approximated by f ( a )+ f ( a ) h for h approximately equal to 0. Note that geometrically this is just saying that the graph of the function is close to the tangent line, for points on the graph near the point of tangency. Example: For x 100, y = f ( x )=sqrt( x ) is close to sqrt(100)+ C ( x –100) = 10+ C ( x –100),

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where C = f (100) = (1/2)(100) –1/2 = 1/20. E.g., sqrt(103) is close to 10+3/20=10.15. Check this mentally: (10.15) 2 = 10 2 + 2(10)(.15) + (.15) 2 = 100 + 3 + (.15) 2 103. This is often expressed notationally using differentials.
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10.24 - Hand back homework collect section notes Section...

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