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This 1 page Class Notes was uploaded by Lennie White PhD on Monday October 5, 2015. The Class Notes belongs to MATH110A at City College of San Francisco taught by GaryLing in Fall. Since its upload, it has received 40 views. For similar materials see /class/219509/math110a-city-college-of-san-francisco in Mathematics (M) at City College of San Francisco.
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Date Created: 10/05/15
MATH 110A LINEAR APPROXIMATION amp DIFFERENTIATION continued De nition Let f be a function and let yfX XE Domf Then the increments AX and Ay are defined by the condition Ax e R such that X AX E Domf and by the equation Ay fx AX fX Note So if f is a differentiable function and yfx then f x DXfx Emm Ijm Al ugtgtlt u x Aan AX De nition Let f be a differentiable function and let y fX XE Domf Then the differential dy is de ned by dy f XAX Note So Ay fx AX fX z f x X AX X f XAX dy See the figure below A and in fact 11m Ay dy 11miiy f XAX 0 In other words dy is the best AXgtO AXgt Ax linear approximation for Ay K Note Suppose yfx and Xgt where f and g are both differentiable functions So y fgt Then using the Chain Rule we get dy f gt g tAt f x dx Y d But then assuming dx 0 we can write dif x And so we can interpret a x tangent line quotient of differentials as a derivative and Vice versa The notation dl as an x alternative to f X is called Leibniz notation Note The symbol d7 also called Leibniz notation is sometimes used to indicate x differentiation with respect to X ie difx DX fx f x x
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