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Thread: Tangent line/plane Approximation- difference between dz and change in z.

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    Tangent line/plane Approximation- difference between dz and change in z.

    Hi,

    I just learned about approximations of f(x,y,z) for a surface is L(x,y,z) = f(a,b,c) + fx(a,b,c)(x-a).....(too lazy to typw out the rest).

    Anyway, there was a question that asked me to approximate the change in z for z= f(2.1,4.3,2.1) . I believe that the change in z is just the above equation, but what is 'dz' and when or how would i be asked about it ? I beliebe that dz is just the differentiation of z and it is just L-f(a,b,c) accordimg to my notes, but im not sure when i would be asked to find dz.
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    Re: Tangent line/plane Approximation- difference between dz and change in z.

    You define f(x, y, z) to be a function of three variables, x, y, and z but then talk about z= f(2.1,4.3,2.1). It looks like the problem has changed notation and is now using "z" to mean the value of f, not the independent variable.
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    Re: Tangent line/plane Approximation- difference between dz and change in z.

    I see, but I would like to know what the difference between the differential of z and the linear approximation deltaZ. My textbook compares both of them and says that differential of Z ~~ deltaZ, but i'm not sure what is the difference in use/calculation.
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    Re: Tangent line/plane Approximation- difference between dz and change in z.

    Given that z= f(x), that z is a function of f, then $\displaystyle \Delta z= f(x+\Delta x)- f(x)$. That is, $\displaystyle \Delta z$ is the change is z if x changes slightly. The derivative, $\displaystyle z'(x)= \frac{dz}{dx}= \lim_{\Delta x\to 0}\frac{\Delta z}{\Delta x}$, the limit of that ratio as the change in x goes to 0 (the "z'(x)" is just an alternate notation for the derivative). Notice that does NOT define "dz" separately from "dx". Strictly speaking the derivative is NOT a fraction. But it can often be treated as a fraction so it is often convenient to define $\displaystyle dz= \left(\frac{dz}{dx}\right)dx= z'(x)dx$. It simply makes no sense to have "dz" without an accompanying "dx".
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