When \(k=1\), we have \(P_a,1(x)=f(a)+f'(a)x\), and so \[R_a,1(h)=f(a+h)-f(a)-f'(a)h.\] Our alternative definition of the derivative tells us that \(\displaystyle\lim_h\to 0\fracR_a,1(h)h = 0.\) Next, we will show that this extends to higher values of \(k\). Then we will generalize Taylor polynomials to give approximations of multivariable functions, provided their partial derivatives all exist and are continuous up to some order.
Taylor Polynomials And Approximations Homework
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