Given f(3)=9 and f(5)=28, using the first two terms of the Taylor series, the approximate value of f'(3) is (Keep 4 decimal points).

Answer:The approximation by Taylor series of f'(3) is 9.5000.Step-by-step explanation:Taking the two first terms for Taylor series gives the following expression:[tex]f(x)=f(x_{o})+f'(x_{o})*(x-x_{o})+e[/tex]Where e represents the error of the approximation and in this case is of second order. For this particular example, it is no required to calculate.  Note that in the equation x is the independent variable, in contrast, xo is the value where all the derivatives of the function are calculated and must be defined. Observe that the value for the first derivative in x equal 3 is wanted to know, so xo will be equal 3.Now to find f'(3) will use the rest of the information that was given:f(5)=28 (x=5)Replacing the previous values in the Taylor expansion:[tex]f(5)=f(3)+f'(3)*(5-3)[/tex][tex]28=9+f'(3)*(5-3)[/tex]Solving the equation:[tex]f'(3)=(28-9)/(5-3)[/tex][tex]f'(3)=9.5[/tex][tex]f'(3)=9.5000[/tex]There was no rounding in the calculated value.