Find the Taylor series expansion (that is, infinite-order approxima-tion) of the function f (x) = ex2 around the point x = 2, and use it toobtain a cubic approximation of this function. What is the Lagrangeform of the remainder term that reconciles the cubic approximationof the function based on its Taylor series expansion and the true valueof the function?

Given the function f (x) = e2x and f (xi) = fi, which of these is the correct 3rd order Taylor seriesexpansion for fi+1 = f (xi + ∆x)?A. fi+1 ≈ e2xi + 4∆xe2xi + 2∆x2e2xi + 43 ∆x3e2xiB. fi+1 ≈ e2xi + 2∆xe2xi + 2∆x2e2xiC. fi+1 ≈ e2xi + 2∆xe2xi + 2∆x2e2xi + 43 ∆x3e2xi

Does Taylor's Theorem with Remainder guarantee that the second Taylor polynomial at x=3 has an error less than 0.0001 in the estimate of 10e3.4 ?

Find the third Taylor polynomial about x = 1 for f(x) =1+2(X-1)-(X-1)^2+(X-1)^3

Does Taylor's Theorem with Remainder guarantee that the second Taylor polynomial of f(x)=2ex at x=1 has an error less than 0.01 in the estimate of 2e1.2 ?

Does Taylor's Theorem with Remainder guarantee that the second Taylor polynomial of f(x)=8ex at x=2 has an error less than 0.1 in the estimate of 8e2.3 ?