Use Taylor's formula for f(x,y) at the origin to find quadratic and cubic approximations of f near the origin. f(x,y) = e In (1 + y) The quadratic approximation is The cubic approximation is

Consider the quadratic function f(x) = αx2 + βx + γ. By Lagrangetheorem we know that for any a, b ∈ R, with a < b, there is a c ∈ (a, b)such thatf(b) − f(a) = f′(c)(b − a).(i) Find c in terms of a and b.(ii) Provide a geometric significance of the result obtained in part (i).

Prove if the following statements are true or false.If true, provide a calculation that justifies the answer.If false, provide a counterexample

5. Sketch the quadratic f(x) = (x-4)² + 1 by moving the 5 main points from the parent function, y = x² in the appropriate manner. 10-