How to solve these two questions of calculus?

1.

It is given that an infinitely many times differentiable function f (x ) has Taylor series 1 f(20 ) = g - act 2 1 18 72 2 2 + {terms of higher degree} (1) 36 valid in an interval containing zero. (a) Find f (0) = f (0) = f"(0) = fll'(0) = (b) The coefficients in the right hand side of (1) are given by the sequence an 1 9.2n, n 2 0. Find the radius of convergence R of the series. R = (c) Use the infinite series on the right hand side of (1) to compute f (1/2). If you find that the series diverges for x = 1/2, then type divergent in the box. f (1/2) = (d) Use the infinite series on the right hand side of (1) to compute f (-1/2). If you find that the series diverges for x = -1/2, then type divergent in the box. f (-1/2) =(4 marks) LEFT PICTURE RIGHT PICTURE Consider a differentiable function F(m, y) of two variables near the point (a, b, F(a, b)) in R3 . In the LEFT PICTURE is shown the cross-section of the graph 2 = F(m, y) with the plane a: = a and the RIGHT PICTURE shows the cross-section of the graph 2 = F(m,y) with the plane y = b. The tangent line to the curve of cross-section at (a, b, F(a,, b)) in the LEFT PICTURE has slope 2.9 , and the tangent line to the cuwe of cross-section at (a, b, F0}, 12)) in the RIGHT PICTURE has slope 2.7. I) Which equation best describes the curve of cross-section shown in red in the LEFI' PICTURE? ___=.(,) II) Which equation best describes the curve of cross-section shown in red in the RIGHT PICTURE? = F I) Find a vector that is parallel to the tangent line to the curve of cross-section at (a, b, F(a, b)) in the LEFT PICTURE. _Im II) Find a vector that is perpendicular to the tangent line to the curve of cross-section at (a, b, F(a, b)) in the RIGHT PICTURE. _n b c) Imogen wrote down an equation for the tangent plane to the surface 2 = F(m, y) at (a, b, F(a,, b)) , but they have forgotten what values they had found for the vector c: b) fl) a. y b - c = 0. z F(a, b) All Imogen can remember is that all the components of c were integers. =-I@ d) Use the linear approximation (i.e. the tangent plane) to estimate AF = F(a, + 0.5, b 0.2) F(a, b). ..-.E What might their vector have been