Practice FT3 Q4 - Looking for a detailed, worked solution with theoretical explanation

LEFT PICTURE RIGHT PICTURE Consider a differentiable function F(x, y) of two variables near the point (a, b, F(a, b)) in RS. In the LEFT PICTURE is shown the cross-section of the graph z = F(x, y) with the plane y = b and the RIGHT PICTURE shows the cross- section of the graph z = F(x, y) with the plane r = a. The tangent line to the curve of cross-section at (a, b, F(a, b)) in the LEFT PICTURE has slope -3.3, and the tangent line to the curve of cross-section at (a, b, F(a, b) ) in the RIGHT PICTURE has slope 2.4 a) i) Which equation best describes the curve of cross-section shown in red in the LEFT PICTURE? Z = F X b ii) Which equation best describes the curve of cross-section shown in red in the RIGHT PICTURE? Z O =F( a b) 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. 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. c) Imogen wrote down an equation for the tangent plane to the surface z = F(x, y) at (a, b, F(a, b)), but they have forgotten what values they had found for the vector c: I - a y - b . C = 0 z - F(a, b) All Imogen can remember is that all the components of c were integers. What might their vector have been? d) Use the linear approximation (i.e. the tangent plane) to estimate AF = F(a + 0.4, b -0.5) - F(a, b). AF ~