MAT2615 Assignment 2 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED

MAT2615 Assignment 2 (DETAILED ANSWERS) 2026 - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED - DISTINCTION GUARANTEED Answers, guidelines, workings and references.. Consider the R2 − R function f defined by f (x, y) = 1 − x2 − y2. Let C be the contour curve of f through the point (1,−1), let L be the tangent to C at (x, y) = (1, 1) and let V be the tangent plane to f at (x, y) = (1, 1). (a) Find the equation of the curve C. (2) (b) Find a vector in R2 that is perpendicular to C at (x, y) = (1, 1). (2) (c) Find the Cartesian equation of the line L. (3) (d) Find a vector in R3 that is perpendicular to the graph of f at the point (x, y, z) = (1, 1, 3).(3) (e) Find the Cartesian equation of the plane V. (3) (f) Draw a sketch to visualize the graph of f , together with appropriate sections of the line L and the plane V. Also show the vectors that you obtained in (b) and (d) on your sketch. (3) Hints • Study Definitions 3.2.5 and 3.2.9. Note that the level of C is given by f (1, 1). • By a vector perpendicular to a curve at a given point, we mean a vector perpendicular to the tangent to the curve at that point. Use Theorem 7.9.1 to find a vector perpendicular to C at the point (1, 1). • Study Remark 2.12.2(1) and use Definition 2.12.1 to find the Cartesian equation of L. Or, equivalently, use Definition 7.9.6. (Note that, in the case n = 2, the formula in Definition 7.9.6 gives a Cartesian equation for a tangent to a contour curve.) • By a vector perpendicular to a surface at a given point, we mean a vector perpendicular to the tangent plane to the surface at that point. Define an R3 − R function g such that the graph of f is a contour surface of g, and then use Theorem 7.9.3 to find a vector perpendicular to V at the point (1, 1, 3). • Use Definition 2.12.1 or Definition 7.9.6 (with g in the place of f ) to find the equation of V, or use Definition 7.5.3. (Read Remark 7.5.4(2).) [16] 2. (Chapter 9) Consider the R2 − R function f defined by f (x, y) = sin x cos y. (a) Find the second order Taylor Polynomial of f about the point π 4 , π 4  . Leave your answer in the form of a polynomial in  x − π 4  and  y − π 4  . (This form is convenient for evaluating function values at points near π 4 , π 4  .) (6) (b) Use your answer to (a) to estimate the value of e0,1 ln 0, 9. Compare your answer with the approximation given by a pocket calculator. (2) [8] 2 Downloaded by Edge Tutor () lOMoARcPSD| MAT2615/AS2/0/2026 3. (Sections 11.1 - 11.3, 7.5 and 9.3) (a) State the Implicit Function Theorem for an equation in the three variables, x y and z.(2) (b) Use the Implicit Function Theorem to show that the equation xyz = cos (x + y + z) has a smooth unique local solution of the form z = g(x, y) about the point (0, 0, π2 ). Then find a linear approximation for g about (0, 0). Hints • Use the method of Example 11.2.6, but take into account that you are dealing with an equation in three variables here and that g in this case is a function of two variables. • Before you apply the Implicit Function Theorem you should show that all the necessary conditions are satisfied. • Study Remark 11.3.3(1) and Remark 9.3.6(2). (6) [8] 4. (Sections 1.3, 11.2 and 11.3) Consider the 2-dimensional vector field F defined by F(x, y) =