Math32a R Kozhan Midterm2 Summary Midterm2 will be focused on the sections listed below and will not explicitly test the knowledge of the material included for Midterm1 However the student is assumed to know it and be able to use it when needed The material for Midterm2 includes Basics of the conic sections Section 10 5 without foci and directrices Section 12 6 Sections 14 1 14 5 Bring your ID card to the exam No calculators no books no notes no cheatsheets no cell phones no computers will be allowed on the exam Below is the summary list of the concepts methods theorems statements and formulas that students should know and understand 1 Know the standard form of conic sections ellipses parabolas and hyperbolas Be able to sketch them which includes finding their vertices and asymptotes for hyperbolas 2 Be able to classify and sketch the graph of a quadratic equation in two variables of the form Ax2 By 2 Cx Dy E 0 3 Know how to parametrize ellipses 4 Know what are the domain and range of a function of severable variables 5 Be able to find and sketch the natural domain of a function i e the maximal domain where the function is defined 6 Know what are the traces of a surface level curves of a function of two variables and level surface of a function of three variables Be able to find classify and sketch them no artistic skills required 7 Know what a cylindrical surface is 8 Know the standard form of quadric surfaces ellipsoid hyperboloid of one sheet hyperboloid of two sheets cone elliptic paraboloid hyperbolic paraboloid Be able to sketch them and find their traces 9 Be able to classify and sketch the graph of a quadratic equation in three variables of the form Ax2 By 2 Cz 2 Dx Ey F z G 0 1 10 Be able to find parametric equations of the axes of symmetry of hyperboloids cones and elliptic paraboloids 11 Know what happens to the graphs in R2 and R3 in the situations of the type described in problems 14 1 69 14 1 70 12 Know the definition of lim x y a b f x y L Just understand it 13 Know how to find limits of the polynomials and rational functions 14 Be able to prove the non existence of limits by exhibiting two paths along which f produces different limiting values 15 Know the definition of a function f x y to be continuous at a point a b 16 Know the definition of a function f x y to be continuous on a set D 17 Know that the composition function g f x y is continuous at a b if f x y is continuous at a b and g t is continuous at f a b 18 Be able to find the domain of continuity of functions in particular polynomials rational functions and composition of functions 19 Know the definition of partial derivatives of a function of several variable Be able to find partial derivatives 20 Know the geometric interpretation of partial derivatives 21 Know and be able to find the higher order partial derivatives of a function of several variables 22 Know the Clairaut s Theorem 23 Be able to find the tangent plane to a surface z f x y at a point x0 y0 24 Be able to find the linearization function L of a function f of several variables at a given point 25 Know the definition of a function f of several variables to be differentiable at a given point 26 Know that if partial derivatives fx x y fy x y exist in some disk around a b and are continuous at a b then f is differentiable at a b 27 Know that a mere existence of fx a b fy a b does not imply that f is differentiable at a b in fact even the existence of all directional derivatives is not sufficient to conclude that f is differentiable 28 Know the definition of the differential of a function of several variables at a given point 29 Be able to compute differentials and know that differential dz is an approximation of the increment z of the function z f x y 30 Be able to use the Chain Rule for functions of several variables dy 31 Be able to find dx at a given point when the dependence of y on x is given in the form F x y 0 z z 32 Be able to find x and y at a given point when the dependence of z on x and y is given in the form F x y z 0 2