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 whenever needed The material for Midterm2 includes from Rogawski s Multivariable Calculus 2nd ed Section 14 4 Basics of the conic sections what was covered in class or Section 12 5 without foci directrices eccentricity Section 13 6 Sections 15 1 15 4 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 Be able to compute curvature of a curve at a given point using any of the formulae the curvature formulae will be given if needed 2 Know that curvature is the reciprocal of the radius of a circle that best approximates the curve 1 0 for a line and 1 R for a circle of radius R the osculating circle In particular it s 3 Be able to find the center of curvature for a 2 dim or a 3 dim curve Be able to parametrize the osculating circle for a 2 dim curve t and the 4 Know how to find the unit tangent vector T t the principle normal vector N binormal vector B t of a curve at a given point Note binormal vector is defined to be T t N t 5 Know what the normal plane and osculating plane are and how to find their equations Note t and B t normal plane is the plane through the point x t y t z t parallel to N and osculating plane is the plane through the point x t y t z t parallel to T t and N t 6 Know the physical interpretation of r t r 0 t r 00 t r 0 t 1 7 Know the standard form of conic sections ellipses parabolas and hyperbolas Be able to sketch them which includes finding their vertices axes focal conjugate and asymptotes for hyperbolas 8 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 9 Know how to parametrize ellipses 10 Know what are the domain and range of a function of severable variables 11 Be able to find and sketch the natural domain of a function i e the maximal domain where the function is defined 12 Know what are the traces of a surface level curves of a function of two variables and level surfaces of a function of three variables Be able to find classify and sketch them no artistic skills required 13 Know what a cylindrical surface is 14 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 15 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 16 Be able to find parametric equations of the axes of symmetry of hyperboloids cones and elliptic paraboloids 17 Know what happens to the graphs in R2 and R3 in the situations of the type described in Problem 1 of HW6 18 Know the definition of lim x y a b f x y L Just understand it 19 Know the definition of a function f x y being continuous at a point a b 20 Know that sum difference product and quotient if denominator doesn t vanish of two continuous functions is continuous In particular know that any polynomial is continuous 21 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 22 Be able to find the domain of continuity of functions in particular of polynomials rational functions and composition of functions 23 Be able to prove the non existence of limits by exhibiting two paths along which f produces different limiting values 24 Be able to use continuity to find certain limits 25 Be able to use polar coordinates to find certain limits 26 Be able to use Squeeze Theorem to find certain limits 27 Know the definition of partial derivatives of a function of several variable Be able to find partial derivatives 28 Know the geometric interpretation of partial derivatives 2 29 30 31 32 33 34 35 36 37 Know and be able to find the higher order partial derivatives of a function of several variables Know the Clairaut s Theorem Be able to find the tangent plane to a surface z f x y at a point x0 y0 Be able to find the linearization function L of a function f of several variables at a given point Be able to use L to approximate f Know the definition of a function f of several variables being differentiable at a given point Know that if partial derivatives fx x y fy x y exist and are continuous in some disk around a b then f is differentiable at a b Know that a mere existence of fx a b fy a b does not imply that f is differentiable at a b Know the definition of the differential of a function of several variables at a given point Be able to compute differentials and know that differential dz is an approximation of the increment z of the function z f x y 3