Math32a Winter 2013 Midterm1 Summary R Kozhan The material for the midterm includes from Rogawski s Multivariable Calculus 2nd ed Sections 13 1 13 5 Sections 12 1 Sections 14 1 14 3 Bring your ID card to the exam No calculators no books no notes no cheatsheets 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 basic concepts of the 2 dim and 3 dim coordinate system coordinate axes coordinate planes 4 quadrants in 2 d 8 octants in 3 d orthogonal projections onto coordinate planes and onto coordinate axes 2 Be able to distinguish between the right handed and left handed orientations of coordinate systems similarly between the left handed and right handed triples of vectors u v w 3 Be able to find distance between two points in R2 and R3 4 Be able to find the midpoint between two points in R2 and R3 5 Be able to draw graphs of basic equations inequalities in R2 and R3 in particular lines circles spheres cylindrical surfaces 6 Know and understand the concepts of a vector components of a vector initial and terminal point of a vector Be able to compute the components of a vector with initial point P1 and terminal point P2 7 Know the definition equivalent vectors 8 Be able to check if two vectors are parallel 9 Know the definition of the length magnitude of a vector 10 Know the Triangle Inequality for vectors 11 Know how to add subtract vectors geometrically triangle law and parallelogram law and algebraically when the components are known Same for multiplication of vectors by a scalar 12 Know that c u c u for any scalar c 13 Know the definition of a unit vector Be able to find unit vector in a given direction 14 Know basic properties of addition and scalar multiplication Thm 1 in 13 1 15 Know the definition of a linear combination of given two vectors v and w 1 16 17 18 19 20 21 22 23 24 25 26 Know the definition of the dot product of two vectors and its basic properties Know that angle between two vectors can be computed by cos u u vv Know that two vectors are perpendicular orthogonal if and only if their dot product is zero Be able to compute the projection of one vector onto another vector Be able to find the decomposition u u u with respect to another vector v note I will also reserve a right to use the notation proj v u for u with respect to v Be able to find the component of one vector along another vector Know how to compute 2 2 and 3 3 determinants Know the definition of the cross product of two vectors and its basic properties Know that the cross product u v is perpendicular to both u and v has length u v sin and that u v u v forms a right handed triple Be able to compute the area of a parallelogram and of a triangle with given vertices Know that two vectors are parallel if and only if their cross product is the zero vector Know the definition of the scalar triple product same as whatever the textbook calls vector triple product Know the formula for computing the scalar triple product in terms of determinant and its geometric interpretation volume of a parallelepiped up to a sign 27 Be able to write vector and parametric equations of a line through a given point parallel to a given vector 28 Be able to write vector and parametric equations of a line through two given points 29 Be able to write vector and parametric equations of a line segment 30 Be able to write an equation of a plane through a given point with a given normal vector 31 Be able to write an equation of a plane through given three points 32 Know what it means for three vectors to be coplanar and be able to check if they are 33 Given an equation of a line in any of the form be able to find its directional i e parallel vector Given an equation of a plane be able to find a vector perpendicular to it normal vector 34 Know how to find the angle between two lines and two planes 35 Be able to find the distance from a point to a line or a plane either via projections or via area volume of parallelogram parallelepiped 36 Be able to solve other geometrical problems such as finding intersection of two planes finding plane through two intersecting lines and others 37 38 39 40 41 2 dy d y dx2 of a parametric 2 dim curve Be able to find derivatives dx Be able to find tangent lines arc length area below the curveof a 2 dim curve Be able to parametrize common curves Be able to parametrize intersection of two given surfaces Know what a vector function is Know the correspondence between a vector function and a parametric curve 2 42 Know the definition of the limit of a vector function Know the definition of the derivative of a vector function 43 Know how to take limit derivative and integral of a vector function 44 Know the Fundamental Theorem of Calculus for vector functions 45 Know that r 0 t is a tangent vector to the curve Be able to write the equation of the tangent line to a given curve in 2 dim and in 3 dim 46 Know that if r t describes the position of a particle at time t then its velocity is r 0 t its speed is r 0 t its acceleration is r 00 t 47 Know the differentiation rules for vector functions p 739 and p 740 48 Know how to find the arc length of a parametric curve in R3 49 Know what the arc length function is and be able to reparametrize a curve with respect to the arc length 3