Math 221 Final Exam Objectives

For the final exam, a well-prepared student should be able to

state a definition for each relevant term (for example, partial derivative and double integral)
understand and draw relevant pictures for geometric definitions of vector, magnitude, addition of vectors, scalar multiplication
compute with component expressions for given vectors in a given basis
give a geometric definition of dot product
compute a dot product using component expressions for two given vectors
use a dot product to compute the angle between two given vectors or find the projection of a given vector in the direction of a second given vector
give a geometric definition of cross product
compute a cross product using component expressions for two given vectors
use the cross product to find a vector perpendicular to two given vectors or to compute the area of a given parallelogram
use algebraic properties and identities for vector addition, scalar multiplication, dot product, and cross product to simplify or rewrite a given vector expression
prove a given vector identity
understand the connections among the various forms for the equation of a plane, including the point-normal form
find the equation of a plane using given information (for example, the coordinates of three points)
determine the domain of a vector-output function and plot or describe the output curve for a simple vector-output function
given a linear vector-output function to parametrize a given line
parametrize a simple curve described geometrically
compute the limit of a vector-output function and analyze continuity of a vector-output function
compute the derivative of a vector-output function
state and use a geometric interpretation of the derivative of a vector-output function
understand and use the relations between position, velocity, and acceleration for an object moving on a line, in a plane, or in space
determine the domain of a function of two variables and plot or describe both level curves and the graph of a simple function of two variables
determine the domain of a function of three variables and plot or describe level surfaces of a simple function of three variables
use path limits to show that a given limit does not exist for a function of several variables
compute the partial derivatives of a function of several variables
read, with understanding, the various notations for partial derivatives
state and use an appropriate interpretation (as slope or rate of change) of the partial derivatives of a function of several variables
determine the equation of a tangent plane for a given function at a given point
determine and use the linear approximation for a given function
state a definition for each relevant term (e.g., gradient or local maximum)
use the appropriate chain rule to compute partial derivatives for a given composition of functions
compute the gradient of a given function
give a geometric or rate of change interpretation of the direction and magnitude of a given gradient vector
compute the directional derivative for a given function, input, and direction
give a geometric or rate of change interpretation of a directional derivative value
find and classify (as local minimizer, local maximizer, or neither) all critical points for a given function on a given domain
find the global minimum and global maximum for a given function on a given domain
understand the geometric basis for the method of Lagrange multipliers
use the method of Lagrange multipliers to solve a constrained optimization problem for a function of two or more variables
solve applied optimization problems
state and understand Fubini's Theorem
set up an iterated integral (in a chosen or specified coordinate system) equal to a double integral for a given function and given domain in the plane
set up an iterated integral (in a chosen or specified coordinate system) equal to a triple integral for a given function and given domain in space
evaluate a given iterated integral
set up a double integral that gives the area of a given region of the plane
set up a double integral or a triple integral that gives the volume of a given region of space
set up an integral that gives the total of a quantity given a density function for that quantity and a region
give a geometric argument for the conversion formulas and the area element for polar coordinates
give a geometric argument for the conversion formulas and the volume element for cylindrical coordinates
give a geometric argument for the conversion formulas and the volume element for spherical coordinates
plot a given planar vector field
compute the line integral of a given vector field and given curve
compute the surface integral of a given vector field and given surface
compute the divergence and curl of a given vector field
give a "fluid flow" interpretation of a given line integral, surface integral, divergence, or curl
determine if a given vector field is conservative
determine the signs of divergence and curl at a given point for a given plot of a planar vector field
find a potential function for a given conservative vector field
state and understand the hypotheses and conclusion of the Fundamental Theorem for Line Integrals
use the Fundamental Theorem for Line Integrals to evaluate the line integral for a given conservative vector field and given endpoints
state and understand the hypotheses and conclusion of Stokes' Theorem
state and understand the hypotheses and conclusion of the Divergence Theorem
use the Divergence Theorem to evaluate the surface integral for a given vector field and a given closed surface