Calculus Volume 3

# Key Terms

boundary point
a point $P0P0$ of $RR$ is a boundary point if every $δδ$ disk centered around $P0P0$ contains points both inside and outside $RR$
closed set
a set $SS$ that contains all its boundary points
connected set
an open set $SS$ that cannot be represented as the union of two or more disjoint, nonempty open subsets
constraint
an inequality or equation involving one or more variables that is used in an optimization problem; the constraint enforces a limit on the possible solutions for the problem
contour map
a plot of the various level curves of a given function $f(x,y)f(x,y)$
critical point of a function of two variables
the point $(x0,y0)(x0,y0)$ is called a critical point of $f(x,y)f(x,y)$ if one of the two following conditions holds:
1. $fx(x0,y0)=fy(x0,y0)=0fx(x0,y0)=fy(x0,y0)=0$
2. At least one of $fx(x0,y0)fx(x0,y0)$ and $fy(x0,y0)fy(x0,y0)$ do not exist
differentiable
a function $f(x,y)f(x,y)$ is differentiable at $(x0,y0)(x0,y0)$ if $f(x,y)f(x,y)$ can be expressed in the form $f(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)+E(x,y),f(x,y)=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)+E(x,y),$
where the error term $E(x,y)E(x,y)$ satisfies $lim(x,y)→(x0,y0)E(x,y)(x−x0)2+(y−y0)2=0lim(x,y)→(x0,y0)E(x,y)(x−x0)2+(y−y0)2=0$
directional derivative
the derivative of a function in the direction of a given unit vector
discriminant
the discriminant of the function $f(x,y)f(x,y)$ is given by the formula $D=fxx(x0,y0)fyy(x0,y0)−(fxy(x0,y0))2D=fxx(x0,y0)fyy(x0,y0)−(fxy(x0,y0))2$
function of two variables
a function $z=f(x,y)z=f(x,y)$ that maps each ordered pair $(x,y)(x,y)$ in a subset $DD$ of $ℝ2ℝ2$ to a unique real number $zz$
generalized chain rule
the chain rule extended to functions of more than one independent variable, in which each independent variable may depend on one or more other variables
the gradient of the function $f(x,y)f(x,y)$ is defined to be $∇f(x,y)=(∂f/∂x)i+(∂f/∂y)j,∇f(x,y)=(∂f/∂x)i+(∂f/∂y)j,$ which can be generalized to a function of any number of independent variables
graph of a function of two variables
a set of ordered triples $(x,y,z)(x,y,z)$ that satisfies the equation $z=f(x,y)z=f(x,y)$ plotted in three-dimensional Cartesian space
higher-order partial derivatives
second-order or higher partial derivatives, regardless of whether they are mixed partial derivatives
interior point
a point $P0P0$ of $RR$ is a boundary point if there is a $δδ$ disk centered around $P0P0$ contained completely in $RR$
intermediate variable
given a composition of functions (e.g., $f(x(t),y(t))),f(x(t),y(t))),$ the intermediate variables are the variables that are independent in the outer function but dependent on other variables as well; in the function $f(x(t),y(t)),f(x(t),y(t)),$ the variables $xandyxandy$ are examples of intermediate variables
Lagrange multiplier
the constant (or constants) used in the method of Lagrange multipliers; in the case of one constant, it is represented by the variable $λλ$
level curve of a function of two variables
the set of points satisfying the equation $f(x,y)=cf(x,y)=c$ for some real number $cc$ in the range of $ff$
level surface of a function of three variables
the set of points satisfying the equation $f(x,y,z)=cf(x,y,z)=c$ for some real number $cc$ in the range of $ff$
linear approximation
given a function $f(x,y)f(x,y)$ and a tangent plane to the function at a point $(x0,y0),(x0,y0),$ we can approximate $f(x,y)f(x,y)$ for points near $(x0,y0)(x0,y0)$ using the tangent plane formula
method of Lagrange multipliers
a method of solving an optimization problem subject to one or more constraints
mixed partial derivatives
second-order or higher partial derivatives, in which at least two of the differentiations are with respect to different variables
objective function
the function that is to be maximized or minimized in an optimization problem
open set
a set $SS$ that contains none of its boundary points
optimization problem
calculation of a maximum or minimum value of a function of several variables, often using Lagrange multipliers
partial derivative
a derivative of a function of more than one independent variable in which all the variables but one are held constant
partial differential equation
an equation that involves an unknown function of more than one independent variable and one or more of its partial derivatives
region
an open, connected, nonempty subset of $ℝ2ℝ2$
given the function $z=f(x,y),z=f(x,y),$ the point $(x0,y0,f(x0,y0))(x0,y0,f(x0,y0))$ is a saddle point if both $fx(x0,y0)=0fx(x0,y0)=0$ and $fy(x0,y0)=0,fy(x0,y0)=0,$ but $ff$ does not have a local extremum at $(x0,y0)(x0,y0)$
surface
the graph of a function of two variables, $z=f(x,y)z=f(x,y)$
tangent plane
given a function $f(x,y)f(x,y)$ that is differentiable at a point $(x0,y0),(x0,y0),$ the equation of the tangent plane to the surface $z=f(x,y)z=f(x,y)$ is given by $z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)z=f(x0,y0)+fx(x0,y0)(x−x0)+fy(x0,y0)(y−y0)$
total differential
the total differential of the function $f(x,y)f(x,y)$ at $(x0,y0)(x0,y0)$ is given by the formula $dz=fx(x0,y0)dx+fy(x0,y0)dydz=fx(x0,y0)dx+fy(x0,y0)dy$
tree diagram
illustrates and derives formulas for the generalized chain rule, in which each independent variable is accounted for
vertical trace
the set of ordered triples $(c,y,z)(c,y,z)$ that solves the equation $f(c,y)=zf(c,y)=z$ for a given constant $x=cx=c$ or the set of ordered triples $(x,d,z)(x,d,z)$ that solves the equation $f(x,d)=zf(x,d)=z$ for a given constant $y=dy=d$
$δδ$ ball
all points in $ℝ3ℝ3$ lying at a distance of less than $δδ$ from $(x0,y0,z0)(x0,y0,z0)$
$δδ$ disk
an open disk of radius $δδ$ centered at point $(a,b)(a,b)$