12.6-13.1 Flashcards
how to find the directional derivative at (a,b) in the direction of (c,d)
gradient evaluated at (a,b) dotted into the unit vector from (a,b) to (c,d)
what must be true of the vector you are taking the directional derivative in the direction of?
It must be a unit vector
when does the funciton have the maximum rate of increase on its directional derivative
when the unit vector u is in the direction of the gradient vector. its max increase value is the magnitude of the gradient
when does the function have the maximum rate of decrease on its directional derivative
when the unit vector u is in the NEGATIVE direction of the gradient vector. its max increase value is the NEGATIVE magnitude of the gradient
how to verify that the gradient vector at a point is orthogonal to the level curve at that point.
- find gradient at point,
- use slope to find a vector proportional to the curve at that point
- the cross product must be 0
how to find the equation of a line tangent to level curve at a point.
fx(x-`a)+fy(y-b)
2 ways a surface can be defined
explicitly z=f(x,y) implicitly f(z,y,z) = 0
how to find the equation of a tangent plane at a given point implicitly
- define it implicitly
2. fx(pt)(x-a)+fy(pt)(y-b)+fz(pt)(y-c)
find all the points on a plane that are horizontal(or fit the bill)
- gradient of the equation
2. set the gradient equal to the qualification vector, In this case <0,0,c>
how to find the equation of a tangent plane at a point when it is defined explicitly
z=fx(pt)(x-a)+fy(pt)(y-b)+f(a,b)
find the linear approximation at a
Lx=fx(pt)(x-a)+fy(pt)(y-b)+f(a,b)
differential
dz = fx(pt)dx+fy(pt)dy
what can the differential do?
approximate the change of things
make sure to study
how to find critical points on an equation
gradient set equal to 0
equation for second derivative test
D(pt) = fxx(pt)fyy(pt) - fxy^2
if D>0 and Fxx<0
local max at a,b
local max at a,b
if D>0 and Fxx<0
D(pt) =0
inconclusive
inconclusive
D(pt) =0
if D>0 and Fxx>0
local min
local min
if D<0 and Fxx>0
D<0
saddle point
saddle point
D<0
how to find the absolute maximums and minimums
- find all the critical points
- find the max and min values on the boundaries
- max value of f at the rest is max. min is relative to same
how to find the critical points on the boundaries
derive and set equal to 0;
legrange multipliers
grad f = lambda grad g
when setting up the legrange multiplieers what cant i forget
The constraint equation is the final equation,
and grad of constraint is also multiplied by lambda
how to do a double integral
integrate the inner part then the outer part. pay atention to the bounds and crap.
what is the average value of a function over a region
(1/area of r )double int(f(x,y) dA
how to find the normal vector of a surface at a point
gradient evaluated at the point
Consider the line perpendicular to the surface z=x2+y2 at the point where x=3 and y=−4. Find a vector parametric equation for this line in terms of the parameter t.
- find z
- find the gradient of the equation.
- parameterize by multiplying t by the grad at that point and then adding the initial value
how to find a linearization at a point
- gradient at the point
2. multiply each gradient by the (variable - initial term.
(1 pt) A car is driving northwest at v mph across a sloping plain whose height, in feet above sea level, at a point N miles north and E miles east of a city is given by
h(N,E)=2000+175N+25E.
(a) At what rate is the height above sea level changing with respect to distance in the direction the car is driving?
rate =
(150/sqrt(2))
feet/mile
(b) Express the rate of change of the height of the car with respect to time in terms of v.
rate =
150v/sqrt(2)
feet/hour
a. ‘
1. directional derivative to find the rate of the height change in that direction
b. multiply a by b
logically speaking how does the gradient relate to the level curves
the gradient points perpendicular to the level curves in the direction of the greatest increase
