Level Curves, Directional Derivatives
The curve of points (x,y) where f(x,y) is some constant value.
What is a level curve?
the rate of change of a multivariable function with respect to one specific variable, while holding all other variables constant
What is a partial derivative?
A plane that touches the surface at the point and is “parallel” to the surface at the point
What is a tangent plane?
A function of summation of products of derivatives.
What is the multivariable chain rule?
A point where the slopes in orthogonal directions are all zero, but which is not local extremum of the function.
What is a saddle point?
The instantaneous rate of change of the function in the direction v through x
What is the directional derivative?
fxy=fyx
What is clairaut's theorem?
approximates a multivariable function near a point using the tangent plane at that point, essentially treating the function as linear in a small region
What is a linear approximation?
Apply chain rule to the following functions: z=x^2y^3, x=scost, y=ssint
(2xy^3)(cost)+(3x^2y^2)(sint)
(2xy^3)(-ssint)+(3x^2y^2)(scost)
fxxfyy-(fxy)^2<0
What is the check for a saddle point?
For a function f(x,y), this vector points in the direction of greatest increase.
What is the gradient vector?
This notation represents the second partial derivative of f with respect to x, then y.
What is fxy?
Linear Approximation Formula
L(x, y) = f(a, b) + fₓ(a, b)(x - a) + fᵧ(a, b)(y - b)
This formula represents the multivariable chain rule for z=f(x,y).
What is
dtdz=fx(dtdx)+fy(dtdy)?
Lagrange multipliers are used to find extrema subject to this.
What is a constraint?
The directional derivative is maximized when the direction vector points in this direction.
What is the direction of the gradient?
A function whose Laplacian equals zero is called this.
What is a harmonic function?
The tangent plane to z=f(x,y) at a point uses the values of fx and this derivative.
What is fy?
When a function depends on three variables that each depend on time, the total derivative includes this many terms.
What are three terms?
This equation must be satisfied when applying Lagrange multipliers.
What is ∇f=λ∇g?
Find the directional derivative.
f(x,y,z)=x^2z+y^3z^2-xyz
v=(-1, 0, 3)
Duf(x,y,z)=-1/root10(2xz-yz)+3/root10(x^2-2y^3z^2-xy)
Given f(x,y)=x^3y+2xy^3, verify Laplace’s Theorem at all points.
What is fxy=fyx=3x^2+6y^2?
Linear approximation is most accurate when (x,y) is __________ to the point of tangency.
What is close?
Let z=x^2y+y^3, where x=cost and y=sint.
Find dz/dt when t=π/4
What is (3sqrt2)/2?
Find the maximum value of
f(x,y)=xy
subject to the constraint
x^2+y^2=8
What is 4?