Calculus 3: Chapter 14

Min/Max

100

The discriminant

What is f_xx f_yy - f_xy² ?

200

It equals 0 at critical points in 3D

What is the gradient ?

300

Discriminant > 0 and f_xx < 0

What is a maximum ?

400

Discriminant > 0 and f_xx > 0

What is a minimum ?

500

Discriminant < 0

What is a saddle point ?

Gradient 1

100

Relationship between grad f and level curves in a contour plot

What is orthogonal?

200

This points in the direction of maximum rate of increase of f(x, y)

What is

grad f  ?

300

 grad f cdot vec u , where vec u is a unit vector

What is the directional derivative D_uf ?

400

Formula for D_uf that includes the angle between direction vector vec u and grad f 

What is 

|| grad f ||  cos theta ?

500

The relationship between D_uf(P) and the angle of inclination psi 

What is

tan psi = D_uf(P)  ?

Lagrange Multipliers

100

The basic Lagrange Multiplier equation

What is

grad f = lambda grad g ?

200

The relationship between  grad f and grad g when f is maximized

What is parallel ?

300

The relationship between grad f and the constraint curve g when f is maximized

What is orthogonal ?

400

When solving a problem, the next step after finding grad f and grad g 

What is: solve both equations for

lambda ?

500

When solving a problem, the next step after finding a relationship between x and y

What is: substitute the relationship between x and y into the constraint equation  ?

Partial Differentiation

100

 lim_(k->0) (f(a, b + k) - f(a, b))/k 

What is f_y ?

200

The slope of the tangent line to the vertical trace curve through the point (a, b, f(a, b)), with y = b

What is f_x ?

300

Clairaut's Theorem

What is:

(del^2f)/(del x del y) = (del^2f)/(del y del x)  ?

400

If  z = e^(-x^2-y^2) f_x =

What is

-2xe^(-x^2-y^2) ?

500

If f(u, v) = cos(u + v²), f_uuv =

What is: f_uuv = 2v sin(u + v²)  ?

Gradient 2

100

 grad(fg) = fgradg + ggrad f 

What is the Product Rule for Gradients ?

200

 grad(F(f(x,y,z))=F'(f(x,y,z)grad f 

What is the Chain Rule for Gradients ?

300

If P is a point on the surface F(x, y, z) = k, the relationship between grad F(P) and the surface

What is orthogonal ?

400

If f(x, y) = cos(x² + y), grad F =

What is <(-2x)(sin(x² + y), -sin(x² + y)> ?

500

 d/dtf(r(t))=gradf cdot r'(t) 

What is the Chain Rule for Paths ?