Calc 3 Exam 3

Front 1

Parametrization

Back 1

r(t) = <x(t), y(t), z(t)>

Front 2

Vector equation

Back 2

r(t) = a

r(t)=<t^2 +1, 3t+4, e^t-1>

a=<2,7,1>

1 vector equation = 3 regular equations.

Front 3

Speed

Back 3

(||r(t+h)-r(t)||)/h

Distance/time elapsed

Front 4

Vector equation of tangent line

Back 4

T(t)=r(t) + t*r'(t)

T(t)= a +tv

Front 5

Length of curve

Back 5

Integral from 0-t of ||r'(t)||dt

Front 6

Partial derivatives

Back 6

Fx and Fy of f(x,y)

Front 7

Tangent plane approx/linear approx

Back 7

z = L(x,y) = f(a,b) + Fx(a,b)(x-a) + Fy(a,b)(y-b)

z = D + A(x-a) + B(y-b)

(Aa + Bb -D) = Ax + By -Z

Normal = <A, B, -1>

Tan plane = <Fx, Fy, -1>

Front 8

Gradient vector

Back 8

Vf(a,b) = <Fx, Fy>

Front 9

Chain rule

Back 9

Vf * r'(t)

Front 10

Directional derivative

Back 10

Z = f(x,y) u = <h, k> unit vector

Duf(a,b) = Vf(a,b) * u

Front 11

Critical points

Back 11

Fx(a,b) = 0 or does not exist

Fy(a,b) = 0 or does not exist

Front 12

Local min and max

Back 12

Max : f(x,y) <= f(a,b); D>0, Fxx(a,b)<0

Min : f(x,y) >= f(a,b); D>0, Fxx(a,b)>0

If D<0 then f has a saddle point

Front 13

Hessian

Back 13

Fxx(x,y), Fxy(x,y)

Fyx(x,y), Fyy(x,y)