MATH 2015 Study Guide - Midterm Guide: Directional Derivative
3 Jul 2018
School
Department
Course
Professor
The projection of vonto uis given by
projvu=u·v
v2v(1)
The equation of a plane πthrough points x0is
n·∆x= 0 for n=
x12 ×x13
∇f(x0)
nω×y12
(2)
The arc length is L=Rb
ar′(t)dt. The normals are
B=T=r′
r′×N=T′
T′(3)
The curvature of a function is
κ(t) = T′(t)
r′(t)=|r′(t)×r′′(t)|
[r′(t)]3(4)
A function is continuous given
lim
x,y→0,0f(x, y) = lim
r→0f(r, θ) = f(0,0) (5)
The directional derivative of a function fat a point
x0in direction of vis
Dv=
max(Dv) = n, n
∇f(x)·(v/v)
(6)
For a function f(x) where xi=xi(s, t), the derivative
df
dt =X∂f
∂xi
·dxi
dt (7)
A function fis differentiable at (a, b) if
lim
x,y→a,b fx(x, y) and lim
x,y→a,b fy(x, y) (8)
exist. The linear and quadratic approximations of a
function are
L(x) = f(x0) + n·∆x(9)
Q(x) = L(x) + 1
2(∆x)THx0(∆x) (10)
The derivative of a function f(x) is
dxi
dxj
=−∂f/∂xi
∂f/∂xj
(11)
1
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
The projection of v onto u is given by x0 in direction of v is projvu = u v v2 v. The equation of a plane through points x0 is (1) (2) (3) max(dv) = n, n. For a function f (x) where xi = xi(s, t), the derivative df dt. A function f is di erentiable at (a, b) if lim x,y a,b fx(x, y) and lim x,y a,b fy(x, y) (8) exist. The linear and quadratic approximations of a n x = 0 for n = x12 x13. The arc length is l = r b. B = (cid:18)t = r a r (t) dt. The curvature of a function is function are. L(x) = f (x0) + n x. 2 ( x)t hx0( x) lim x,y 0,0 f (x, y) = lim r 0 f (r, ) = f (0, 0) (5) The derivative of a function f (x) is.
