Math 119 - Week 2

Gradient vector =

A vector with all single order partial derivatives, acts as the derivative for multi-variable functions

Given function f :

∇f = [∂f/∂x, ∂f/∂y]

Multi variate chain rule equation =

∇f(r(t))*r'(t)

or

dz/dt = ∂z/∂x(dx/dt) + ∂z/∂y(dy/dt)

What does parametrizing curves do =

Allows curves to be split up into multiple single-variable functions, expressed as a path travelled over time instead of a static curve

Tangent plane approximation =

To approximate (x, y, z)

df = f_x(x_0, y_0)dx + f_y(x_0, y_0)dy

Where (x_0, y_0, z_0) is the centre of approx (easy value to calculate)

dx = (x - x0) and dy = (y - y0)

z = z_0 + df

How to calculate greatest percentage error =

Given V = πr^2h and r has max 2% error and h has max 0.5% error

Calculate dV/V and sub in dr/r and dh/h error values into the inequality for maximum

dV/V = 2dr/r + dh/h <= 0.045 or 4.5% error max

differential formula =

Let f(x,y) be the function and with the point (a, b)

df = f_x(a, b)dx + f_y(a, b)dy

what is relative change of X =

dX/X

how to prove limits EXIST/DNE =

DNE = show that the limit gives different values through different paths

EXISTS = algebra or squeeze theorem

3 step process for sketching parametric curves =

Find intercepts (x = 0 or y = 0)

Find points where curve is horizontal/vertical

• Take dy/dt / dx/dt

  • horizontal when numerator = 0

  • vertical when denominator = 0

Get test points

HAVE ARROWS TO LABEL DIRECTION

length of a parametric arc formula =

area under a parametric curve formula =

given ax + by + cz = n, what is the normal vector to the plane =

<a, b, c>