Final Notes about Gradients (Geometric Properties of the gradient)
 As we are moving along a curve in space,∇f is perpendicular to r̂’
Linear Approximations
 By using the tangent line of an earlier point, we can approximate a later point on a curve
 Maybe that isn’t a great method of approximation… so let’s use higher derivatives to help!
 This is Taylor Series!
 What’s cool, is we can also approximate a “tangent plane” in a similar way!
 Usually, a first order approximation is… good enough. More so would be more accurate, but a pain to write out.
The Gradient is related to linear approximation!
Extreme Values (Minimums and Maximums)
For a single variable function y=f(x)…

Extreme Values of Y at x =c If f(x)<f(c) for x near c, you have a local max If f(x)>f(c) for x near c, you have a local min

Extreme Values occur at 3 types of points
 Critical Points ⇒ df/dx = 0
 Singular Points ⇒ df/dx = undefined
 Boundaries ⇒ The ends
But what if we don’t have a graph?
 To find critical points:
 Take the first derivative of a function
 Set df/dx =0 and solve for X
 To determine minimum or maximum
 Take the second derivative of a function. This studies the rate of change of the rate of change
 If d^{2}f/dx^{2} < 0 ⇒ You have a local max
 If d^{2}f/dx^{2} > 0 ⇒ You have a local min
 If d^{2}f/dx^{2} = 0 ⇒ Your curve is flat, and you don’t have enough information!
 If you have a case where d^{2}f/dx^{2} < 0 at some point before the critical point, and where If d^{2}f/dx^{2} > 0 after the critical point, or vice versa, you have an inflection point
For a multivariable function y=f(x,y,z,…)…
Types of Extreme Values
 Critical Points ⇒ D_{u}f (The directional derivative) = 0 for all û (for all directions)
 Singular Points ⇒ D_{u}f (The directional derivative) = undefined
 Boundaries ⇒ The ends
Finding critical points: ∇f = 0, solve for points
$(∂x∂f ,∂y∂f ,∂z∂f )=(0,0,0)$Second Order Directional Derivative! (very bad)
Remember:

The Gradient nabla operator (∇) vectorizes scalars. That dot product multiplied by the nabla is a scalar, being vectorized. It doesn’t visually mean anything, but makes things

U_{x} and U_{y} are direction vectors

To determine minimum or maximum
 Take the second derivative of a function. This studies the rate of change of the rate of change
 If d^{2}_{u}f< 0 for all û (for all directions) ⇒ You have a local max
 If d^{2}_{u}f > 0 for all û (for all directions) ⇒ You have a local min
 If d^{2}_{u}f< 0 for all û (for all directions) = 0 ⇒ Your curve is flat, and you probably have a saddle point (think the dip in a horse saddle)
Restrictions

If you are given a restriction, such as x=2, the easiest way to solve the problem is to reduce the dimensionality.
 Just plug whatever your restriction is into the function, and solve for the other variable’s critical point.