MATH 2411 - Harrell
Making the grad
Lecture 9
Copyright 2013 by Evans M. Harrell II.
This week’s learning plan
  A review, and more, of the gradient, the chain
rules, the directional derivative, and tangent
spaces.
  Word problems using these concepts.
  Matching a function to better than first order with
Taylor’s theorem in many variables.
  Max and min problems in many variables.
In our last episode...
We used partials and the gradient to
estimate functions, find tangent planes,
and such mathematical problems. How
about some practical...
Word problems
 Suppose that price of your widgets is P
(t), you are selling at a rate of R(t) per
month, and your expenses are
F(t) + c(t) R(t)
 How rapidly is your profit changing, if P
= 2, R = 3000, F = 2500, c = 1, Pʹ′(t) = .1, Rʹ′(t)
= -20, c(t) = .05, and Fʹ′(t) = 5 ?
Word problems
 Suppose that price of your widgets is P
(t), you are selling at a rate of R(t) per
month, and your expenses are
F(t) + c(t) R(t)
 Profit = PR - F - c R
d Profit/dt = R Pʹ′ + (P - c) Rʹ′ - Fʹ′ - cʹ′R
Another word problem
 Suppose the temperature in a plate is T
(x,y) = 4 x2 – 2 x y – 4 y2, and that an
object moves in a circle,
r(t) = 2 cos(t) i + 2 sin(t) j .
At what rate is the temperature
changing?
Another word problem
 Suppose the temperature in a plate is T
(x,y) = 4 x2 – 2 x y – 4 y2, and that an
object moves in a circle,
r(t) = 2 cos(t) i + 2 sin(t) j .
At what rate is the temperature
changing?
Directional derivative
 Choose a direction, by taking a unit
vector u.
 Measure rise/run over a small distance
in the direction u.
The chain rule(s)
1. (d/dt) u(r(t)) = ∇u(r(t)) • rʹ′(t)
 Just like 1-D, but with a dot
 scalar t → vector r → scalar u.
The chain rule(s)
1 (d/dt) u(r(t)) = ∇u(r(t)) • rʹ′(t)
 In components:
du/dt = (∂u/∂x) dx/dt + (∂u/∂y) dy/dt
The chain rule(s)
 What about u(x,y), where x and y
depend on s and t?
 For example, a change of variables.
The chain rule(s)
 What about u(x,y), where x and y
depend on s and t?
(∂u/∂s) = (∂u/∂x)(∂x/∂s) + (∂u/∂y)(∂y/∂s)
 Remember: Add up all the possible
routes for connecting u to the
independent variable.
Some geometric problems you
can work out with the gradient
 Tangent plane to a surface in 3D. For
example, the ellipsoid x2+y2+z2/4 = 3 at
(1,1,2)
If there is only one variable, there is a
Taylor-made formula that is better than
the tangent plane (when f has continuous
derivatives up to order k+1 for x near x0).
What would this be like with many
variables?
It should involve all the partials, because, for
example at second order, f(x,y) = xy, Taylor
can’t be right with only the appearance of
fxx = 0 and fyy =0.
And math is beautiful, so there should be some
sort of pattern.
Notice there is double counting in
this “Hessian” term.