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Math 2433
Section 26013
MW 1-2:30pm GAR 205
Bekki George
bekki@math.uh.edu
639 PGH
Office Hours:
11:00 - 11:45am MWF or by appointment
Class webpage:
http://www.math.uh.edu/~bekki/Math2433.html
Quiz due dates and sections covered are now posted on class
webpage!
Popper07
1. Find the gradient of
f (x, y, z) =
3x
+ ln z
y
15.2 Gradients and Directional Derivatives
Properties of gradients:
Directional Derivatives:
f 'u gives the directional derivative of f in the direction u. In other words,
f u′ gives the rate of change of f in the direction of u.
f 'u = ∇f (x) i u
Example:
1. Find the directional derivative at the point P in the direction indicated.
f (x, y) = x 2 + 3y 2 at P(1,1) in the direction of i − j
2
2
2.Find the directional derivative for f (x, y) = x + 3y at
⎛ 3⎞
R ⎜ 2, ⎟
⎝ 2⎠ .
⎛
1⎞
Q ⎜ −1, ⎟
2⎠
⎝
towards
Note that the directional derivative in a direction u is the component of the
gradient vector in that direction.
Important:
Example:
Find a unit vector in the direction in which f increases most rapidly at P and
give the rate of change of f in that direction; find a unit vector in the
direction in which f decreases most rapidly at P and give the rate of change
of f in that direction.
f ( x, y) = y 2e2 x at P(0,1)
Popper07
2. What is the direction in which the function
f (x, y) = yx 2 −
x
y2
increases most
f (x, y) = yx 2 −
x
y2
decreases
rapidly at the point (-2, 1)?
3. What is the direction in which the function
most rapidly at the point (-2, 1)?
15.3 - The Mean Value Theorem; Chain Rules
What was the MVT for functions of one variable?
3
Example: Let f (x, y) = x − xy and let a = (0,1) and b = (1,3). Find a point c
on the line segment ab for which the mean value theorem (for several
variables) is satisfied.
A nonempty open set U (in the plane or in three-space) is said to be
connected if any two points of U can be joined by a polygonal path that lies
entirely in U.
Thm – Let U be an open connected set and let f be a differentiable function
on U. If ∇ f (x) = 0 for all x in U, then f is constant on U.
Thm - Let U be an open connected set and let f and g be differentiable
functions on U.
If ∇ f (x) = ∇ g(x) for all x in U, then f and g differ by a constant on U.
Examples:
1. Find
d
⎡ f (r(t)) ⎤⎦
dt ⎣
given f (x, y) = 6x + y, r(t) = t i + 7t j
2. Find the rate of change of f with respect to t along the given curve.
f (x, y) = x2y, r(t) = e t i + e−t j
3. Find the rate of change of f with respect to t along the given curve.
Other chain rules:
If
Then
Example:
4. u = x − 3xy + 2 y
2
2
x(t) = cost
du
y(t) = sint . Find dt
And if
Then
Example:
5. u = x − xy + z
2
2
du
x = scost y = sin(t − s) z = t sin s . Find dt
Popper07
4. Suppose that
g(t) = f (r(t)) .
How would you find
g ′(t) ?
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