MATH 251 – LECTURE 11
JENS FORSGÅRD
http://www.math.tamu.edu/~jensf/
This week: 12.4–6
webAssign: 12.4–6, due 2/15 11:55 p.m.
Friday:
Kevin.
Next week: 12.7
webAssign: 12.7, opens 2/15 12 a.m.
Friday 2/19:
Midterm 1: Covering chapters 11 and 12.
Help Sessions:
M W 5.30–8 p.m. in BLOC 161
Office Hours:
BLOC 641C
M 12:30–2:30 p.m.
W 2–3 p.m.
or by appointment.
Directional derivatives
The surface z = f (x, y) can be viewed as a level curve of the function F (x, y, z) = z − f (x, y).
Exercise 1. If you are at the point (x0, y0, f (x0, y0)), in which direction should you move so that the function
F increases the fastest?
Exercise 2. Recall from previous lectures: what is the normal vector of the surface z = f (x, y) at the point
(x0, y0, f (x0, y0))?
The Gradient
The gradient of F (x, y, z) is always a normal vector to the level curves of the function F .
Exercise 3. Let f (x, y) = x2 + y 2. Sketch the level curve of f passing through the point (1, 1). What is
∇f (1, 1)?
Local maximum and minimum
Definition 4. A function f (x, y) is said to have a local maxima at a point (a, b) if f (a, b) ≥ f (x, y) for all
(x, y) close to (a, b).
Definition 5. A function f (x, y) is said to have a local minima at a point (a, b) if f (a, b) ≤ f (x, y) for all
(x, y) close to (a, b).
Theorem 6. If f (x, y) is differentiable at a local maximum (or minimum) (a, b), then ∇f (a, b) = h0, 0i.
Local maximum and minimum
Definition 7. A point (a, b) is said to be a critical point of f (x, y) if either ∇f (a, b) = h0, 0i, or one of the
first partial derivatives of f does not exists at (a, b).
Exercise 8. Find all critical points of the function f (x, y) = x2 + xy + 4y.
Local maximum and minimum
Exercise 9. Find all critical points of the function f (x, y) = ex
3 +4xy
.
Local maximum and minimum
Assuming that f is smooth at the critical point (a, b), then there are three types of critical points.
Local maxima
Local minima
Saddle points
Local maximum and minimum
Theorem 10 (Second derivative test). Assume that f is twice differentiable at a critical point (a, b). Let
00
00
fxx(a, b) fxy
(a, b) 00
00
00
D(a, b) = 00
= fxx
(a, b)fyy
(a, b) − (fxy
(a, b))2
00
fyx(a, b) fyy (a, b)
00
1) if D(a, b) > 0 and fxx
(a, b) < 0 then (a, b) is a local maximum.
00
2) if D(a, b) > 0 and fyy
(a, b) > 0 then (a, b) is a local minimum.
3) if D(a, b) < 0 then (a, b) is a saddle point.
Exercise 11. Find and classify the critical points of f (x, y) = x2 + y 2.
Local maximum and minimum
Exercise 12. Find and classify the critical points of f (x, y) = −x2 − y 2.
Exercise 13. Find and classify the critical points of f (x, y) = x2 − y 2.
Local maximum and minimum
Exercise 14. Find the point on the plane x − y + z = 7 that is closest to (1, 4, 6).