|
| |
 | There will be review sessions. Details to be announced. |
| There will be an early final on Tuesday, 6/13, from
12-3pm in BE 295. Sign up for it in class. |
| Midterm stats are posted. Follow the 22 link on the
left, then click on exam stats. |
| The seventh homework assignment is due on Tuesday,
6/6,
11pm. You can download it from
here.
The assignment covers problems from 15.1-15.4 and 15.6 |
| The last homework assignment is due on Tuesday,
6/13,
11pm. You can download it from
here.
The assignment covers problems from 15.7 and 15.8. |
| Graphs for Maximum/Minimum Problems
a) Hammock
b)#30 (from 14.6)
c) A surface with one critical
point, a relative max but no absolute max or min.
d)The surface z^2=xy+1.
e) A surface with only saddle
points.
f)) Just an absolute
minimum. |
| Graphs on differentiability:
a) Example of a
function with partials at (0,0) but that is not
differentiable at (0,0).
b) Second example of a function
with partials at (0,0) but that is not
differentiable at (0,0). It's crinkled.
c) Example of a
function that is differentiable, but it's partials are not continuous. |
| Graphs on functions of several variables
a)No limit at the
origin
b) Limit exists at
origin.
c) Hyperboloid with
tangent plane.
d)Two hills.
e)
Eggcarton.
f) A
wiggly surface. |
| Graphs of planes, cylinders and quadric surfaces
a) The circular cylinder
x^2+y^2=1
b) The parabolic cylinder
z=x^2
c) Ellipsoid
d) Elliptic Paraboloid
e) Hyperbolic
Paraboloid
f) Cone
g) Hyperboloid
of one sheet
h) Hyperboloid
of two sheets
i)
Moving from one sheet through a cone to two two sheets.
k) intersection of the two planes
x+4y-3z=1 and -3x+6y+7z=0 ; The picture is
here.
l)The parametric equation of the line (which is this
intersection) is
x= 46t+ 1/3, y=2t+1/6, z= 18t ; The picture is
here. |
|
Polar Coordinates/Polar Curves:
a)Flower
b)Henri's Butterfly, again
c) Oscar's Butterfly
d) Lemniscate
e) Archimedean Spiral.
Cylindrical Coordinates/Spherical Coordinates:
a)Sphere in
spherical coordinates.
b)Vase in spherical
coordinates, on table
c)Hyperboloid of
two sheets, cylindrical coordinates
d)Cone in
cylindrical coordinates.
e)Cylinder
with cylindrical coordinates
f) Ellipsoid
with cylindrical coordinates.
Guess What?
a)guess?
Vector functions and space curves 1:
a) Helix r(t)=<cos
t, sin t, t> in box.
b) Helix r(t)=<cos
t, sin t, t> with x,y,z coordinate system
c) Semicubical Parabola r(t)=<t^3,
t^2> in the box
d) Semicubical
Parabola r(t)=<t^3, t^2> with x,y,z axis
e) Twisted Cubic r(t)=<t,
t^2, t^3> in the box
f) Twisted Cubic r(t)=<t,
t^2, t^3> together with the x,y,z axis (sans labels though)
g) Cylinder y=x^2 intersecting the cylinder z=x^3 has the twisted cubic as its intersection.
Vector functions and space curves 2:
a) Twisted Cubic pointwise
b) C1, C 2, both
c) Twisted Cubic with
tangent line. |
| Links to the vector applets are here:
1) vector sum
2) dot or scalar product
3) vector or cross
product
4) What good are dot
products?
5) Parallelepiped
(its volume is computed by the scalar triple product) |
|
