Math 141 - Week 5 Notes Monday, February 8, 1999
Topics for this week -
- Tangent planes and differentials
- More about partial derivatives -- the chain rule
- Directional derivatives, the gradient of a function, normal
lines
Examples -
- Find the differential of z = 3x^{2}y - y^{3}. Then find
the tangent plane to the graph of z at the point (3,2,46). Also find the equations
of the normal line at that point.
- If x = t^{2} and y = 3t^{3}, and z = x^{2} -
y^{3}, then calculate dz/dt.
- Find the gradient of the function z in the first example above. At (3,2), what is
the directional derivative of z in the direction of the vector <1,1> ? In what direction
does z increase the fastest at that point? How fast?
Maple - To plot curves in three dimensions, use "spacecurve"
(pretty much like parametric plotting in 2D). For graphs of functions of
two variables, use "plot3d", for implicitly defined functions use
"implicitplot3d" -- Note that "spacecurve" and "implicitplot3d" are in the
"plots" library.
Fifth Homework Assignment - due on FRIDAY, February
19.
- Reading: In Chapter 12, read sections 12.4, 12.5,
12.6.
- More reading: From the Lab Manual, the sections on 3d
plotting and implicit 3d plotting (pp 117-124). Also look at the solved
problems on pp 169-171.
- Chapter 12 problems: Make sure you can do all of the Core
problems for sections 12.4 12.5 and 12.6. But write up
only the
following to be handed in:
- Section 12.4, p. 793: #2,8 (plot with Maple),12,18,20,28,32,35
- Section 12.5, p. 800: # 6, 10, 20, 36, 42
- Section 12.6, p. 810: # 4, 6, 12, 18, 26, 38, 48 (Draw the ellipsoid, the given plane,
and the tangent planes with Maple)