Write-ups due Monday April 18.

Monday April 10

Solve, write up and turn in:

1.
Consider the function f(x,y) and let g(t) = f( x0+at, y0+bt )  (where a,b are numbers).
a.
Calculate dg/dt  and d^2 d/ dt^2.
b.
Write the Taylor polynomial of degree 2 of g at t=0 and the expression for the remainder.

2. The same problem for f(x,y,z) and
g(t) = f( x0+at, y0+bt, z0+ct ) .

Wednesday April 12
Solve:
Sec. 19.7:
1, 4

Solve, write up and turn in:
Sec. 19.7:
9a,c, 10, 19, 21
Sec. 19.10:
2, 3, 13, 14
1.
Find the linear approximation at (1,0) for the function
f(x,y)=the integral from 0 to x of  sin(t^3y^2)  dt
2.
Complete the squares to show that

fxx/2 x^2 +fxy x y +fyy/2 y^2 equals
A^2/(2fxx)  +D/(2 fxx)  y^2
where D is the discriminant and A=fxx x+fxy y.

Thursday April 13

Solve, write up and turn in:
Sec. 19.10:
11, 13

Sec. 19.8: 1, 2,  4b (but see Change below)  , 17, 19, 20

Friday April 14
Check out the
Exteme Value Theorem and examples.
Solve, write up and turn in:
1.
Consider the function      f(x,y,z)=x^2+y^2-xz        on the domain
x>0, y>0, z>0, x+y+z<1.
Sketch the domain and find the absolute maximum and minimum of the function.

2.  Consider the function f(x,y)=1+square root of (4x^2+y^2)   on the domain  x^2+y^2<1.
Sketch the graph of the function and name this surface.
Find the absolute minimum and maximum of this function on the stated domain.
Explain why absolute extrema necessarily exist for this function on this domain.

Change: problem 4b page 701 became a bonus problem worth extra 5 points; you can turn it in now,

or on April 24.