CALCULUS 265 Honors
Fall Semester 2000
8:00 to 8:50 MTTF
Carver 0002

Instructor: Hentzel
Office: 432 Carver
Office Hours: 9:00-10:00 MTTF

E-mail  hentzel@iastate.edu
phone  294-8141
Website

http://www.math.iastate.edu/hentzel/honors.265 <##### NEW ADDRESS ####

Thursday, September  7:  Section 12.9

Review for tomorrow's test Friday, September 8.

The gradient.

1.  What is grad(f) = the gradient of the function f(x,y,z,w).

   ans:    (df/dx,df/dy,df/dz,df/dw).

                                    2 m s^2
2.  Find the gradient of   hp =   -------------- 
                                    550 t^3

3.  The gradient tells how to change the variables to make the function

increase the fastest.    

4.  new({x,y,z}) =  {x,y,z} - delta*grad(x,y,z)     (where delta is small)

will iterate till the point {x,y,z} is a minimum of the function.


5.  At the minimum of the function (or the maximum, or at a saddle) the

gradient is zero.  (means there is no direction to go to increase, or decrease

the function.

6.  The tangent plane to the function z = f(x,y) at (xo,yo,zo) is:

                    z - zo = A(x-xo) + b(y-yo)

where  A = dz/dx (xo,yo,zo) and B = dz/dy (xo,yo,zo).

7.  Find the tangent plane to  z = x^2 + 3xy  at (1,2,7).

8.  The implicitly defined function given by  f(x,y,z) = c  represents

a surface in three space.

9.  Graph  x^2 + y^2 = 36.   What is the function f in this implicitly

defined system.  What is the "surface" in this instance.

10. Since the gradient of f points in the direction of maximal increase of

the function and the surface is the points of no increase at all, the

gradient points away from the surface.  The gradient is normal to the

surface.



11.  For x^2 + y^2 = 100  graph the gradient at points (0,10), (6,8) and (8,6).

12.  Write the equation of the tangent plane to  x^3 + 3 x^2 y + y z^3 = 5 at

(1,1,1).

13.  Contrast the two ways to write the tangent plane to   z = x^2 + xy - 5

depending on whether the function is written as  

                   z = x^2 + xy - 5  
           or   
                   x^2 + xy -z =  5 

Which one is the correct tangent plane?

14.  Curves in space are expressed by giving two equations and asking

for the points of intersection.

15.  What is the curve of intersection of  3x + 5y + 2z = 8 and 5x + 4y + z = 16?

16.  Explain why

              New[(x,y,z)] =   (x,y,z) + delta* grad(f) x grad(g)

gives a succession of points on the intersection of the two curves f(x,y,z) = c

and g(x,y,z) = d. 

17.  For the horsepower equation

                      2 m s^2
              hp = ------------- 
                     550 t^3

m = 3200    +/-  16 lbs

s = 2000    +/-  10 feet

t = 20      +/-  0.5 seconds

Compute the horse power and estimate the error in horse power due to the

error in measurements.  Which error is measurement leads to the largest

error?

19.  Find the maximum of the function   f(x,y) = 3 x^2 + 2 x y 

on and within the rectangle  0<=x<=2 and 0<=y<=2.

20.  Find the maximum of the function f(x,y) = 3 x^2 + 2 x y

on and within the circle  0<= x^2 + y^2 <= 100.

21.  Find the directional derivative of z = x^3 + 3 x^2 y + y^3

in the direction (3,5).

22.  In what direction should one go to increase the value of the

function z = x^3 + 3 x^2 y + y^3 the fastest from (1,1,5).

23.  If fxx = 3 fyy = 5 and fxy = 4 at a point (xo,yo,zo) is the

point a max, min, or neither.  Explain how you got your answer and

what the surface looks like at (xo,yo,zo).