MA 5901 - Homework: Exam questions for MA 3160 - Thomas Grassl

Section 12.4 - Linear  functions

a) Find  the equation for the linear function  f(x, y)  that fits the following conditions:

(1) (Subscript[f, x]) (x, y) = 3
(2) f intersects the yz plane in z = 2  y + 3

b) Compute g(x + (Subscript[g, y]), y - (Subscript[g, x])) for all  functions of the type g(x) = c + mx + ny and express your result in terms of g(x). What does this  mean for the function f(x) in  problem (a)?

Solution:

a)

Using condition  (1) [2 points]

y = c + mx + ny
(Subscript[f, x]) (x, y) = 3
⟹ m = 2

Possible problem: Is m = (Subscript[f, x])?

Using condition  (2) [2 points]

f intersects yz plane in z = 2  y + 3
⟹ f (0, y) = 
      3 + 0  x + 2  y
⟹c = 3,  n = y

Possible problem: Does yz plane mean x = 0?

Combining  the results [1 point]

f (x, y) = 3 + 3  x + 2  y

Possible problem: Copying mistakes; what is m and what is n?

b)

g (x, y) = c + mx + ny
⟹g (x + (Subscript[g, y]), y - (Subscript[g, x])) = 
      g (x + n, 
            y - m)                                                                   [
          1 point]
    =c + m (x + n) + 
          n (y - m) = c + mx + mn + ny -  nm = 
          c + mx + 
            ny                  [
              1 point]
⟹g (x + n, y - m) = 
      g (x, y)                                                                                    [1 point]
As the function in (a) is a linear function,  
    this also holds for
the resulting function in (a) .                                                                             [2 points]

Possible problems: Some students will not have a solution to problem (a) and might therefore think, that they can't solve
                               this one.
                               What do (Subscript[g, x])and (Subscript[g, y]) mean for a function g(x, y) = c + mx + ny?
                                Is such a function a linear function?
                                Student does not get to the result that g(x + n, y - m) = g(x, y), but he recognizes that f is a linear function and 
                                therefore whatever result he got must also hold for f. Then full 2 points for the conclusion.

Section 14.4 - Gradients and directional derivatives in the plane

For the function f(x, y) = 2  (xy^2) + y
a) Find  the partial derivatives (Subscript[f, x ]) and (Subscript[f, y]). 
b) What is the maximum rate of change in (1,-1)?
c) Find  the directional derivative (Overscript[(Subscript[f, u]), ⇀])(-5, 1) for the function f with (Overscript[u, ⇀]) = 
        8  (Overscript[i, ⇀]) + 6  (Overscript[j, ⇀]).

Solution:

a)

(Subscript[f, x]) (x, y) = 
      2  (y^2)                             [
          1  points]
(Subscript[f, y]) (x, y) = 
      4  xy + 1                      [
          1  points]

Possible problems: Students might mix up (Subscript[f, x])and (Subscript[f, y]);
                               What am I doing with the y, when I'm interested in (Subscript[f, x])?

b)

maximum rate of change  = ||grad f||                               [1 point]
grad f (1, -1) = 
      4  (Overscript[i, ⇀]) - 
        3  (Overscript[j, ⇀])                                                            [
            2 point]
||grad f (1, 2)|| = (Sqrt[(4^2) + (3^2)]) = (Sqrt[25]) = 
          5                          [
            1 point]

Possible problems: How do I find the maximum rate of change?
                               What is the gradient?
                               What is the absolute value of the gradient?
                                Some students might have wrong results from part (a). But if they do b) right with their results, then full 4 points.

c)

||(Overscript[u,  ⇀]) || = (Sqrt[64 + 36]) = (Sqrt[100]) = 10 ⟹ 
                use (Overscript[v,  ⇀])  := (((Overscript[u,  ⇀]) )/(||(Overscript[u,  ⇀]) ||)) = (8/10) (Overscript[i,  ⇀])  - (6/10) (Overscript[j,  ⇀])  = (4/5) (Overscript[i,  ⇀])  - (3/5) (Overscript[j,  ⇀])       [
                      2 points]
(Overscript[(Subscript[f, v]), ⇀]) (-5, 
          1) = (Subscript[f, x]) (-5, 1) ((4/5)) + (Subscript[f, y]) (-5, 
            1) (-(3/5))                                                                        [1 point]
                     =2 ((1^2)) ((4/5)) + (4 (-5) (1) + 
                1) (-(3/5)) = ((8 + 57)/5) = (65/5) = 
            13                               [
              1 point]

Possible problems: What is the directional derivative?
                                Some students might have wrong results from part (a). But if everything is done correct with those results: full 4 
                                points.