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The condition in this result is satisfied for any polynomial, and for all the other differentiable functions that appear in this tutorial.
1.4 Exercises on multivariate calculus
1. Determine whether each of the following sets is open, closed, both open and closed, or neither open nor closed.
a. {(x, y): x2 + y2 < 1}
b. {x: x is an integer}
c. {(x, y): 0 < x < 1 and y = 0}.
2. For each of the following functions, find the partial derivatives f ¢1, f ¢2, and f ²12.
a. f (x1, x2) = 2x13 + x1x2
b. f (x1, x2) = (x1 + 2)/(x2 + 1)
3. For the production function f (K, L) = 9K1/3L2/3, find the marginal products of K and L (i.e. the partial derivatives of the function with respect to K and with respect to L).
1.4 Solutions to exercises on multivariate calculus
1.
a. Every point in this set is an interior point, so the set is open. The boundary of the set is {(x, y): x2 + y2 = 1}; no point in the boundary is thus a member of the set, so the set is not closed.
b. No point in the set is an interior point, so the set is not open. Every point in the set is a boundary point, so the set is closed.
c. Note that the set is a line segment in two-dimensional space. Thus no member of the set is an interior point, and hence the set is not open. (No disk centered at a point in the set is contained entirely within the set.) The set of boundary points of the set is {(x, y): 0 £ x £ 1 and y = 0}. The points (0, 0) and (1, 0) are not members of the set, so the set is not closed.
2.
a. f ¢1(x1, x2) = 6x12 + x2; f ¢2(x1, x2) = x1; f ²12(x1, x2) = 1.
b. f ¢1(x1, x2) = 1/(x2 + 1); f ¢2(x1, x2) = -(x1+2)/(x2 + 1)2; f ²12(x1, x2) = -1/(x2 + 1)2.
3. f K(K, L) = 3K-2/3L2/3; f L(K, L) = 6K1/3L-1/3.
1.6 Graphical representation of functions
Diagrams are very helpful in solving many mathematical problems involving functions. They are especially helpful in solving optimization problems, which occur throughout economic theory. Learning how to graphically represent functions will help enormously in understanding the material in this tutorial.
Functions of a single variable
A function of a single variable is most usefully represented by its graph.
Linear functions
A linear function of a single variable has the form
f (x) = ax + b.
(Such a function is sometimes called "affine" rather than linear, the term "linear" being reserved by some mathematicians for functions of the form f (x) = ax.) The graph of this function is a straight line with slope a; its value when x = 0 is b. Two examples are shown in the following figure.
Quadratic functions
A quadratic function of a single variable has the form
f (x) = ax2 + bx + c
with a ¹ 0. The graph of such a function takes one of the two general forms shown in the following figure, depending on the sign of a.
Functions of two variables
The graph of a function of two variables is a surface in three dimensions. This surface may be represented in a perspective drawing on a piece of paper, but for many functions the drawing (a) is difficult to execute and (b) hides some features of the function---only parts are visible. Computer software allows one to construct such drawings easily, from many different view
