MATH 19520/51 Class 4

Transcription

1 MATH 19520/51 Class 4 Minh-Tam Trinh University of Chicago

2 1 Functions and independent ( nonbasic ) vs. dependent ( basic ) variables. 2 Cobb Douglas production function and its interpretation. 3 Graphing multivariable functions. 4 Level curves ( indifference curves ). 5 Limits in several variables. 6 Continuity in several variables.

3 Functions and Variables For Stewart, a function of n variables is a formula that expresses a quantity in terms of n other numbers: for example, (1) V (r, h) = 1 3 πr2 h or income = revenue expense

4 Functions and Variables For Stewart, a function of n variables is a formula that expresses a quantity in terms of n other numbers: for example, (1) V (r, h) = 1 3 πr2 h or income = revenue expense Above, V and income are called the dependent or basic or bound variables. r, h, revenue, expense are called the independent or nonbasic or unbound variables. 1/3 and π are constants.

5 If f is a function of n variables, then the domain of f is the set of points (x 1,..., x n ) R n where f (x 1,..., x n ) is well-defined.

6 If f is a function of n variables, then the domain of f is the set of points (x 1,..., x n ) R n where f (x 1,..., x n ) is well-defined. For Stewart, the range of f is the set of values that f produces as output. If the domain is D, then the range is (2) {f (x 1,..., x n ) R : (x 1,..., x n ) D}. Other people call this the image of f.

7 Example (Cobb Douglas Production Function) A useful function in economics: (3) P(L, K) = bl α K 1 α.

8 Example (Cobb Douglas Production Function) A useful function in economics: (3) P(L, K) = bl α K 1 α. Above, 1 The dependent variable P stands for the total production ($) of an economic system.

9 Example (Cobb Douglas Production Function) A useful function in economics: (3) P(L, K) = bl α K 1 α. Above, 1 The dependent variable P stands for the total production ($) of an economic system. 2 The independent variables L and K stand for total labor (person-hours) and invested capital ($), respectively.

10 Example (Cobb Douglas Production Function) A useful function in economics: (3) P(L, K) = bl α K 1 α. Above, 1 The dependent variable P stands for the total production ($) of an economic system. 2 The independent variables L and K stand for total labor (person-hours) and invested capital ($), respectively. 3 b and α are constants depending on empirical data.

11 Example (Cobb Douglas Production Function) A useful function in economics: (3) P(L, K) = bl α K 1 α. Above, 1 The dependent variable P stands for the total production ($) of an economic system. 2 The independent variables L and K stand for total labor (person-hours) and invested capital ($), respectively. 3 b and α are constants depending on empirical data. The domain of P(L, K) is {(L, K) R 2 : L, K 0} because labor and capital are nonnegative quantities.

12 Graphs of Multivariable Functions Example Find the domain and range of f (x, y) = 1 xy.

13 Graphs of Multivariable Functions Example Find the domain and range of f (x, y) = 1 xy. The formula is well-defined as long as xy 0, meaning we have both x 0 and y 0. So the domain is (4) {(x, y) R 2 : x 0 and y 0}.

14 Graphs of Multivariable Functions Example Find the domain and range of f (x, y) = 1 xy. The formula is well-defined as long as xy 0, meaning we have both x 0 and y 0. So the domain is (4) {(x, y) R 2 : x 0 and y 0}. This is the (x, y)-plane with the x- and y-axes removed.

15 Graphs of Multivariable Functions Example Find the domain and range of f (x, y) = 1 xy. The formula is well-defined as long as xy 0, meaning we have both x 0 and y 0. So the domain is (4) {(x, y) R 2 : x 0 and y 0}. This is the (x, y)-plane with the x- and y-axes removed. We never have f (x, y) = 0, but if a 0, then f (1/a, 1) = a. So the range is R \ {0}, the set of nonzero real numbers.

16 Example Find the domain and range of f (x, y) = log 1 + x 2 + y 2.

17 Example Find the domain and range of f (x, y) = log 1 + x 2 + y 2. The formula is well-defined for all (x, y), so the domain of f is all of R 2.

18 Example Find the domain and range of f (x, y) = log 1 + x 2 + y 2. The formula is well-defined for all (x, y), so the domain of f is all of R 2. If t = 1 + x 2 + y 2, then t can take any value in the interval [1, ) and only those values. So log 1 + x 2 + y 2 can take any value in the interval [0, ) and only those values. So the range of f is [0, ).

19 The graph of f (x, y) = log 1 + x 2 + y 2 has radial symmetry:

20 Level/Indifference Sets Example Where is f (x, y) = log 1 + x 2 + y 2 equal to 1?

21 Level/Indifference Sets Example Where is f (x, y) = log 1 + x 2 + y 2 equal to 1? This happens when 1 + x 2 + y 2 = e. The set of points where f (x, y) = 1 is (5) {(x, y) R 2 : x 2 + y 2 = e 2 1}, the circle of radius e 2 1 centered at the origin.

22 The level curves where log 1 + x 2 + y 2 = 1, 2,..., 6:

23 In general, if f is a function of n variables, then the level set or indifference set of f corresponding to a value a in its range is: (6) f 1 (a) = {(x 1,..., x n ) R n : f (x 1,..., x n ) = a}. Level sets are subsets of the domain of f. Intuitively, a function is indifferent to movement within a level set.

24 Em x ẋ 1 Em(x, ẋ)

25 1 If f is a function of two variables, then level sets usually look like curves. (E.g., the previous slide.)

26 1 If f is a function of two variables, then level sets usually look like curves. (E.g., the previous slide.) 2 If f is a function of three variables, then level sets usually look like surfaces.

27 1 If f is a function of two variables, then level sets usually look like curves. (E.g., the previous slide.) 2 If f is a function of three variables, then level sets usually look like surfaces. This is why we talk about level curves and level surfaces.

28 Limits Let f be a function of n variables. Suppose D R n is the domain of f and a = (a 1,..., a n ) is a point in or on the boundary of D.

29 Limits Let f be a function of n variables. Suppose D R n is the domain of f and a = (a 1,..., a n ) is a point in or on the boundary of D. The limit of f at a is a value L such that, for any margin-of-error ɛ > 0, we can find a radius δ > 0 small enough that (7) f ( x) is within distance ɛ of L whenever x a is within distance δ of a. In this case, we write L = lim x a f ( x).

30 Example Let f (x, y) = sin(x2 + y 2 ) x 2 + y 2.

31 Example Let f (x, y) = sin(x2 + y 2 ) x 2 + y 2. The function f is well-defined everywhere except (x, y) = (0, 0). It turns out (8) lim f (x, y) = 1. (x,y) (0,0)

32 Example Let f (x, y) = sin(x2 + y 2 ) x 2 + y 2. The function f is well-defined everywhere except (x, y) = (0, 0). It turns out (8) lim f (x, y) = 1. (x,y) (0,0) Using Taylor series, one can show that (9) 1 ɛ < f (x, y) < 1 + ɛ whenever (x, y) (0, 0) is within distance δ = 4 ɛ of (0, 0).

33 Warning! Just like in the single-variable case, limits need not exist.

34 Warning! Just like in the single-variable case, limits need not exist. Example (Stewart, 14.2, Example 2) xy Does the limit of f (x, y) = x 2 exist at (0, 0)? + y2

35 Warning! Just like in the single-variable case, limits need not exist. Example (Stewart, 14.2, Example 2) xy Does the limit of f (x, y) = x 2 exist at (0, 0)? + y2 Approach along the x-axis: f (t, 0) 0 as t 0. Approach along the y-axis: f (0, t) 0 as t 0. Approach along the line where x and y are equal: f (t, t) 1 2 as t 0. So lim (x,y) (0,0) f (x, y) does not exist.

36 Continuity We say that f is continuous at a if and only if the following hold: 1 f ( a) exists. 2 lim x a f ( x) exists. 3 f ( a) = lim x a f ( x).

37 Continuity We say that f is continuous at a if and only if the following hold: 1 f ( a) exists. 2 lim x a f ( x) exists. 3 f ( a) = lim x a f ( x). In words, a belongs to the domain of f, and the value of f does not jump as we approach a from any direction.

38 Example What value(s) of a make (10) f (x, y) = { e xy (x, y) (0, 0) a (x, y) = (0, 0) continuous?

39 Example What value(s) of a make (10) f (x, y) = { e xy (x, y) (0, 0) a (x, y) = (0, 0) continuous? We see that lim (x,y) (0,0) f (x, y) = 1. Therefore, f is continuous if a = 1, and discontinuous otherwise.