math 19520/51 class 4 - university of...

MATH 19520/51 Class 4

Minh-Tam Trinh

University of Chicago

2017-10-02

Minh-Tam Trinh MATH 19520/51 Class 4

1 Functions and independent (“nonbasic”) vs. dependent(“basic”) variables.

2 Cobb–Douglas production function and its interpretation.3 Graphing multivariable functions.4 Level curves (“indi�erence curves”).5 Limits in several variables.6 Continuity in several variables.

Functions and Variables

For Stewart, a function of n variables is a formula that expresses aquantity in terms of n other numbers: for example,

V (r , h) =13πr2h or income = revenue – expense(1)

Above,

V and income are called the dependent or basic or boundvariables.

r , h, revenue, expense are called the independent or nonbasic orunbound variables.

1/3 and π are constants.

Functions and Variables

For Stewart, a function of n variables is a formula that expresses aquantity in terms of n other numbers: for example,

V (r , h) =13πr2h or income = revenue – expense(1)

Above,

V and income are called the dependent or basic or boundvariables.

r , h, revenue, expense are called the independent or nonbasic orunbound variables.

1/3 and π are constants.

If f is a function of n variables, then the domain of f is the set ofpoints (x1, . . . , xn) ∈ Rn where f (x1, . . . , xn) is well-defined.

For Stewart, the range of f is the set of values that f produces asoutput. If the domain is D, then the range is

{f (x1, . . . , xn) ∈ R : (x1, . . . , xn) ∈ D}.(2)

Other people call this the image of f .

If f is a function of n variables, then the domain of f is the set ofpoints (x1, . . . , xn) ∈ Rn where f (x1, . . . , xn) is well-defined.

For Stewart, the range of f is the set of values that f produces asoutput. If the domain is D, then the range is

{f (x1, . . . , xn) ∈ R : (x1, . . . , xn) ∈ D}.(2)

Other people call this the image of f .

Example (Cobb–Douglas Production Function)

A useful function in economics:

P(L,K) = bLαK1–α .(3)

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) ∈ R2 : L,K ≥ 0} because laborand capital are nonnegative quantities.

P(L,K) = bLαK1–α .(3)

an economic system.

2 The independent variables L and K stand for total labor(person-hours) and invested capital ($), respectively.

P(L,K) = bLαK1–α .(3)

Graphs of Multivariable Functions

Example

Find the domain and range of f (x, y) =1xy.

The formula is well-defined as long as xy , 0, meaning we haveboth x , 0 and y , 0. So the domain is

{(x, y) ∈ R2 : x , 0 and y , 0}.(4)

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 therange is R \ {0}, the set of nonzero real numbers.

Example

{(x, y) ∈ R2 : x , 0 and y , 0}.(4)

Example

{(x, y) ∈ R2 : x , 0 and y , 0}.(4)

Example

{(x, y) ∈ R2 : x , 0 and y , 0}.(4)

Example

Find the domain and range of f (x, y) = log√1 + x2 + y2.

The formula is well-defined for all (x, y), so the domain of f is allof R2.

If t =√1 + x2 + y2, then t can take any value in the interval [1,∞)

and only those values. So log√1 + x2 + y2 can take any value in the

interval [0,∞) and only those values.

So the range of f is [0,∞).

Example

The graph of f (x, y) = log√1 + x2 + y2 has radial symmetry:

https://academo.org/demos/3d-surface-plotter/

Level/Indi�erence Sets

Example

Where is f (x, y) = log√1 + x2 + y2 equal to 1?

This happens when√1 + x2 + y2 = e. The set of points where

f (x, y) = 1 is

{(x, y) ∈ R2 : x2 + y2 = e2 – 1},(5)

the circle of radius√e2 – 1 centered at the origin.

Level/Indi�erence Sets

Example

Where is f (x, y) = log√1 + x2 + y2 equal to 1?

This happens when√1 + x2 + y2 = e. The set of points where

f (x, y) = 1 is

{(x, y) ∈ R2 : x2 + y2 = e2 – 1},(5)

the circle of radius√e2 – 1 centered at the origin.

The level curves where log√1 + x2 + y2 = 1, 2, . . . , 6:

https://www.desmos.com/calculator

In general, if f is a function of n variables, then the level set orindi�erence set of f corresponding to a value a in its range is:

f –1(a) = {(x1, . . . , xn) ∈ Rn : f (x1, . . . , xn) = a}.(6)

Level sets are subsets of the domain of f .

Intuitively, a function is “indi�erent” to movement within a levelset.

−0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 −2

Em(x, x)

http://pgfplots.net/tikz/examples/contour-surface/

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

2 If f is a function of three variables, then level sets usually looklike surfaces.

This is why we talk about “level curves” and “level surfaces.”

Limits

Let f be a function of n variables. Suppose D ⊆ Rn is the domainof f and ®a = (a1, . . . , an) 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-errorε > 0, we can find a radius δ > 0 small enough that

f (®x) is within distance ε of L whenever®x , ®a is within distance δ of ®a.

In this case, we write L = lim®x→®a f (®x).

Limits

Let f be a function of n variables. Suppose D ⊆ Rn is the domainof f and ®a = (a1, . . . , an) 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-errorε > 0, we can find a radius δ > 0 small enough that

f (®x) is within distance ε of L whenever®x , ®a is within distance δ of ®a.

In this case, we write L = lim®x→®a f (®x).

Example

Let f (x, y) =sin(x2 + y2)x2 + y2

The function f is well-defined everywhere except (x, y) = (0, 0). Itturns out

lim(x,y)→(0,0)

f (x, y) = 1.(8)

Using Taylor series, one can show that

1 – ε < f (x, y) < 1 + ε whenever(x, y) , (0, 0) is within distance δ = 4

√ε of (0, 0).

Example

Let f (x, y) =sin(x2 + y2)x2 + y2

lim(x,y)→(0,0)

f (x, y) = 1.(8)

√ε of (0, 0).

Example

Let f (x, y) =sin(x2 + y2)x2 + y2

lim(x,y)→(0,0)

f (x, y) = 1.(8)

√ε of (0, 0).

Warning!

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

Example (Stewart, §14.2, Example 2)

Does the limit of f (x, y) =xy

x2 + y2exist at (0, 0)?

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 ast → 0.

So lim(x,y)→(0,0) f (x, y) does not exist.

Warning!

x2 + y2exist at (0, 0)?

2 ast → 0.

Warning!

x2 + y2exist at (0, 0)?

2 ast → 0.

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 doesnot jump as we approach ®a from any direction.

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 doesnot jump as we approach ®a from any direction.

Example

What value(s) of a make

f (x, y) ={exy (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 ifa = 1, and discontinuous otherwise.

Example

What value(s) of a make

f (x, y) ={exy (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 ifa = 1, and discontinuous otherwise.

math 19520/51 class 4 - university of...

Documents