Chapter 5

Question 1

What does a production function indicate?

Answer 1

how much output is produced by input
input=independent variable=argument
output= dependent variable

Question 2

What factors determine production?

Answer 2

employees, machinery, land ownership, raw materials

Question 3

What is z in a function?

Answer 3

z= dependent variable
in an function, it is assigned to a number pair (x,y)
graphically it represents the height of a functions

Question 4

What is an intersection curve (intersections)?

Answer 4

A section through a 3D space for functions with two independent variables

Vertical intersection (x or y plane)
Horizontal intersection (z plane)
z=f(x,y)

Question 5

What do the variables of a production function represent?

Answer 5

X=employees
Y= machines
Z=output

Question 6

What is a contour line?

Answer 6

Happens when z= f (x,y0)

-Horizontal slice in the z plane, parallel to x, y-plane. It indicates the different combinations of x and y for a certain output

Question 7

What is a partial derivative?

Answer 7

When one independent variable is assumed to be a constant

Question 8

What is the derivative of ln (u)?

Answer 8

U’/U

Question 9

What is a direct second order partial derivative?

Answer 9

fxx and fyy

Question 10

What is a mixed second order partial derivative

Answer 10

fxy and fyx

Question 11

When is optimization used in math?

Answer 11

To determine extreme points (min/max) in multivariable functions

Question 12

What must be considered in optimization problems?

Answer 12

The constraint

Question 13

What are the sufficient conditions for a max in a multivariable function?

Answer 13

fxx (x0,y0) <0 ,fyy (x0,y0) <0 and fxx(x0,y0) *fyy (x0,y0) > fxy (x0,y0)^2

Question 14

What are the sufficient conditions for a min in multivariable functions?

Answer 14

fxx (x0,y0)>0 ,fyy (x0,y0) >0 and fxx(x0,y0) *fyy (x0,y0) > fxy (x0,y0)^2

Question 15

How do you find critical point in optimization?

Answer 15

fx (x0,y0)=0
fy (x0,y0)=0

Question 16

When do you have a saddle point in a multivariable function?

Answer 16

fx=fy=0
And
fxx*fyy<fxy^2

Question 17

When to use system of equations?

Answer 17

When you have 2 equations with 2 unknowns

Question 18

In business, what factors/variables do you maximize in optimization?

Answer 18

profit, production level, benefits, utility

Question 19

In business, what factor/variable do you minimize in optimization?

Answer 19

Costs

Question 20

What are restrictions called in business math?

Answer 20

Constraints

Question 21

What are the formulas to find max and min in optimization with constraints?

Answer 21

max f(x,y) subject to g(x,y)=c
min f(x,y) subject to g(x,y)=c

C is a constant and a real number

Question 22

What are independent variables/inputs also called?

Answer 22

function arguments

Question 23

How do you solve optimization problems and what is the goal in otimization ?

Answer 23

through functions with several independent variables

-find min cost for producers and max utility for consumers

Question 24

What is the dependent variable?

Answer 24

z

Question 25

What are the independent variables?

Answer 25

x, y

Question 26

What is the function value equal to?

Answer 26

range of values

Question 27

What is the function equation of 2 independent and 1 dependent variable?

Answer 27

z= f (x,y)

Question 28

What is system is used to represent a 3D space?

Answer 28

cartesian coordinate system

Question 29

What happens when z= f (x0,y)

Answer 29

x remains constant graphically, it is the vertical slice through x0, parallel to (y,z) plane -it shows how much output changes as machines (y) increase while employees (x) stay the same

Question 30

What happens when z= f (x,y0)

Answer 30

y remains constant graphically, it is the vertical slice through y0, parallel to (x,z) plane -it shows how much output changes as employees (x) increase while machines (y) stay the same

Question 31

What kind of numbers are x,y,z?

Answer 31

any real numbers

Question 32

What does x1 or index 1 indicate when there are several independent vairables

Answer 32

the (1) n number of variable

Question 33

Where can differential calculus be used in business math?

Answer 33

in functions with multiple indepent variables

Question 34

Why do we call it partial derivative?

Answer 34

Because we differentiate 1 variable of a function with 2 independent variables e.g.: f (x, y0) you differentiate z with respect to x y becomes a constant

Question 35

What is f(x,y)

Answer 35

The function value at points (x,y)

Question 36

What does it mean when the domain is R^2? What does the exponent mean?

Answer 36

All real numbers may be used as numerical values for 2 independent variables

Question 37

What is the f ‘ (x) and f ‘ (y) of e^x^2y

Answer 37

fx = 2xye^x^2y fy = x^2e^x^2y

Question 38

What does f ‘ (x) mean?

Answer 38

Shows how much z (output changes) when you move x by 1 unit and y is constant

Question 39

What does f ‘ (y) mean?

Answer 39

Shows how much z (output changes) when you move y by 1 unit and x is constant

Question 40

What does the partial derivative of one constant tell you?

Answer 40

The instant rate of change in one direction (either x or y)

Question 41

What is another name for mixed second order partial derivative?

Answer 41

Cross partial derivative

Question 42

What does the partial differential dfx tell you?

Answer 42

The small change in the function value. When only x changes in a small amount and y stays constant

Question 43

What does the partial differential dfy tell you?

Answer 43

The small change in the function value. When only y changes in a small amount and x stays constant

Question 44

What is total differential

Answer 44

When both x and y change. It is the sum of the partial differentials

Question 45

What are the additional rules when differentiating functions with 2 independent variables? 4 n

Answer 45

-1st & 2nd order partial derivatives can be interpreted graphically - a total derivative can be formed by chain rule -functions can be implicitly differentiated along the contour line -functions can be checked for convexity and concavity

Question 46

What are some optimization problems in business with constraints?

Answer 46

Minimize cost at a production level Maximize profit at a total cost Maximize benefits at a budget

Question 47

What is a langrage function?

Answer 47

A function that helps link the objective function and the constraint, so this function can be used to solve an optimization problem

Question 48

What is lamda λ also called?

Answer 48

Langrage multiplier