Math for Econ II, Written Assignment 2 Solutions (28 points)
1. (4 pts) Find all unit vectors which are perpendicular to h2, 7i.
(by perpendicular, we mean that the vectors meet at a right angle)
Solution: We use the formula h2, 7i u = |h2, 7i|u| cos . If
Math for Econ II, Written Assignment 6 (28 points)
Due Friday, October 24
Jankowski, Fall 2014
Please write neat solutions for the problems below. Show all your work. If you only write the answer with no work,
you will not be given any credit.
Write your
ins5-2h22-28
In Problems ?, find the area of the regions between the curve
and the horizontal axis
5-2h22
5-2h23
5-2h24
5-2h25
5-2h26
5-2h27
5-2h28
5-2h29
2
f (x)
1
x
22. Under y = 6x3 2 for 5 x 10.
1 Assignment 8 (28 points)
for Econ II, Written
23. Unde
5.3
Worksheet 2
Multivariable optimization using Lagrange multipliers
1. Consider
he15-miscw21
maxi- 15. f (x,
y) = x3 +f (x,
y, y)x =
+ y2xy
1under the constraint 5x + 4y = 100. Does f have a minimum value subject to the
constraint?
Find
all
points
sati
1
Worksheet 1, Math for Economics II
Multivariable
review,
linearization,
tangent
planes,
gradients
A manufacturer sells two goods, one at a price of $3000 a unit
and the
other at a price
of $12,000
a unit.
A
quantity q1 of the 1.
firstAgood
and q2 of the
Continuity (7.8)
Trush, Math for Econ 1
September 22, 2016
Trush, Math for Econ 1
Continuity (7.8)
Warm-up
Consider the function f graphed below.
limx4 f (x) and f (4) both exist, but arent equal.
(limx4 f (x) = 1.5, f (4) = 3). We had to lift our pen off
The Chain Rule (6.8 MFE, 2.5 E-Book)
Math for Economics 1
October 4, 2016
Math for Economics 1
The Chain Rule (6.8 MFE, 2.5 E-Book)
Here it is
Proposition
If f and g are differentiable, then so is the function F = f g , and
F 0 (x) = f 0 (g (x) g 0 (x).
L
A variety of essential functions
(4.6-4.10 and 10.5 in MFE book, 1.2 in e-book)
September 7, 2016
A variety of essential functions(4.6-4.10 and 10.5 in MFE book, 1
Some basic terminology we will discuss:
Revenue, cost, profit
Compound interest
Present and
Exponential and Logarithmic Fcns (6.10, 6.11)
October 6, 2016
Exponential and Logarithmic Fcns (6.10, 6.11)
Exponential Functions
Recall that if a > 1, the exponential function f (x) = ax is an
increasing function with range (0, ):
Note its derivative is
Higher-Order Derivatives (6.9)
Math for Economics 1
October 4, 2016
Math for Economics 1
Higher-Order Derivatives (6.9)
2nd derivative
Differentiate f 0 (x) to get the 2nd derivative of f , written f 00 (x).
Example: If f (x) = x 3 , then f 0 (x) = 3x 2 ,
Partial Derivatives (11.2, 11.3, 11.8)
Jankowski, Math for Economics I
December 4, 2013
Jankowski, Math for Economics I
Partial Derivatives (11.2, 11.3, 11.8)
In the past weve xed one variable and varied the other a bit.
Now, we x a variable and dierentia
Limits (6.5)
Jankowski, Math for Econ 1
September 23, 2013
Jankowski, Math for Econ 1
Limits (6.5)
In order to understand the concept of a derivative (sections 6.2
through 6.4), we must understand limits.
Jankowski, Math for Econ 1
Limits (6.5)
What is a
Functions, and some applications
Jankowski, Math for Econ 1
September 30, 2013
Jankowski, Math for Econ 1
Functions, and some applications
This PDF covers a variety of topics from sections 4.1 through 4.5.
Supply and demand is addressed briey in the book
Implicit Dierentiation (7.1)
Jankowski, Calculus I
October 17, 2013
Jankowski, Calculus I
Implicit Dierentiation (7.1)
Motivation
Sometimes we arent given y as an explicit function of x.
x 2 + y 2 = 1,
y 3 + 4x 2 y = 10,
etc.
How do we nd y , if we cant e
Higher-Order Derivatives (6.9)
Jankowski, Math for Economics 1
October 17, 2013
Jankowski, Math for Economics 1
Higher-Order Derivatives (6.9)
2nd derivative
Dierentiate f (x) to get the 2nd derivative of f , written f (x).
Example: If f (x) = x 3 , then
Transforming graphs and functions; inverses
Jankowski, Math for Econ 1
September 24, 2013
Jankowski, Math for Econ 1
Transforming graphs and functions; inverses
Shifting graphs (5.1)
If we are given a function y = f (x), then to graph:
1
y = f (x) + c, sh