DEL MATH REFRESHER: MODULE 1 - Duke University

DEL MATH

REFRESHER: MODULE

1Kristofor Voss (kristofor.voss@duke.edu)

Summer 2011

Goal: Provide a quick refresher of mathematical concepts useful in DEL-MEM quantitative courses.

The refresher was requested by former students.

Topics suggested by current instructors based on their courses. Application of the methods will occur in your coursework.

Initial familiarity with these topics will vary among students.

Please ask questions at the Q&A session at Duke.

Be patient with yourself and with me!

Module 1

Introductions

Order of Operations

Graphing

Lines

Areas

Exponents and Logarithms

Intros: What do I study?

(1) How can we employ statistical

models to predict the effect of

mountaintop removal mining and

valley-fill (MTM-VF) on stream

biological communities in the Central

Appalachians?

(2) Do biological communities exhibit

a threshold response to mining? If so,

at what level and how do we

appropriately characterize the

community threshold?

(3) What is the landscape effect of

multiple mining areas on biological

communities?

(4) How do headwaters, isolated as a

result of mining practices, compare to

non-isolated headwaters with regard

to biological composition?

(5) How does mining impact the

biological/functional trait composition

of the community? What is the

implication for ecosystem function?

Algebra Refresher Part I

Order of Operations

Parentheses

Exponents & Roots

Multiplication &

Division

Addition &

Subtraction

PEMDAS Practice

x-y Coordinate Plane

x-axis horizontal, abscissa,

independent variable

right = +, left = -

y-axis vertical, ordinate,

dependent variable

up = +, down = -

ordered pair (point): (x,y) origin: (0,0)

mostly in quadrant I

Distance Between Points

Horizontal distance:

Vertical distance:

Straight line distance:

What are the three distances between the 2 points plotted to the left?

Linear Equation

Linear Equation:

What is the equation for the line at right?

What is the y value when x = 4.2? [Interpolation]

Changing Intercepts

Slope Constant

•Changing the

intercept of a line

while keeping the

slope constant

produces a new line

that is parallel to the

original.

•Changing intercepts

“shifts” lines on the

graph.

Changing Slopes

Intercept Constant

•Changing the slope of a

line while keeping the

intercept constant

produces a new line with

a different steepness.

•Changing slopes

“rotates” lines on the

graph through the y-

intercept.

•Lines with

•positive slopes rise

•negative slopes fall

Horizontal & Vertical Lines

•Horizontal lines

(green) possess

zero slope (m = 0)

and a constant y-

value.

• y = constant

•Vertical lines

(magenta) do not

have a slope and

possess a constant

x value.

• x = constant

Finding the equation of a line

(1)

•STEP 1: Find slope

•STEP 2: Substitute

the x and y

coordinates and

slope into

y = mx + b and

solve for b.

Finding the equation of a line

(2)

What is the equation

of the line formed by

the two blue points?

Finding the equation of a line

(3)

ANSWER

Finding the intersection point from

two non-parallel lines

•The intersection point

occurs where the x and

y coordinates of both

lines are the same.

•The intersection point is

known as the solution

to the two linear

equations.

•Several methods are

possible.

Finding areas: rectangles

Finding areas: triangles

Finding areas: trapezoids

Exponential Functions and

Logarithmic Functions

An exponential

function is a function

in which the variable

is in the exponent:

A logarithmic

function is the

inverse of an

exponential function:

Exponential and Logarithmic

Properties

Exponential

Properties (with

same base):

Logarithm

Properties (with

same base):

Log and Exponential Practice

If y = 72x , what is the value of x when y = 55?

If 3y = log10 x, what is the value of x when y = 1.8?