del math refresher: module 1 - duke university
TRANSCRIPT
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?