Calculus 1.2The Parthenon,
Photo by Vickie Kelly, 2006
Built in 1897 for Tennessee’s centennial celebration.
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Photo by Vickie Kelly, 2006
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We will start our study of calculus by spending a few days in a quick review of several pre-calculus topics.
Whether you are comfortable with all of these pre-calculus topics or have forgotten many of them and have to learn them now, it is okay either way.
The most important thing to remember in your study of calculus is this:
Don’t panic!
Calculus is a challenging class for everyone.
If you ask questions, keep up with homework, and study for quizzes and tests, this will be the best math class you have ever taken, and you will be proud of your success.
and …
College will be way more fun if you have taken calculus in high school and don’t have to stress over passing your first college math class!
Domain and Range
Functions have an independent variable (often x) and a dependent variable (often y).
In this discussion we are going to use x for the independent variable and y for the dependent variable, but we could use other letters.
domain: The set of all possible x values for a function.
range: The set of all possible y values for a function.
We can specify the domain of a function, or we can look at the nature of the function to determine the natural domain of the function.
Example:
When finding the domain, it is often easier to think of what values that we cannot use for x.
In this case, x cannot equal zero. The domain is:
or
We usually use interval notation.
These are open intervals (with parentheses) because we don’t count the zero. Also notice the symbol for union.
This is set builder notation.
Find the domain and range of:
Looking at the graph, we see that we can have every y value except zero, so the range is:
or
Another example:
Since we cannot take the square root of a negative number, the domain is:
This is a half-open interval.
We use a square bracket on the left boundary point because we count the zero. (The interval is closed at zero.)
When infinity is a boundary, it is always an open interval on that end, with parentheses.
Find the domain and range of:
Looking at the graph, we see that there are no negative y values, so the range is:
Another example:
Since we cannot take the square root of a negative number, the domain is:
Find the domain and range of:
Symmetry
When we graph the function , we see that it has
y-axis symmetry.
When we square the x, any negative sign cancels out, so changing the sign of x does not change the y value.
Any polynomial function with only even exponents behaves the same way, and has y-axis symmetry.
This is , which has an even exponent.
Any function with y-axis symmetry is called an even function.
For example, is an even function.
So is .
origin symmetry.
Changing the sign of x changes the sign of y.
In other words, if (x,y) is on the graph, so is (-x,-y).
Any function with origin symmetry is called an odd function.
Polynomial functions with exponents that are both even and odd have no symmetry.
For example,
is an odd function.
Of course, a graph with x-axis symmetry is not a function at all!
Fails the vertical line test!
Piecewise Functions
While many functions can be defined by a single formula, others are defined by applying different formulas to different parts of their domains.
Just graph each piece separately, for each part of the domain.
Example:
Example:
There are two pieces to this graph, so we need two equations.
Paying attention to the open and closed circles, we get the piecewise function:
p
This simplifies to: