Any function is graphed with a few basic steps.
- Put the function into the calculator.
- Set the graphing window.
- Graph the function.
The exact sequence of steps is demonstrated in the handout below.
Category: Chapter 10
How Do I Make A Table Of Values In Excel?
Excel may be used to create tables of values for function at equally spaced inputs or at input that are not equally spaced.
A table at equally spaced inputs is useful for creating a graph. The handout below shows how to do this in Excel 2007. The process is almost identical in other versions of Excel.
We can make a table at arbitrary input also. The handout below demonstrates this process. This is very useful for evaluating limits in calculus.
How Do You Prove That A Function Is Continuous At Some Point?
Health insurance, taxes and many consumer applications result in a models that are piecewise functions. To prove these functions are continuous at some point, such as the locations where the pieces meet, we need to apply the definition of continuity at a point.
A function f is continuous at a point x = a if each of the three conditions below are met:
i. f (a) is defined
ii. $latex \displaystyle \underset{x\to a}{\mathop{\lim }},f(x)$ is defined
iii. $latex \displaystyle \underset{x\to a}{\mathop{\lim }},f(x)=f(a)$
In the problem below, we ‘ll develop a piecewise function and then prove it is continuous at two points.
Problem A company transports a freight container according to the schedule below.
- First 200 miles is $4.00 per mile
- Next 300 miles is $3.00 per mile
- All miles over 500 is $2.50 per mile
Let C(x) denote the cost to move a freight container x miles.
a. Find a piecewise function for C(x).
For this function, there are three pieces. The first piece corresponds to the first 200 miles. The second piece corresponds to 200 to 500 miles, The third piece corresponds to miles over 500.
The board below show the function.
Let’s break this down a bit. In the first section, each mile costs $4.50 so x miles would cost 4.5x.
In the second piece, the first 200 miles costs 4.5(200) = 900. All miles over 200 cost 3(x-200). This gives the sum in the second piece.
In the third piece, we need $900 for the first 200 miles and 3(300) = 900 for the next 300 miles. In addition, miles over 500 cost 2.5(x-500).
b. Prove that C(x) is continuous over its domain.
Each piece is linear so we know that the individual pieces are continuous. However, are the pieces continuous at x = 200 and x = 500?
Let’s look at each one sided limit at x = 200 and the value of the function at x = 200.
Since these are all equal, the two pieces must connect and the function is continuous at x = 200. At x = 500,
so the function is also continuous at x = 500.
This means that the function is continuous for x > 0 since each piece is continuous and the function is continuous at the edges of each piece.
Death and Piecewise Functions
Although this may seem a little gruesome, it is not uncommon for businesses to give discounts for volume sales. In this case, a mortician charges less per pound for bodies weighing more than a certain amount.
The local mortician charges by the pound for embalming according to the following table:
Find a piecewise linear function that models the cost as a function of weight.
How Do I Write Down a Piecewise Function For Postage?
In of April 2015, the US Postal Service established new postal rates for first class mail. The postage charged for first class mail is a function of its weight. The US Postal Service uses this table to describe the rates.
Problem Convert this table to a piecewise defined function that represents first class postage for letters weighing up to 3 ounces, using x as the weight in ounces and P as the postage in cents.
Solution This one always causes many questions. I suggest trying a bunch of different inputs (weights) and seeing how it works…then try to come up with the formula.
So, I created a table of possible weights. I made sure to include weights that were fractions of an ounce (something other than 1, 2, or 3). This allows us to understand what that phrasing means.
Now let us try to add in some corresponding postage amounts. Suppose a letter weighed 0.5 ounces. It would fall into that first part of the function “First ounce or fraction of an ounce”. So it would cost 49 cents. The same would be true of a letter weighing 0.75 or 1 ounce…both would cost 49 cents. We can add some numbers to the table.
In fact, any letter weighing 1 ounce of less (and greater than 0) would cost 49 cents.
Now what happens when the letter weighs a little more than 1 ounce? For a letter weighing 1.5 ounces, the first ounce would cost 49 cents and since the letter falls into another “additional ounce or fraction of an ounce”, the total cost would be 49 + 22 = 71. Any letter weighing more than 1 ounce up to 2 ounces would have the same exact cost, 71 cents. Now we can update our table:
As soon as we increase the weight to the next ounce, another 22 cents is added. So a letter weighing 2.5 or 3 ounces would cost 49 + 22 + 22 = 93.
The key thing to note is that for each ounce, the postage stays constant until the next ounce. The correct piecewise function needs to take this into account.
Start your piecewise function with where the pieces are valid:
Each piece corresponds to where the postage is constant and were the rates change. For instance, at x = 1, the postage changes from 41 to 71 cents since we have gone to a new ounce. In each weight interval, the postage is constant according to the table. This give us the function
A graph of the postage function P(x) looks like the one below.
