On many problems, the most challenging aspect is keeping track of the variables and what they represent. When the variables are scaled in hundreds, thousands, or even millions, we need to pay careful attention.
Scaling means that the quantity has been divided by some amount. For instance, we can write the amount $14,400,000 as 14.4 million dollars.
To scale $14,400,000 in millions, divide by 1,000,000 and then add the word “million” after the result. We could also scale in thousands by dividing by 1000. This gives 14,400 thousand dollars.
In calculus, we will need to take a function f (x) and write out f (x+h) for that function. Let’s look at how to do this properly.
To do a problem like this, you need to understand exactly what the x in f (x) represents and what the f represents. Let’s look at the functionf (x) = x2 – x. A function is a process. In this case, it is the process of
Square the input
Take the result and subtract the input
Notice that there is no mention of the x in the formula. That is because it is a placeholder representing the input. There is nothing special about x. We could have just as easily used a different letter as a placeholder for the input. If I had wanted to call the input t, I would have written
f (t) = t2 – t
If the input had been represented by the word dog, I would have written
f (dog) = dog2 – dog
The input variable is simply a placeholder…if a number is put in its place like 7, we get
f (7) = 72 – 7 = 42
Notice that the process is the same. Square the input and subtract the input from the result. In this case, the input is 7 so we are squaring 7 and then subtracting 7 from the result.
Many students are confused byf(x+h). Now the input is represented by x+h instead of x. This means we need to square it and then subtract x+h from the result.
f (x+h) = (x+h)2 – (x+h)
We can simplify this by foiling out the square,
(x+h) = (x+h)(x+h) = x2 +2xh + h2
And removing the parentheses after the subtraction we get
f (x+h) = x2 +2xh + h2 – x – h
The handout below has more examples with this function.
Graphing an absolute value function can be a bit deceiving. Depending on the technology you use, the graph you get may not actually represent the function well.
Let’s graph the function
using WolframAlpha and Desmos. These two online graphing tools are both free to use and can produce excellent graphs.
To graph this function in WolframAlpha, go to the website and type this in the box on the screen.
Both the numerator and denominator need to be in parentheses. The absolute value function in WolframAlpha is “abs”. Putting this in front of (x+4) means the absolute value of the quantity x + 4.
Press return to give the following result.
The graph consists of a horizontal section at y = -1 and another at y = 1. These sections are connected by a vertical line at x = -4. This is problematic since this is not a function…it does not pass the vertical line test at x = -4.
As shown in the video above, the graph of this function looks like this in Desmos.
This looks similar to the WolframAlpha version, except that the tow horizontal pieces are not connected. As noted in the video,
is undefined at x = -4. This is because x = -4 causes the denominator to be zero.
This might not seem like a big deal. But if you were determining whether the function was continuous at x = -4, the two graphs would lead to different conclusions. The WolframAlpha graph would lead you to think the function is continuous. Desmos would give the opposite conclusion.
In this case, Desmos gives a more accurate graph since it shows the discontinuity at x = -4. An even better version of this graph would be to include open circles at x = -4.
This not only shows the discontinuity, but also indicates that the function is undefined at x = -4. To put these on the graph I downloaded the image and then added the circles in an image editing program like Paint.
Our goal in this activity is to find the slopes of tangent lines at several x values. The handout below has four pages. On the first and third pages contain the graphs along with a table of x values. Pages two and four contain empty graphs.
Print out this PDF file. To fill out the tables on pages one and three, draw tangent lines on the x values indicated in the tables. Now use the grid to estimate the slope of these tangent lines. Enter those values in the second column of the table.
Once you have filled out the table, graph the ordered pairs on the corresponding blank graph. In Example 1, you should see a very obvious pattern ( a line) to the data points you just graphed. In Example 2, there is also a pattern (a parabola) although it may not be as obvious.
Each of the ordered pairs in the table gives the derivative of the graph you took it off of. The video below illustrates this process.
This technique of drawing the derivative is not a very effective method for finding the derivative of a function. It gives us the graph, but not necessarily the formula. In the rest of Section 11.4, you’ll learn how to find the derivative using the definition with limits. You will also learn some shortcuts to take the derivative.