Category Archives: Graphing

Graphing – an Application of the Derivative.

Posted on by
2

Graphing and the analysis of graphs given (1) the equation, (2) a graph, or (3) a table of values of a function and its derivative(s) makes up the largest group of questions on the AP exams. Most of the other applications of the derivative depend on understanding the relationship between a function and its derivatives.

Here is a list of posts on these topics. Since this list is rather long and the topic takes more than a week to (un)cover,

Tangents and Slopes

Concepts Related to Graphs

The Shapes of a Graph 

Open or Closed?  Concerning intervals on which a function increases or decreases.

Extreme Values

Concavity

Joining the Pieces of a Graph

Using the Derivative to Graph the Function

Real “Real life” Graph Reading

Comparing the Graph of a Function and its Derivative  Activities on comparing the graphs using Desmos.

Writing on the AP Calculus Exams   Justifying features of the graph of a function is a major point-earner on the AP Exams.

Reading the Derivative’s Graph Summary and my most read post!

 

 

 

Revised from a post of October 10, 2017

 

Difference Quotients

Posted on by
Reply

Difference quotients are the path to the definition of the derivative. Here are three posts exploring difference quotients.

Difference Quotients I  The forward and backward difference quotients

Difference Quotients II      The symmetric difference quotient and seeing the three difference quotients in action.  Showing that the three difference quotients converge to the same value.

Seeing Difference Quotients      Expands on the post immediately above and shows some numerical and graphical approaches using calculators or the Desmos graph

 Tangents and Slopes You can use this Desmos app now to preview some of the things that he tangent line can tell us about the graph of a function or save (or reuse) it for later when concentrating on graphs. Discuss slope in relation to increasing, decreasing, concavity, etc.

At Just the Right Time

Stamp out Slope-intercept Form

 

 

 

 

Updated from a post of 9-5-2017

 

Local Linearity

Posted on by
Reply

If you use your calculator or graphing program and zoom-in of the graph of a function (with equal zoom factors in both directions), the graph eventually looks like a line: the graph appears to be straight. This property is called Local Linearity. The slope of this line is the number called the derivative. (There are exceptions: if the graph never appears linear, then no derivative exists at that point.) Local Linearity is the graphical manifestation of differentiability. 

To find this slope, we need to zoom-in numerically. Zooming-in numerically is accomplished by finding the slope of a secant line, a line that intersects the graph twice near the point we are interested in. Then finding the limit of that slope as the two points come closer to our point. This limit is the derivative. It is also the slope of the line tangent to the function at the point. 

While limit is what makes all of the calculus work, people usually think of calculus as starting with the derivative. The first problem in calculus is finding the slope of a line tangent to a graph at a point and then writing the equation of that tangent line. The slope is called the derivative and a function whose derivative exists is said to be differentiable. 

This week’s posts start with local linearity and tangent lines. They lead to the difference quotient and the equation of the tangent line.

Local Linearity I

Local Linearity II      Working up to difference quotient. The next post explains this in more detail.

Tangent Lines approaching difference quotients on calculator by graphing tan line.

Next week: Difference Quotients.

 

 

 

Revised from a post of August 29, 2017

 

Derivative Applications – Graphing

Posted on by
Reply

Graphing and the analysis of graphs given (1) the equation, (2) a graph, or (3) a table of values of a function and its derivative(s) makes up the largest group of questions on the AP exams. Most of the other applications of the derivative depend on understanding the relationship between a function and its derivatives.

Here is a list of posts on these topics. Since this list is rather long and the topic takes more than a week to (un)cover, I will leave it as the lede post for the next two weeks.

Tangents and Slopes

Concepts Related to Graphs

The Shapes of a Graph 

Open or Closed?  Concerning intervals on which a function increases or decreases.

Extreme Values

Concavity

Joining the Pieces of a Graph

Using the Derivative to Graph the Function

Real “Real life” Graph Reading

Comparing the Graph of a Function and its Derivative  Activities on comparing the graphs using Desmos.

Writing on the AP Calculus Exams   Justifying features of the graph of a function is a major point-earner on the AP Exams.

Reading the Derivative’s Graph Summary and my most read post!

Graph Analysis (Type 3)

Posted on by
1

The long name is “Here’s the graph of the derivative, tell me things about the function.”

Most often students are given the graph identified as the derivative of a function. There is no equation given and it is not expected that students will write the equation (although this may be possible); rather, students are expected to determine important features of the function directly from the graph of the derivative. They may be asked for the location of extreme values, intervals where the function is increasing or decreasing, concavity, etc. They may be asked for function values at points.

The graph may be given in context and student will be asked about that context. The graph may be identified as the velocity of a moving object and questions will be asked about the motion and position. (Type 2)

Less often the function’s graph may be given and students will be asked about its derivatives.

What students should be able to do:

  • Read information about the function from the graph of the derivative. This may be approached by derivative techniques or by antiderivative techniques.
  • Find and justify where the function is increasing or decreasing.
  • Find and justify extreme values (1st and 2nd derivative tests, Closed interval test aka.  Candidates’ test).
  • Find and justify points of inflection.
  • Find slopes (second derivatives, acceleration) from the graph.
  • Write an equation of a tangent line.
  • Evaluate Riemann sums from geometry of the graph only.
  • FTC: Evaluate integral from the area of regions on the graph.
  • FTC: The function, g(x), maybe defined by an integral where the given graph is the graph of  the integrand, f(t), so students should know that if,  \displaystyle g\left( x \right)=g\left( a \right)+\int_{a}^{t}{f\left( t \right)dt} then  {g}'\left( x \right)=f\left( x \right)  and  {{g}'}'\left( x \right)={f}'\left( x \right). In this case students should write {g}'(t)=f\left( t \right) on their answer paper, so it is clear to the reader that they understand this.

Not only must students be able to identify these things, but they are usually asked to justify their answer and reasoning. See Writing on the AP Exams for more on justifying and explaining answers.

The ideas and concepts that can be tested with this type question are numerous. The type appears on the multiple-choice exams as well as the free-response. They have accounted for almost 20% of the points available on recent tests. It is very important that students are familiar with all the ins and outs of this situation.

As with other questions, the topics tested come from the entire year’s work, not just a single unit. In my opinion many textbooks do not do a good job with these topics.

Study past exams; look them over and see the different things that can be asked.

For some previous posts on this subject see October 15, 17, 19, 24, 26 (my most read post), 2012 and  January 25, 28, 2013

An activity on this topic is here. The first pages are the teacher’s copy and solution. Then there are copies for Groups A, B, and C. Divide your class into 3 or 6 or 9 groups and give one copy to each. After they complete their activity have the students compare their results with the other groups.

Added 4-1-17

Next Posts:

Tuesday March 14: Area and Volume (Type 4)

Friday March 17: Table and Riemann sums (Type 5)

Tuesday Match 21: Differential Equations (Type 6)

Friday March 24: Others (Type 7: related rates, implicit differentiation, etc.)

 

 

 

How to Tell your Asymptote from a Hole in the Graph.

Posted on by
1

The fifth in the Graphing Calculator / Technology series

(The MPAC discussion will continue next week)

Seeing discontinuities on a graphing calculator is possible; but you need to know how a calculator graphs to do it. Here’s the story:

The number you choose for XMIN becomes the x-coordinate of the (center of) the pixels in the left most column of pixels. The number you choose for XMAX is the x-coordinate of the right most column of pixels. The distance between XMIN and XMAX is divided evenly between the remaining pixels so that all the pixels are evenly spaced across the screen (the same distance apart). The rows of pixels are done the same way evenly spacing them between YMIN and YMAX.

This spacing is usually not at “nice” values as can be seen by just moving the cursor across the screen and noticing the x-values or y-values at the bottom of the screen.

The cursor is located one pixel to the right of the y-axis and one pixel above the x-axis in the “standard” window of a TI-8x. Note the coordinates of that pixel at the bottom of the screen.

The cursor is located one pixel to the right of the y-axis and one pixel above the x-axis in the “standard” window of a TI-8x. Note the coordinates of that pixel at the bottom of the screen. These are the distances between the pixels.

To draw a graph, the calculator takes the x-coordinate of each pixel, calculates the corresponding y-value and turns on the pixel in that column with closest y-pixel-coordinate. If set in a connect mode, the calculator turns on several pixels in adjacent columns so that the y-values seem to connect; this is why the graph often looks jagged in steep sections of the graph. If you are in DOT mode, this does not happen and only one pixel in each column is on.

If you move the cursor over one of the points on a graph, you will see the pixel coordinates, NOT the actual y-coordinates. Use TRACE to see the actual y-coordinate. This is why when finding intersections, you should not just move the cursor over the point, but rather use “intersect” to see the actual y-value of the function.

If the function is undefined for some x-pixel value, then no pixel will turn on in that column. If the function is undefined for some value between the pixel values, then nothing happens because the calculator has not evaluated the function there, so the graph seems to be continuous.

Vertical “asymptotes” are the result of the calculator not evaluating the function at the undefined value; rather it connects the value on one side of the asymptote off the bottom of the screen with the next value on the other side of the asymptote off the top of the screen. If the asymptote appears exactly at a pixel value, then no “asymptote” will appear and that column of pixels will have no pixel turned on. (Some newer calculators and newer operating systems on older calculators have made adjustments so that the “asymptotes” do not show up. In some systems this feature can be turned on or off.)

The function $latex \displaystyle y=\frac{3\left( x-2 \right)}{\left( x-2 \right)\left( x+2 \right)}$ in the standard window. The vertical line is not really the asymptote and the “hole” at (2, 0.75) is not seen.

The function \displaystyle y=\frac{3\left( x-2 \right)}{\left( x-2 \right)\left( x+2 \right)} in the standard window. The vertical line is not really the asymptote and the “hole” at (2, 0.75) is not seen.

A removable discontinuity, a hole in the graph (really a skipped pixel), can be seen, if it occurs at a pixel value. Since in most examples the hole is at an integer or other “nice” number, you will not see them in the “standard” window. Use a “decimal” window, which has been chosen in advance so the x-values of the pixels are integers and nice decimals. (To see this, in a decimal window move the cursor around and notice the pixel coordinates).

The other thing you can do is adjust the XMIN and XMAX values so that the distance between them will land on integer values. (Nice project for your class – the number of pixels can be found in the guidebook, or you can count them. In the old days, before decimal windows, this was necessary – it was called finding a “friendly window.”)

The function $latex \displaystyle y=\frac{3\left( x-2 \right)}{\left( x-2 \right)\left( x+2 \right)}$ in the “decimal” window. The “asymptote” is not shown and the “hole” at (2, 0.75) is visible.

The function \displaystyle y=\frac{3\left( x-2 \right)}{\left( x-2 \right)\left( x+2 \right)} in the “decimal” window. The “asymptote” has disappeared and the “hole” at (2, 0.75) is now visible.

Zooming in or out may change these values so the hole or asymptote disappears.

For a related idea see the post My Favorite Function

 

 

PLEASE NOTE: I have no control over the advertising that appears on this blog. It is provided by WordPress and I would have to pay a great deal to not have advertising. I do not endorse anything advertised here. I noticed that ads for one of the presidential candidates occasionally appears; I certainly do not endorse him.

 

 

 

 

 

 

 

 

 

Comparing the Graph of a Function and its Derivative

Posted on by
4

The fourth in the Graphing Calculator / Technology series

Comparing the graph of a function and its derivative is instructive and necessary in beginning calculus. Today I will show you how you can do this first with Desmos a free online graphing program and then on a graphing calculator. Desmos does this a lot better than graphing calculators, because of the easy use of sliders. CAS calculators also have sliders but they are not as easy to use as Desmos.

Let’s get started. Instead of presenting you with a completed Desmos graph, I will show you how to make you own. One of the things I have found over the years is that it takes some mathematical knowledge to make good demonstration graph and that in itself if useful and instructive. Hopefully, you and your students will soon be able to make your own to show exactly what you want.

Open Desmos and sign into your account; if you don’t have one then register – its free and you can keep your results and even share them with others.

In the first entry line on the left, enter the equation of  the function whose graph you want to explore. Call it f(x); that is enter f(x) = your function. Later you will be able to change this to other functions and investigate them, without changing anything else.

On the second line enter the symmetric difference quotient as

\displaystyle s\left( x \right)=\frac{f\left( x+0.001 \right)-f\left( x-0.001 \right)}{2\left( 0.001 \right)}

Instead of a variable h, as we did in our last post in this series, enter 0.001. This will graph the derivative without having to calculate the derivative. Of course, you could enter the derivative here if your class has learned how to calculate derivatives. If so, you will have to change this line each time you change the function.

In order to closely compare the function and its derivative, on the next line enter the equation of a vertical segment from a point on the function (a, f(a)) to a point on the derivative (a, s(a)). Desmos does not have a segment operation, but here is how you graph a segment. In general, a segment from (a, b) to (c, d) is entered as the parametric/vector function

\left( a\cdot t+c\cdot \left( 1-t \right),b\cdot t+d\cdot \left( 1-t \right) \right),\ 0\le t\le 1

The a, b, c, and d may be numbers or functions. Since our segment is vertical the first coordinate will have a = c and will reduce to a. Here’s what to enter on the third line:

\left( a,f\left( a \right)\cdot t+s\left( a \right)\cdot \left( 1-t \right) \right)

(Notice that there is no x in this expression; t is the variable. Also, the f(a) and s(a) may be interchanged.)

When you push enter, you will be prompted to add a slider for a: click to add the slider. A line will appear under the expression which will allow you to set the domain for t: click the endpoints and enter 0 on the left and 1 on the right, if necessary.

That’s it. You’re done. Use the slider to move around the graphs.

Using the graphs

Discuss with your class, or better yet divide them into groups and let them discuss, what they see. Since at this point they are probably new to this provide some hints such as “What happens on the graph of  f when s is 0?” or “What is true on s when f is increasing?” or “What happens to the function at the extreme values of the derivative?” Prompt the students to look for increasing and decreasing, concavity, points of inflection, and extreme values. All the usual stuff. Work from the function to the derivative and from the derivative to the function.

Have your students formulate their results as (tentative) theorems.  You actually want them to make some mistakes here, so you can help them improve their thinking and wording. For example, one result might be:  If the function is increasing, then the derivative is positive. By changing the first function to an example like f(x) = x3 or f(x) = x + sin (x). Help them see that non-negative might be a better choice.

You might try giving different groups different functions and let them compare and contrast their results.

This is very much in line with MPACs 1, 2, 4, and 6.

You can do the same kind of thing with graphing calculators. That is, you can graph the function and its derivative or a difference quotient. The difference is that graphing calculators do not have sliders.

Extra feature: Desmos will graph a point if you enter the coordinates just like you write them: (a, b). The coordinates may be numbers or functions or a combination of both. Try adding two points to your graph one at each the end of the segment between the graphs that will move with the same slider.

f(x) = x + 2sin(x) and its derivative.

f(x) = x + 2sin(x) and its derivative.