<Text-field style="_cstyle284" layout="Heading 1"><Font bold="true">Derivatives, Slopes, Tangent Lines, and Making Movies</Font></Text-field>

<Text-field style="_cstyle284" layout="Heading 1"><Font bold="true">Derivatives, Slopes, Tangent Lines, and Making Movies</Font></Text-field>

Note: You may notice differences between this Maple worksheet and the equivalent Mathematica notebook. These differences were introduced to preserve the content of these modules and were necessary because of major functional differences between Maple and Mathematica.

<Text-field style="Heading 1" layout="Heading 1">Introduction</Text-field>

OBJECTIVE: To visualize the derivative and the linearization of a function at a point. In this module, we explore the derivative as the slope of a nonlinear function and find the equation of the line tangent to a curve at a point. You will learn how to plot the curve and selected tangents on the same graph. In addition, you will see how to use Maple to make a movie animation by generating a sequence of plots, each showing a different tangent to the curve. When the sequence of graphs is animated, the tangent lines appear to roll along the graph of the function.

<Text-field style="_cstyle262" layout="Heading 1"><Font size="18">Technology Guidelines</Font></Text-field>

NOTE: If you have just finished a worksheet, restart Maple before executing a new worksheet. TO OPEN SECTIONS, Click on the PLUS sign at the left hand side of the screen or select Expand All Sections from the View drop down menu.

TO STOP AN EXECUTION Click on STOP button from the toolbar.

ORDER OF EXECUTION Execute commands in the order given. Do not skip any Maple Input lines within a given worksheet

Alternatively, you can execute the entire worksheet by selecting the Execute Worksheet command from the Edit drop down menu.

SAVING WORKSHEETS.

You can save anytime to any directory you choose, and it is wise to save often. EXPERIENCING MAJOR PROBLEMS Save if appropriate, and then sut down Maple and start it up again.

<Text-field style="Heading 1" layout="Heading 1">You Try It</Text-field>

First, work through Parts I - V with the example function f (x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE=, and then repeat the steps in Parts I - V for some functions that you select. Here are some suggestions. 1. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiM0YnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEvLTJ+PH49fnh+PH49fjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn

2. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSRzaW5GJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkjbW9HRiQ2LVEwJkFwcGx5RnVuY3Rpb247RidGNy8lJmZlbmNlR0Y2LyUqc2VwYXJhdG9yR0Y2LyUpc3RyZXRjaHlHRjYvJSpzeW1tZXRyaWNHRjYvJShsYXJnZW9wR0Y2LyUubW92YWJsZWxpbWl0c0dGNi8lJ2FjY2VudEdGNi8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRk4tSShtZmVuY2VkR0YkNiQtRiM2JC1GLDYlUSJ4RicvRjVRJXRydWVGJy9GOFEnaXRhbGljRidGN0Y3RjdGK0Y3 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUS4wfjx+PX54fjx+PX4yRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVExJkludmlzaWJsZVRpbWVzO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNiVRJSZwaTtGJy9GNUZCRj5GPkYrRj4= 3. 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 for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JVEvLTJ+PH49fnh+PH49fjJGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjNRJ25vcm1hbEYn 4. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkmbXNxcnRHRiQ2Iy1JI21pR0YkNiVRInhGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRicvRjZRJ25vcm1hbEYn for 0 < LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2Ji1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LVElJmxlO0YnL0Y4USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQi8lKXN0cmV0Y2h5R0ZCLyUqc3ltbWV0cmljR0ZCLyUobGFyZ2VvcEdGQi8lLm1vdmFibGVsaW1pdHNHRkIvJSdhY2NlbnRHRkIvJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZRLUkjbW5HRiQ2JFEiNEYnRj5GPkYrRj4= Note that we do not include x = 0. Do you know why?

<Text-field style="Heading 1" layout="Heading 1">Part I: The Derivative at a Point</Text-field>

Define a nonlinear function of your choice and call it f (x) and graph it. For an example, we choose the function f (x) = LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJW1zdXBHRiQ2JS1GLDYlUSJ4RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW5HRiQ2JFEiMkYnL0Y7USdub3JtYWxGJy8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGQUYrRkE= for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbWlHRiQ2JFEtLTJ+PD1+eH48PX4yRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnRi8= (To put in a different function and domain, change the entries in the following input cell.)

restart; with(plots): x0:=-2; xf:=2; f:=x->x^2;

plot(f(x), x=x0..xf, labels=["x","y"], color=blue);

Now pick a point on the function and use the definition of the derivative to find the slope of the function's graph at the point you pick, calling it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEibUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRKHRhbmdlbnRGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ==. We choose x = 1 for our example.

mtangent:=limit((f(1+h)-f(1))/h,h=0);

<Text-field style="Heading 1" layout="Heading 1">Part II: The Linearization of a Function</Text-field>

Form a new function for the line that is tangent to the function that you chose in the "You Try It" section at that point you picked in Part I, and call it LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUklbXN1YkdGJDYlLUkjbWlHRiQ2JVEieUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1GIzYkLUYvNiVRKHRhbmdlbnRGJ0YyRjUvRjZRJ25vcm1hbEYnLyUvc3Vic2NyaXB0c2hpZnRHUSIwRidGPQ== = L(x). This function is called the linearization of y = f (x) at the point (1, f (1)).

L:=x->f(1)+mtangent*(x-1);

Plot the function and the tangent line on the same graph.

plot({f(x), L(x)}, x=x0..xf, labels=["x","y,ytangent"], color=[blue, red]);

<Text-field style="Heading 1" layout="Heading 1">Part III: The Derivative Function</Text-field>

Form the derivative function that will give the slope of the tangent to your chosen function at any point with coordinates (x, f (x)). Call the derivative function yprime(x), and graph it.

yprime:=limit((f(x+h)-f(x))/h,h=0);

Graph yprime(x).

plot(yprime, x=x0..xf, labels=["x", "yprime"], color=COLOR(RGB,0,1,0));

<Text-field style="Heading 1" layout="Heading 1">Part IV: A Whole Bunch of Tangents</Text-field>

Form a new Maple function that gives the equation of the line tangent to y = f (x) at the point (a, f (a)). Call the new function tanline(x,a).

tanline:=(x,a)->f(a)+eval(yprime,x=a)*(x-a);

Test your tanline(x,a) for several values of a by plotting the tangent lines and y = f (x) together on the same graph. First, use the tanline(x,a) function and a short loop to generate a list of equations for the tangents to the curve at points (a, f (a)), for values of a varying from -2 to 2 in increments of 0.1. Then graph the tangent lines and y = f (x) together on the same graph.

listoftangentline:=seq(evalf(tanline(x,x0+(xf-x0)*a/20)), a=0..20);

p1:=plot({listoftangentline}, x=x0..xf, color=blue): print(p1);

p2:=plot(f(x), x=x0..xf, thickness=[5], labels=["x","y"]): print(p2);

print(plots[display]({p1,p2}));

<Text-field style="Heading 1" layout="Heading 1">Part V: Making Movies</Text-field>

The following command generates a sequence of graphs that you can animate. Click on the graph below to bring up the animation toolbar at the top of the worksheet.

tanlist:=[listoftangentline]: b:=1: tanplot := []: for a from evalf(x0) to evalf(xf) by evalf((xf-x0)/20) do tanplot := [op(tanplot), plot([f(x),tanlist[b]], x=x0..xf,color=[red,blue])]: b:=b+1: od: display(seq(plot([f(x),listoftangentline[i]], x=x0..xf,labels=["x","y"],color=[red,blue]),i=1..21),insequence=true);