<Text-field style="Heading 1256" layout="Heading 1256">Convergence of Secant Slopes to the Derivative Function</Text-field>

<Text-field style="Heading 1256" layout="Heading 1256">Convergence of Secant Slopes to the Derivative Function</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 secant line between successive points on a curve, observe what happens as the distance between successive points becomes small, compare the graph of successive secant line slopes with the graph of the derivative function, and apply the definition of the derivative in computing it.

One of the central problems of calculus is measuring and quantifying slopes on the graphs of nonlinear functions. In this module, you will explore two methods for approximating these slopes and see how the approximations approach the derivative function in the limit. The first method is to extract a sample of evenly spaced points from the graph of a function y = f(x), calculate the slopes of the secant lines between the sample points, and graph the secant slope values together with the derivative function f '(x). You will see that as the number of sample points increases and the horizontal spacing between consecutive points decreases, the set of discrete secant-slope values gets closer to the derivative function. The second method for approximating the slopes on the graph of a function is to take every value of x in the domain of the function y = f(x) and calculate the slope of the secant line between the two points [x, f(x)] and [x+h, f(x+h)] for a fixed nonzero value of h. This gives a new function of x and h, namely, the function of secant slopes for any x in the domain of f. This secant-slope function is plotted together with the derivative f '(x) . You will see that as the value of h gets closer to 0, the secant slope function gets closer to f '(x). Finally, you will see how to use Maple to take the limit of the secant-slope function as h approaches 0.

Before you begin this module, we recommend that you refer to "Tangents and Secants," a JAVA applet included in this supplement. This applet allows you to explore how the slopes of secant lines converge to the slope of the line tangent to a curve at a point.

<Text-field style="_cstyle266" layout="Heading 1"><Font size="18">Technology Guidelines</Font></Text-field>

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<Text-field style="Heading 1" layout="Heading 1">Part I: Secant Slopes from a Discrete Set of Sample of Points</Text-field>

The function DerivApprox( ) extracts n evenly spaced sample points from a differentiable function and calculates the slopes of the secant lines between consecutive pairs of sample points. It then plots the function, the sample points, and the secant lines on one graph. A second graph is plotted to show the slopes of the secant lines in comparison with the derivative of the function. As the number of sample points increases, the secant slopes get closer to the derivative function.

Note: DerivApprox( ) was written specially for this worksheet and is not a built-in Maple command. You should not worry about understanding the commands in the second of the next two cells, that is, unless you are interested in doing so. These commands define the DerivApprox( ) function, which is used in the remainder of this worksheet. All that you need to do is execute the commands in the next two cells so that DerivApprox( ) will work.

restart;

DerivApprox := proc(f, var, n) local xval, plist, slist, i, j, plotA, plotB, plotC, plotD, plotE, plotF; plist := []; for i from 0 to n-1 do xval := evalf(op(1, rhs(var))+i/(n-1)*(op(2,rhs(var))-op(1,rhs(var)))); plist := [op(plist), [xval, subs(lhs(var)=xval, f)]]; od; plotA := plots[pointplot](plist, style=LINE, color=RED): plotB := plots[pointplot](plist, symbol=CIRCLE, color=BLUE): plotC := plot(f, var, color=BLACK): print(plots[display]([plotA,plotB,plotC], title = "y, sample points & secants")); lprint(`The derivative of `, f, ` is `, diff(f, lhs(var))); slist := []; for j from 0 to n-2 do xval := evalf(op(1, rhs(var))+j/(n-1)*(op(2,rhs(var))-op(1,rhs(var)))); slist := [op(slist), [xval+0.000001*(op(2,rhs(var))-op(1,rhs(var))), (plist[j+2,2]-plist[j+1,2])/(plist[j+2,1]-plist[j+1,1])]]; od; plotE := plots[pointplot](slist, symbol=CIRCLE, color=RED): plotF := plot(diff(f, lhs(var)), var, discont=true, color=BLACK); print(plots[display]([plotE,plotF], title= "y' and secant slopes")); end:

The arguments of DerivApprox( f, x, a, b, n) are: f, the function you wish to analyze; x, the independent variable; a, the beginning of the domain interval; b, the end of the domain interval; and n, the number of evenly spaced sample points. The following example illustrates how DerivApprox( ) works. All that you need to do is change the arguments.

DerivApprox(x*sin(x), x=-2*Pi..2*Pi, 13);

<Text-field style="Heading 1" layout="Heading 1">You Try It: On Differentiable and Non-Differentiable Functions</Text-field>

1. Change the value of n [the last argument of DerivApprox( )] and describe what you see. To do this systematically, you might want to repeatedly double the number of sample points. That is, let n = 6, 12, 24, 48, 96, ... .

2. Try DerivApprox( ) on some of differentiable functions that you pick. Be sure to use several sample sizes so that you can see how the secant slopes function converges on the derivative. Here is an example of a function that has a very interesting derivative. Do you see what makes this function special?

f:=x-> exp(x); plot(f(x),x=-2..2,labels=["x","y"]);

DerivApprox(f(x), x=-2..2, 6);

3. Try DerivApprox( ) on some functions that are not differentiable at one or more points. Here is an example.

f:=x->piecewise(x<=2, 2*x, x>2, 4*x-x^2); plot(f(x), x=0..4, labels=["X","Y"]);

DerivApprox(f(x), x=0..4, 6);

The DerivApprox( ) command gives us a formula for the derivative function that is printed above. Is it correct in this case? To answer this question you should consider what happens at x =2.

<Text-field style="Heading 1" layout="Heading 1">Part II: Secant Slopes at Every Point</Text-field>

Another approach to approximating the slopes on the graph of a function is to take every value of x in the domain of the function y = f(x) and calculate the slope of the secant line between the two points [x, f(x)] and [x + h, f(x + h)] for a fixed nonzero value of h. This gives a new function of x and h, namely, the function of secant slopes for any x in the domain of f. We form this secant-slopes function in the next section.

restart; f:=x->x*sin(x); fprime:=diff(f(x),x); msecant:=(f(x+h)-f(x))/h;

Now, we graph the secant slopes for a specified h value together with the derivative function.

h:=1: plot([msecant, fprime], x=-2*Pi..2*Pi, labels=["X","Y' & m[sec]"], color=[red,blue], thickness=[1,2], discont=true);

The red curve gives the secant slope for each value of x, with h=1, and the green curve gives the value of the derivative, that is, the slope of the tangent to the graph of f(x) for each value of x.

<Text-field style="Heading 1" layout="Heading 1">You Try It: On Differentiable and Non-Differentiable Functions</Text-field>

1. See what happens to the two graphs in the preceding section when you decrease the value of h. Also, try some negative values of h. To be systematic, you should repeatedly cut h in half. For example, you could use h = -1, -0.5, -0.25, -0.125, ... for negative values of h. Describe what you observe.

2. Repeat the steps in Part II on some of differentiable functions that you pick. Be sure to decrease values of h so that you can see how the secant-slopes function, msecant( ), converges on the derivative function as h approaches 0. Here is an example.

restart: f:=x->sin(x); fprime:=diff(f(x),x); msecant:=(f(x+h)-f(x))/h;

h:=1: plot([msecant, fprime], x=0..2*Pi, labels=["X","Y' & m[sec]"], color=[red, blue], thickness=[1,2]);

3. Repeat the above steps on some functions that are not differentiable at one or more points. Here is an example.

f:=x->piecewise(x<=2, 2*x, x>2, 4*x-x^2); plot(f(x), x=0..4, labels=["X", "Y"]); fprime:=diff(f(x),x); msecant:=(f(x+h)-f(x))/h;

h:=1;

plot([msecant, fprime], x=-2*Pi..2*Pi, labels=["X","Y' & m[sec]"], color=[red, blue], thickness=[1,2], discont=true);

Note that we make the msecant( ) graph a little thicker so that it is not hidden behind the graph of the derivative function.

Maple's derivative command, diff( ), gives us a formula for the derivative function above. Is it correct in this case? To answer this question, you should consider what happens at x = 2.

<Text-field style="Heading 1" layout="Heading 1">Part III: Taking It to the Limit</Text-field>

The slope of the tangent to the graph of y = f(x) for each value of x is given by 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 provided, of course, that the limit exists. This gives the derivative function f '(x). In the next group of commands, we use this definition to find the derivative of a specific function.

First, we pick a function.

restart: f:=x->a*x^2+b*x+c;

Next, we calculate the secant slopes, that is, the difference-quotient.

simplify((f(x+h)-f(x))/h);

And we take the limit to form the derivative function.

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

<Text-field style="Heading 1" layout="Heading 1">You Try It: Finding Derivatives Using the Definition </Text-field>

Repeat the steps in Part III to find the derivatives of several functions that you pick. Here are some suggestions.

1. 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

2. 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

3. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2KEYrLUYjNiYtRiw2JFEiZkYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RMCZBcHBseUZ1bmN0aW9uO0YnL0Y3USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQS8lKXN0cmV0Y2h5R0ZBLyUqc3ltbWV0cmljR0ZBLyUobGFyZ2VvcEdGQS8lLm1vdmFibGVsaW1pdHNHRkEvJSdhY2NlbnRHRkEvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZQLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JFEieEYnRjZGNi8lJ2l0YWxpY0dGQUY9RjYtRjo2LlEiPUYnRmVuRj1GP0ZCRkRGRkZIRkpGTC9GT1EsMC4yNzc3Nzc4ZW1GJy9GUkZbby1GIzYmLUYsNiVRJHNpbkYnRmVuRj1GOUZTRjZGK0Y2RitGNg==

4. 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 where A, a, B, and b are constants.

5. 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

6. 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

7. 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 where C and a are constants.