Example 1 - Tangent Line by Zooming

Consider the graph of the function

>	f1 := 3*x^3+9*x^2-7: f(x) = f1;

at the point (1, 5 ).

>	x1 := 1: y1 := eval( f1, x=x1 ):

The graph of the function on the interval [ -3, 2 ] and the point confirm that this point is on the graph.

>

h_list := [seq( 2^(3-i), i=1..10 )]: P1 := [ seq( Tangent( f1, x=x1,                           x1-h..x1+h/4, output=plot, showtangent=false,                           title=sprintf("Graph of y=%a on [%6.3f,%6.3f]",                                         f1, x1-h, x1+h/4),                           view=[x1-h..x1+h/4,DEFAULT] ),              h=h_list )]: P1[1];

Now, look at the graph of the function over shorter and shorter intervals around , that is, zoom in on the point ( 1, 5 ) on the graph.

>	for i from 2 to nops(P1) do   P1[i]; end do;

Notice that the graph of the function becomes straighter each time the viewing interval is cut in half. In the last few views the function appears to be a straight line. When this happens, the straight line that appears is the tangent line to the graph of the function at the point. This straightening of the graph of the function is even more apparent when tangent line is included in the plots.

>

P1t := [ seq( Tangent( f1, x=x1,                            x1-h..x1+h/4, output=plot, showtangent=true,                            title=sprintf("Graph of y=%a on [%6.3f,%6.3f]",                                          f1, x1-h, x1+h/4),                            view=[x1-h..x1+h/4,DEFAULT] ),               h=h_list )]: for i from 1 to nops(P1t) do   P1t[i]; end do;

The derivative of a function at a point exists when the graph of the function has a tangent line at that point. The value of the derivative at this point is the slope of the tangent line.

In this case the equation of the tangent line to the graph of    at the point ( 1, 5 ) is

>	y = Tangent( f1, x=x1, output=line );

so the slope of the tangent line is

>	m = Tangent( f1, x=x1, output=slope );

>

Example 2 - Corners

Consider the graph of the function

>	f2 := sqrt(abs(x^2-4)+9): f(x) = f2;

at the point ( 2, 3 ):

>	x2 := 2: y2 := eval( f2, x=x2 ):

>

The graph of the function on the interval [ -3, 7 ] and the point confirm that this point is on the graph.

>	myTangent( f2, x=x2, -3..7, view=[DEFAULT, 0..7] );

Zooming in on the portion of the graph near ( 2, 3 ) leads to the plots

>	P2 := [ seq( myTangent( f2, x=x2, x2-h..x2+h ),              h=h_list ) ]: for i from 1 to nops(P2) do   P2[i]; end do;

Notice that the graph of the function has a very pronounced corner that does not disappear as the viewing window becomes smaller and smaller. At no pont will the function appear to be a (single) straight line through the point ( 2, 3 ).

The fact that there is a corner in the graph of  at the point ( 2, 3 ) means there is no tangent line to this function at ; this function is not differentiable at . This conclusion is confirmed with the Tangent  command as follows:

>	Tangent( f2, x=x2 );

Error, (in Student:-Calculus1:-Tangent) the slope is not defied at the point `x` = 2

Example 3 - Jumps and Cusps

In addition to a corner, derivatives do not exist when the graph has a cusp or a jump at the point in question. Jumps are problematic because the tangent line does not have any jumps and so it is impossible to find a single straight line that approximates the graph of  on both sides of the point.

>	f3a := piecewise( x<1, 1, x>1, 2 ): f(x) = f3a;

>

x3a := 1:

>	myTangent( f3a, x=1, -1..3 );

>	P3a := [ seq( myTangent( f3a, x=x3a, x3a-h..x3a+h ),               h=h_list ) ]: for i from 1 to nops(P3a) do   P3a[i]; end do;

An example with a cusp is.

>	f3b := abs(x)^(1/3): `f`(x) = f3b;

at the point ( 0, 0 ):

>	x3b := 0: y3b := eval( f3b, x=x3b ):

>	myTangent( f3b, x=x3b, -3..7, view=[-3..7,0..2] );

While there certainly appears to be a sharp point (cusp) at the origin, it is still prudent to zoom in on this point.

>	P3b := [ seq( myTangent( f3b, x=x3b, x3b-h..x3b+h ),              h=h_list ) ]: for i from 1 to nops(P3b) do   P3b[i]; end do;

>

Corners, cusps, and jumps are the pictures you should have in mind when you think about a function that does not have a derivative.

Example 4 - Tangent Line as Limit of Secant Lines

Note that the zooming process is a limit as the viewing window size decreases to zero. An alternate way of looking at tangent lines is to define

Consider the function

>	f := (x^3-5)*(x^2-1)/(x^2+1): 'f'(x) = f;

at the point with :

>	x1 := 2; y1 := eval( f, x=x1 );

The graph of the function and the point ( 2,  ) suggests that there should be a tangent line at this point.

>	myTangent( f, x=x1, -2..3 );

We could find the tangent line by zooming. The equation of this tangent line is

>	y = Tangent( f, x=x1, output=line );

Example 5 - An Oscillating Position

An object is traveling in a straight line through the origin in such a manner that its directed distance from the origin is given by

>	f5 := sin( (t^2+1)/200 ): s(t) = f5;

where t measures time in seconds. Positive and negative distances are on opposite sides of the origin. The graph of the object's motion for the first minute shows that the object passes through the origin 5 times while moving back and forth between two positions  and  .

>	plot( f5, t=0..60, title="Object's position during first minute" );

The first time the object reaches position s=1 occurs between  and . Maple finds this time with much more accuracy:

>	t1 := fsolve( f5=1, t ): t[1] = t1;

>

The velocity of the object when it passes through the origin is

   .