Derivatives with Maple

Maple can differentiate a function f, using the command [> D(f);    For example if we define the function  then we can differentiate that function by entering the above command.

>

restart:

>	f:=x->x^3-2*x^2 +3;

>

D( f) ;

D(f) should be treated as a new function in Maple's memory which is the derivative of f.  It can be treated just like any other function, so it can be evaluated, plotted, and even differentiated:

>

D(f)(2);

>

D(f)(x);

>	plot(D(f)(x), x=-2..2);

>

D(D(f));

Warning:   The function we are differentiating is f, not f(x), so we enter D(f), not   D(f(x)).  (Try this and see what happens.)

Since f and D(f) are closely related functions, it is interesting to plot them on the same display

>	plot([f(x),D(f)(x)], x=-2..2);

Project - Part 1

Let k  denote the number of letters in your first name. Define the function .

1.)   Have Maple calculate D(f) and D(D(f)).  Have Maple evaluate D(f)(3) and D(D(f))(3).

2.)   Make a plot of both the function f ( x ) and the derivative of f ( x ) in the same window. Use the solve  command to find the intervals on which the derivative is positive and those on which it is negative. Explain how the sign of the derivative is related to the graph of f ( x ).

3.)   In question 2 you used the solve command to find the x-coordinates of all of the points on the graph of   f ( x ) where the derivative equals 0. Next find the y-coordinates of these points on the graph of f(x).

Tangent Lines

The derivative of f ( x ) at a value, say x  = c , gives the slope of the line tangent to the graph of f ( x ) when x  = c. Using the point-slope  formula we obtain an equation for this tangent line:

which can be rewritten as

.

If f  has been defined as a function in Maple then one can define the line tangent to the graph of f  when x  = c  as a function named t1 with the command  t1:=x->D(f)(c)*(x-c)+f(c);  For example, let's enter the function we were working with in the above section and find its tangent line when x=2.

>

restart;

>	f:=x->x^3-2*x^2 +3;

>

c:=2;

>	t1:=x->D(f)(c)*(x-c)+f(c);

>

t1(x);

>	plot([f(x),t1(x)], x=-3..4);

>

Project - Part 2

1.)   Using the same function as in part 1,  i.e.   ,  find an equation of the line tangent to the graph of f ( x ) when x  = 3. (So you need to use the above command defining t1,  but you want to first enter c:=3.)  After this, have Maple display the expression for the tangent line by entering the command t1(x); .

2.)   Plot both the function and the tangent line in the same window.

3.)   Use the solve command to find the x-coordinate of the point where the tangent line intersects the x-axis.

4.)   The tangent line intersects the graph of f ( x ) at exactly one point other than the point of tangency. Make sure that you adjust your window in question 2.) to show this.  Using the solve command, find both coordinates of this point.

Part 3 (Advanced)

1.)  Using the same function as in parts1 and 2, find an equation of a line passing through the point (-2,-0.4) that is also tangent to the graph of f ( x ). Note that the point (-2,-0.4) is not on the graph of f ( x ).

2.)  Plot both the function and this tangent line in the same window.

Part 4 (More Advanced)

A line is said to be normal  to the graph of a function f ( x ) when x  = c  if that line is perpendicular to the line tangent to the graph of f ( x ) when x  = c .

1.)   Using the same function as in  the above parts, determine an equation for the line normal to the graph of f ( x ) when x  = -2. Plot the graphs of f ( x ), the tangent line at x  = -2 and the normal line at x  = -2 in the same window. In order to see perpendicularity between the tangent and normal lines add the option "scaling = constrained" to your plot command. This will force the same scales on both the x  and y  axes.