The **implicitdiff** command can be used to find derivatives of
implicitly defined functions. Suppose we wanted to use implicit
differentiation to find
for the relation

Then we first define our relation and give it a label for later use.

> f:=x^2*y^2+y^3=0;Note that the

> implicitdiff(f,y,x);

The result of the command is the derivative,
. The first argument is the relation
that you want to differentiate implicitly. The **second** argument to the
`implicitdiff` command is where you tell Maple what the
**dependent variable** is. The remaining argument is to
specifying the derivative you want.

To compute numerical values of derivatives obtained by implicit differentiation, you have to use the subs command. For example, to find the value of at the point you could use the following command.

> subs([x=1,y=-1],implicitdiff(f,y,x));

> with(plots):The

> f := x^2*y^2+y^3 = 0;g := x^2*sin(y) = 1; > implicitplot([f,g],x=-10..10,y=-10..0);You can add options to the plot command to make the picture more clear. Execute the following commands to see the improvements.

> implicitplot([f,g],x=-10..10,y=-10..0,numpoints=10000); > implicitplot([f,g],x=-10..10,y=-10..0,numpoints=10000,color=[``Aqua'', ``Magenta'']); > implicitplot([f,g],x=-10..10,y=-10..0,numpoints=10000,color=[``Aqua'', ``Magenta''],scaling=constrained);To find where the graphs intersect you can use the

> a := fsolve({f, g}, {x = -5 .. 0, y = -5 .. 0}); > b := fsolve({f, g}, {x = -5 .. 0, y = -5 .. 0}); > c := fsolve({f, g}, {x = 0 .. 5, y = -5 .. 0}); > d := fsolve({f, g}, {x = 0 .. 5, y = -10 .. 5});You could then use these points to find the slopes of

> afslope := subs(a, implicitdiff(f, y, x)); > agslope := evalf(subs(a, implicitdiff(g, y, x))); > afline := afslope*(x-a[1])+a[2]; > agline := agslope*(x-b[1])+b[2]; > implicitplot([g, f, afline, agline], x = -10 .. 10, y = -10 .. 5, numpoints = 10000, color = ["Aqua", "Magenta", "DarkOliveGreen", "Blue"], scaling = constrained);

- Enter the following equation as expression in Maple:
.
- A)
- Plot the equation using the domain and the range . Looking at the graph, how many points have the x value ?
- B)
- Using the
`solve`command find the y values of the point(s). State your point(s) (x,y) in text. - C)
- Find the slope of the tangent at each point in part C) and name them and

- Using the same expression :
- A)
- Looking at your graph in exercise 1, is the concavity positive or negative at each point whose x value is .
- B)
- Find of each of the points.

- Using the same expression :
- A)
- Write the equation of the tangent lines for your two points. Make sure to give them each a name.
- B)
- Plot the expression and the two tangent lines.

2011-10-19