1. Consider the expression
whose graph is an ellipse. We wish to find the slopes of the tangent lines to the graph of this ellipse at the point x = 1.
a) Solve the above expression for the
variable y. How many solutions do
you obtain? Graph each of these
solutions separately. What do you
notice when these graphs are viewed together on the same set of coordinate
axes?
b) While the ellipse itself is not the
graph of a function (why?), each of the solutions for y you obtained in (a)
does represent a function (why?).
In order to obtain an answer to our original question regarding the
slopes of the tangent lines at x = 1, describe what you would do with the
functions obtained in (a), and carry out this operation.
2. Suppose we wish to find the slope formula for the tangent lines to the curve
Explain what happens when you attempt to solve this
equation for the variable y as you did in (1a) above.
Example:
In (1) the equation can be solved
'explicitly' for y and hence 
can be
determined explicitly. In
contrast, the equation in (2) cannot be solved explicitly for y. In this case, we can still find the
slope of the desired tangent line by using 'implicit' differentiation. Here is one way to accomplish this
'implicit' differentiation using Maple:
>yprime:=implicitdiff(y^5-x^2*y^3-32=0,
y, x);
By using the command above, you are asking Maple to find the derivative of y with respect to x.
3. For each of the following problems,
find 
implicitly. If possible, find 
explicitly as
well.
e) x sin(y) = 1
4. Find 
for the curves
in (3a) - (3c). What information
about the graph of each expression does 
convey?
5. Summarize what you have learned in
problems (1) - (4). What is the
role of implicit differentiation; i.e., what does it allow us to do in terms of
differentiation that we have been unable to do previously? Describe implicit differentiation from
a graphical viewpoint. If one can
find 
explicitly for
an expression, can one find 
implicitly as
well? What about vice versa?
6. Let m be a real number. Show that there are at most two points on the curve
where the tangent line has slope m. Interpret this geometrically. How are the points related?
7. a) Plot curve 1 and curve 2 together on the same set of axes.
Curve
1: 
Curve
2: 
To do this, use the 'implicitplot' command as follows:
>with(plots);
>implicitplot({3*x-2*y+x^3-x^2*y=0,
x^2-2*x+y^2-3*y=0},x=-5..5, y=-5..5);
b) Prove that the curves 1 and 2 are perpendicular at the origin (i.e., their tangent lines are perpendicular).