p. 106. An equation in which y is not ex pressed explicitly
in terms of x may determine one or
more functions . Any such function, where y is defined implicitly as a function
of x, is called an
implicit function.
A good example is the relation in x and y which defines
the equation of a circle such as
x2 + y 2= 9. We know that this relation is not a function
since a circle fails the vertical line test .
Furthermore, when we solve for y in terms of x we get two
functions , 
.
We may enter these into the calculator as Y 1= √(9 – X^2) and Y2=- √(9 – X^2).
This is an alternative method of graphing a circle . Previously we had used the
parametric
equations x = 3cos t and y = 3sin t .
We know how to take the derivative of each of the two
explicit functions. For Y1 the derivative
is
. Implicit differentiation often al lows us
to find the
derivative more easily. We will use implicit differentiation to find 
when x2 + y 2= 9 .
First take the derivative with respect to x of each term and apply the chain
rule:
Check with
the result above to see that they are equivalent .
Reality check : Find
for each of these points on the graph of x2 + y 2= 9
and verify visually
that the result is plausible:
Now let’s find the second derivative for x2
+ y 2= 9 and use it to locate where the circle is
concave up and where it is concave down.
Apply the quotient rule to 
Thus when y > 0, 
and
the graph is concave down. When y < 0, 
and
the graph
is concave up.
Study these examples in the book:
Example 2, page 106. This is a conic section . Can you identify which type?
Example 3, page 107. Note use of the power rule .
Example 2, page 117, to find the equation of a tangent line.
Example 4, page 110 to find a second derivative. Note the use of the notation y’
and y’’.
Exercises page 108: 3, 5, 7, 9, 11, 21, 27, 29
Page 111: 25, 27 page 119: 8 and 12
