Sec.3.5 - Families of Implicit Curves

Applied Project in Sec.3.5, Calculus by Stewart
Chinese version 隱式曲線集合

In this project you will explore the changing shapes of implicitly defined curves as you vary the constants in a family, and determine which features are common to all members of the family.

Question 1

Consider the family of curves $$ y^2-2x^2(x+8)=c[(y+1)^2(y+9)-x^2] $$

(a) By graphing the curves with $c=0$ and $c=2$ , determine how many points of intersection there are. (You might have to zoom in to find all of them.)

(b) Now add the curves with $c=5$ and $c=10$ to your graphs in part (a) . What do you notice? What about other values of $c$?

Solution

(a) There appear to be nine points of intersection. The “inner four” near the origin are about $(\pm0.2, -0.9)$ and $(\pm0.3,-1.1)$. The “outer five” are about $(2.0,-8.9)$, $(-2.8,-8.8)$, $(-7.5,-7.7)$, $(-7.8,-4.7)$, and $(-8.0,1.5)$.

(b) We see from the graphs with $c=5$ and $c=10$, and for other values of $c$, that the curves change but the nine points of intersection are the same.

Question 2

(a) Graph several members of the family of curves $$ x^2+y^2+cx^2 y^2=1 $$ Describe how the graph changes as you change the value of $c$.

(b) What happens to the curve when $c=-1$? Describe what appears on the screen. Can you prove it algebraically?

(c) Find $y'$ by implicit differentiation. For the case $c=-1$, is your expression for $y'$ consistent with what you discovered in part (b)?

Solution

(a) If $c=0$, the graph is the unit circle. As $c$ increases, the graph looks more diamondlike and more crosslike (see the graph for $c\geq 0$)

For $-1<c<0$(see the graph), there are four hyperboliclike branches as well as an ellipticlike curve bounded by $|x|\leq1$ and $|y|\leq1$ for values of $c$ close to $0$. As $c$ gets closer to $-1$, the branches and the curve become more rectangular, approaching the lines $|x|=1$ and $|y|=1$.

For $c=-1$, we get the lines $x=\pm1$ and $y=\pm1$. As $c$ decreases, we get four test-tubelike curves (see the graph) that are bounded by $|x|=1$ and $|y|=1$, and get thinner as $|c|$ gets larger.