Equations involving `x` and `y` define curves in the `xy`-plane. For example,
the graph of the equation `(x^2)/(3^2) + (y^2)/(4^2) = 1` is the set of all
points `(s, t)` in the plane such that substituting `s` for `x` and `t` for `y`
in the equation produces a true statement. In the applet, make `a = 3`, `b =4`,
and `c = 0` to see what this curve looks like. By varying the parameters,
we can see a family of curves, all of which go through the four points `(+-a, 0)`
and `(0, +-b)`. Given a curve, how can we find the slope of a line tangent to
a point on the curve?
- Circle. Make `a = 5`, `b = 5`, and `c = 0`. Notice that the point
`(4,3)` is on the circle. What is the slope of the line tangent to the circle
- Ellipse. Make `a = 4`, `b = 4`, and `c = -0.1`
(you can fine-tune `c` by clicking the dot on the slider then using
the left and right arrow keys).
- What is the slope of the line through `(0, 4)`? (The applet shows the
answer to be `4/5`, but use implicit differentiation to determine
whether or not this is the exact answer or an approximation.)
- Find the two points where the tangent line is horizontal.
- Parallel(?) Lines. The graph of the equation `x^2 - xy + 1/4 y^2 = 1`
appears to be a pair of parallel lines. See this by setting `a = 1`, `b = 2`, and `c = -1`.
Can you prove that these lines are really parallel, or show that they are not?
Use implicit differentiation to find the slope of the curve
`(x^2)/(a^2) + cxy + (y^2)/(b^2) = 1` at any point `(x, y)` on the curve. Keep
`a`, `b`, and `c` as parameters, so you'll treat them like numbers as you differentiate.
Enter your answer, in terms of `a`, `b`, `c`, `x`, and `y`:
When you click submit, you'll see the value of your expression in the
bottom left corner of the applet. The `x` and `y`
values are the `x` and `y` values of the point on the curve. See how the value of
your expression compares to the value of the slope computed by the applet.