Question

The Cissoid of Diocles (dates from about 200 B.c.) (a) Find equations for the tangent and normal to the cissoid of Diocles, $$y^{2}(2-x)=x^{3}$$ at the point (1, 1) as pictured below. (b) Explain how to reproduce the graph on a grapher.

Short answer

The equation of the tangent to the cissoid of Diocles at the point (1,1) is y = x and the equation of the normal is y = 2 - x. To graph the cissoid of Diocles on a grapher, input the equation \(y^{2}(2-x)=x^{3}\) into the function input area, and plot the lines y = x and y = 2 - x as well.

Step-by-step solution

Derive the function

Given the equation \(y^{2}(2-x)=x^{3}\), we start by differentiating both sides with respect to x. Doing this gives us:2yy' (2-x) - y^2 = 3x^2. This is the equation for the slope of the tangent line to the curve at any point (x, y).

Calculate the derivative at (1,1)

Now we will evaluate the derivative at the point (1,1) which gives us a particular value. Substituting x = 1, y = 1 into the derivative equation we obtain, 2(2-1)-1 = 1. This is the slope of the tangent line at the point (1,1).

Find the equation of the tangent

The equation of the tangent line at (1,1) is then given by,(y - 1) = 1(x - 1), which simplifies to y = x.

Find the slope and equation of the normal

The slope of the normal is the negative reciprocal of the slope of the tangent. Therefore, in this case, the slope of the normal is -1. Hence, the equation of the normal line at (1,1) is then given by,(y - 1) = -1(x - 1), which simplifies to y = 2 - x.

Reproduce the graph on a grapher

To reproduce the graph of \(y^{2}(2-x)=x^{3}\) on a grapher, simply type or paste the equation into the function input area of any graphing software. Some software may require rearranging the equation into an explicit function of y. Also, many graphers will allow you to plot more than one function at once. So, you can also plot the equations y = x and y = 2 - x to see the tangent and normal lines at the point (1,1) respectively.