Chapter 4.2, Problem 45E

a.

To determine

To find: the slope of the given figure−eight−shaped curve $y^4=y^2-x^2$ at the points shown on the graph that follows.

Answer to Problem 45E

  $\begin{array}{l}\frac{dy}{dx}{)}_{\frac{\sqrt{3}}{4},\frac{\sqrt{3}}{2}}=-1.\\ $ {(\frac{dy}{dx})}_{\frac{\sqrt{3}}{4},\frac{1}{2}}=\sqrt{3.}$\end{array}$

Explanation of Solution

Given information: The given figure−eight−shaped curve $y^4=y^2-x^2$ is shown below.

  

Calculation:

The given equation of the curve is ,

  $y^4=y^2-x^2$ .

Differentiate with respect to x.

  $\begin{array}{l}\frac{d}{dx}(y^4)=\frac{d}{dx}(y^2-x^2).\\ 4y^3\frac{dy}{dx}=2y\frac{dy}{dx}-2x\frac{dx}{dx},\frac{dx}{dx}=1.\\ 4y^3\frac{dy}{dx}-2y\frac{dy}{dx}=-2x.\\ (4y^3-2y)\frac{dy}{dx}=-2x.\\ \frac{dy}{dx}=\frac{-2x}{(4y^3-2y)}\mathrm{...............................}(1)\\ \text{putting}x=\frac{\sqrt{3}}{4}\text{and}y=\frac{\sqrt{3}}{2}.\\ $ {\left(\frac{dy}{dx}\right)}_{\frac{\sqrt{3}}{4}\cdot \frac{\sqrt{3}}{2}}=\frac{-2\left(\frac{\sqrt{3}}{4}\right)}{4{\left(\frac{\sqrt{3}}{2}\right)}^3-2\left(\frac{\sqrt{3}}{2}\right)}=\frac{-\left(\frac{\sqrt{3}}{2}\right)}{\frac{\sqrt{3}}{2}}=-1$$ \text{putting}x=\frac{\sqrt{3}}{4}\text{and}y=\frac{1}{2}.$$ {\left(\frac{dy}{dx}\right)}_{\frac{\sqrt{3}}{4}\cdot \frac{1}{2}}=\frac{-2\left(\frac{\sqrt{3}}{4}\right)}{4{\left(\frac{1}{2}\right)}^3-2\left(\frac{1}{2}\right)}=\frac{-\left(\frac{\sqrt{3}}{2}\right)}{\frac{-1}{2}}=\sqrt{3}$\end{array}$

b.

To determine

To graph: the curve using parametric mode and the two pairs of parametric equations. Specify a window and a parametric interval.

Answer to Problem 45E

Parametric interval is $0\le t\le 1$ and Window is square window.

Explanation of Solution

Given information: Given parametric equations are ,

  $\begin{array}{l}{x}_1(t)=\sqrt{t^2-t^4},y_1(t)=t,\\ x_2(t)=-\sqrt{t^2-t^4},y_2(t)=t,\end{array}$

Calculation:

Given parametric equations are ,

  $\begin{array}{l}{x}_1(t)=\sqrt{t^2-t^4},y_1(t)=t,\\ x_2(t)=-\sqrt{t^2-t^4},y_2(t)=t,\end{array}$

The graph of given parametric equations is shown below.

  

Parametric interval is $0\le t\le 1$ and Window is square window.