Question

(a) The curve with the equation \({y^2} = 5{x^4} - {x^2}\)is called akampyle of Eudoxus. Find and equation of the tangent line to this curve at the point\(\left( {1,2} \right)\)

(b) Illustrate part\(\left( a \right)\)by graphing the curve and the tangent line on a common screen. (If your graph device will graph implicitly defined curves, then use that capability. If not, you can still graph this curve by graphing its upper and lower halves separately.)

Short answer

1. The equation of the tangent line to the curve\({y^2} = 5{x^4} - {x^2}\)is\(y = \frac{9}{2}x - \frac{5}{2}\,\,{\rm{or,}}\,\,2y = 9x - 5\).
2. The graph of part (a) representing the curve and the tangent line is

Step-by-step solution

Definition of implicit differentiation

In implicit differentiation, we separate each side of an equation with two variables by treating one of the variables as a function of the other.

Given Parameters

Given the equation of curve as \({y^2} = 5{x^4} - {x^2}\).

The equation of the tangent line to the curve is to be found and also a graph is to be drawn that is represents the curve and the tangent line.

(a) Simplify the given equation of curve.

Given equation:\({y^2} = 5{x^4} - {x^2}\)

Find the square root on both sides of equation\({y^2} = 5{x^4} - {x^2}\)

\(\begin{array}{c}\sqrt {{y^2}} = \sqrt {5{x^4} - {x^2}} \\y = \sqrt {5{x^4} - {x^2}} \end{array}\)

Further solve to get simplified.

\(y = x\sqrt {5{x^2} - 1} \)

Differentiate the equation \((1)\) to get the value of \(y'\)

Differentiate the equation\(y = x\sqrt {5{x^2} - 1} \)with respect to\(x\).

\(\begin{array}{c}y' = \sqrt {5{x^2} - 1} + \frac{{x \times 10x}}{{2\sqrt {5{x^2} - 1} }}\\ = \sqrt {5{x^2} - 1} + \frac{{x \times 5x}}{{\sqrt {5{x^2} - 1} }}\\ = \frac{{5{x^2} - 1 + 5{x^2}}}{{\sqrt {5{x^2} - 1} }}\\ = \frac{{10{x^2} - 1}}{{\sqrt {5{x^2} - 1} }}\end{array}\)

Let the equation of tangent be\(y = mx + c\) …… (2)

where\(m\)is slope of line which can be calculated by differentiating the given equation of curve and\(x,y\)are coordinates of point where the tangent line intersects the curve.

\(\begin{array}{c}{\rm{m = y'}}\\{\rm{ = }}\frac{{{\rm{10}}{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}}}{{\sqrt {{\rm{5}}{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}} }}\\{{\rm{m}}_{{\rm{x = 1}}}}{\rm{ = }}\frac{{{\rm{10 \times 1 - 1}}}}{{\sqrt {{\rm{5 - 1}}} }}\\{\rm{ = }}\frac{{\rm{9}}}{{\rm{2}}}\end{array}\)

and\(x,y = 1,2\)

Substitute the above values in equation (2).

\(\begin{array}{l}2 = \frac{9}{2} + c\\c = - \frac{5}{2}\end{array}\)

The required equation of tangent line is \(y = \frac{9}{2}x - \frac{5}{2}\,\,{\rm{or,}}2y = 9x - 5\)

(b) Plot the scenario in part (a) using graphing calculator and represent the given curve and tangent line.

1. Therefore, for part (a) The equation of the tangent line to the curve \({y^2} = 5{x^4} - {x^2}\)is\(y = \frac{9}{2}x - \frac{5}{2}\,\,{\rm{or,}}2y = 9x - 5\)

And part (b) The graph of part (a) representing the curve and the tangent line is