14. [-/2 Points] DETAILS MY NOTES SCALCET9 3.5.037, A graphing calculator is recommended. (a) The curve with equation $$ y^{2}=5 x^{4}-x^{2} $$ is called a kampyle of Eudowus. Find an equation of the tangent line to this graph at the point ( -1 , 2), $$ y=\square $$ (b) Illustrate part (a) by graphing the curve and the tangent line on a commen screen. (Graph the implicitly defined curve if possible, of you can graph the upper and lower halves separately.)

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**Step 1. Differentiate the equation implicitly**

The given curve is
$$
y^2 = 5x^4 - x^2.
$$
Differentiate both sides with respect to $$x$$:
$$
2y \frac{dy}{dx} = 20x^3 - 2x.
$$
Solving for $$\frac{dy}{dx}$$ gives
$$
\frac{dy}{dx} = \frac{20x^3 - 2x}{2y} = \frac{10x^3 - x}{y}.
$$

**Step 2. Evaluate the derivative at the point $$(-1, 2)$$**

Substitute $$x = -1$$ and $$y = 2$$:
$$
\frac{dy}{dx}\Big|_{(-1,2)} = \frac{10(-1)^3 - (-1)}{2} = \frac{10(-1) + 1}{2} = \frac{-10+1}{2} = \frac{-9}{2}.
$$
Thus, the slope of the tangent line is
$$
m = -\frac{9}{2}.
$$

**Step 3. Write the equation of the tangent line**

Using the point-slope form with point $$(-1,2)$$:
$$
y - 2 = -\frac{9}{2}(x + 1).
$$
Solving for $$y$$:
$$
y = -\frac{9}{2}(x+1) + 2.
$$
Expanding,
$$
y = -\frac{9}{2}x - \frac{9}{2} + 2 = -\frac{9}{2}x - \frac{9}{2} + \frac{4}{2} = -\frac{9}{2}x - \frac{5}{2}.
$$

**Final Answer for (a):**

$$
y = -\frac{9}{2}x - \frac{5}{2}
$$

**Step 4. Graphing Instructions for (b)**

To illustrate the result:
1. Graph the curve defined by
   $$
   y^2 = 5x^4 - x^2.
   $$
   Since the curve is defined implicitly, you may graph the upper half $$ y = \sqrt{5x^4 - x^2} $$ and the lower half $$ y = -\sqrt{5x^4 - x^2} $$ separately.
2. Graph the tangent line
   $$
   y = -\frac{9}{2}x - \frac{5}{2}.
   $$
3. Verify that the tangent line touches the curve at the point $$(-1, 2)$$.

By following these steps on your graphing calculator or graphing software, you will see the tangent clearly touching the curve at $$(-1, 2)$$.
**Part (a): Equation of the Tangent Line** The equation of the tangent line to the curve $$ y^2 = 5x^4 - x^2 $$ at the point $$(-1, 2)$$ is: $$ y = -\frac{9}{2}x - \frac{5}{2} $$ **Part (b): Graphing Instructions** 1. **Graph the Curve:** - Plot the upper half: $$ y = \sqrt{5x^4 - x^2} $$ - Plot the lower half: $$ y = -\sqrt{5x^4 - x^2} $$ 2. **Graph the Tangent Line:** - Plot $$ y = -\frac{9}{2}x - \frac{5}{2} $$ 3. **Verify the Tangent Point:** - Ensure the tangent line touches the curve at $$(-1, 2)$$. By following these steps, you'll see the tangent line intersecting the curve at the specified point.

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