Question

1. the witch of agnesi the graph of $y = \frac{a^{3}}{x^{2}+a^{2}}$, where $a$ is a constant, is called the witch of agnesi (named after the 18th - century italian mathematician maria agnesi).

a. let $a = 3$ and find an equation of the line tangent to $y = \frac{27}{x^{2}+9}$ at $x = 2$.
b. plot the function and the tangent line found in part (a).

Explanation:

Step1: Find the derivative of the function

We use the quotient - rule. If $y=\frac{u}{v}$, then $y^\prime=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u = 27$, $u^\prime=0$, $v=x^{2}+9$, $v^\prime = 2x$. So $y^\prime=\frac{0\times(x^{2}+9)-27\times(2x)}{(x^{2}+9)^{2}}=-\frac{54x}{(x^{2}+9)^{2}}$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $y^\prime$. $y^\prime|_{x = 2}=-\frac{54\times2}{(2^{2}+9)^{2}}=-\frac{108}{169}$.

Step3: Find the $y$ - value when $x = 2$

Substitute $x = 2$ into $y=\frac{27}{x^{2}+9}$. $y=\frac{27}{4 + 9}=\frac{27}{13}$.

Step4: Use the point - slope form of a line

The point - slope form is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(2,\frac{27}{13})$ and $m =-\frac{108}{169}$.
$y-\frac{27}{13}=-\frac{108}{169}(x - 2)$
$y-\frac{27}{13}=-\frac{108}{169}x+\frac{216}{169}$
$y=-\frac{108}{169}x+\frac{216}{169}+\frac{27\times13}{13\times13}$
$y=-\frac{108}{169}x+\frac{216 + 351}{169}$
$y=-\frac{108}{169}x+\frac{567}{169}$

For part (b), to plot the function $y=\frac{27}{x^{2}+9}$ and the tangent line $y =-\frac{108}{169}x+\frac{567}{169}$, we can use graphing software like Desmos or a graphing calculator. We first input the function $y=\frac{27}{x^{2}+9}$, which is a bell - shaped curve. Then we input the tangent line equation $y =-\frac{108}{169}x+\frac{567}{169}$, which is a straight line that touches the curve at the point $(2,\frac{27}{13})$.

Answer:

a. The equation of the tangent line is $y=-\frac{108}{169}x+\frac{567}{169}$
b. Use graphing software or calculator to plot $y=\frac{27}{x^{2}+9}$ and $y=-\frac{108}{169}x+\frac{567}{169}$