Question

a graphing calculator is recommended.
the graph of the function (f(x)=\frac{x}{1 + x^{2}}) is called a serpentine.
(a) find an equation of the tangent line to this curve at the point ((2,0.40)).
(y=)
(b) illustrate part (a) by graphing the curve and the tangent line on the same screen.

Explanation:

Step1: Find the derivative of $f(x)$

Use the quotient - rule. If $f(x)=\frac{u}{v}$ where $u = x$ and $v=1 + x^{2}$, then $f^\prime(x)=\frac{u^\prime v - uv^\prime}{v^{2}}$. Here, $u^\prime=1$ and $v^\prime = 2x$. So $f^\prime(x)=\frac{1\cdot(1 + x^{2})-x\cdot(2x)}{(1 + x^{2})^{2}}=\frac{1 + x^{2}-2x^{2}}{(1 + x^{2})^{2}}=\frac{1 - x^{2}}{(1 + x^{2})^{2}}$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $f^\prime(x)$. $f^\prime(2)=\frac{1-2^{2}}{(1 + 2^{2})^{2}}=\frac{1 - 4}{(1 + 4)^{2}}=\frac{-3}{25}=-0.12$. This is the slope $m$ of the tangent line at the point $(2,0.40)$.

Step3: Use the point - slope form of a line

The point - slope form is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(2,0.40)$ and $m=-0.12$. So $y-0.40=-0.12(x - 2)$.

Step4: Simplify the equation

$y-0.40=-0.12x + 0.24$, then $y=-0.12x+0.64$.

Answer:

$y=-0.12x + 0.64$