Chapter 3.3, Problem 28E

To determine

The equation for the line tangent to the curve at the given point.

Answer to Problem 28E

The required equation is $y=2x+4$ .

Explanation of Solution

Given information:

The curve: $y=\frac{x^4+2}{x^2}$

The point: $x=-1$

Formula used:

Differentiation formula:

  $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\prime }v-v^{\prime }u}{v^2}$

Point-slope form of the equation of a line:

  $y-y_1=m\left(x-x_1\right)$

Calculation:

The curve is $y=\frac{x^4+2}{x^2}$ .

And, the point is $x=-1$ .

Substitute $-1$ for x in the given curve.

  $\begin{array}{c}y=\frac{{\left(-1\right)}^4+2}{{\left(-1\right)}^2}\\ =\frac{1+2}{1}\\ =\frac{3}{1}\\ =3\end{array}$

So, the point through which the tangent line passes is $\left(-1,3\right)$ .

To find the slope of the tangent line to the given curve, differentiate the above curve.

Use the differentiation formula $\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u^{\prime }v-v^{\prime }u}{v^2}$ .

  $\begin{array}{c}\frac{dy}{dx}=\frac{d}{dx}\left(\frac{x^4+2}{x^2}\right)\\ =\frac{x^2\frac{d}{dx}\left(x^4+2\right)-\left(x^4+2\right)\frac{d}{dx}\left(x^2\right)}{{\left(x^2\right)}^2}\\ =\frac{x^2\left(4x^3\right)-\left(x^4+2\right)2x}{x^4}\\ =\frac{4x^5-2x^5-4x}{x^4}\end{array}$

Simplify the expression.

  $\begin{array}{l}\frac{dy}{dx}=\frac{4x^5-2x^5-4x}{x^4}\\ \frac{dy}{dx}=\frac{2x^5-4x}{x^4}\\ \frac{dy}{dx}=\frac{2x^5}{x^4}-\frac{4x}{x^4}\\ \frac{dy}{dx}=2x-4x^{-3}\end{array}$

Substitute $-1$ for x in the above derivative.

  $\begin{array}{c}\frac{dy}{dx}=2\left(-1\right)-4{\left(-1\right)}^{-3}\\ =-2-\frac{4}{{\left(-1\right)}^3}\\ =-2+4\\ =2\end{array}$

Thus, the slope of the tangent line is 2.

Now, to find the equation of the tangent line use the point-slope form of an equation of a line $y-y_1=m\left(x-x_1\right)$ .

Substitute $-1$ for $x_1$ , 3 for $y_1$ and 2 for m in the above formula.

  $\begin{array}{c}y-3=2\left(x-\left(-1\right)\right)\\ y-\cancel{3}+\cancel{3}=2x+1+3\\ y=2x+4\end{array}$

Hence, the required equation is $y=2x+4$ .