Question

this is section 2.5 problem 44: for the function f(x)=x^2 - \frac{1}{x}:(a) find the equation of the tangent line at the point x = - 1: y=-x + 1.(b) find the equation of the tangent line at the point x = 1: y=.(c) find the equation of the tangent line at the point x = 2: y=\frac{17}{4}x - 5. hint: follow example 6. resources ebook

Explanation:

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

First, rewrite $f(x)=x^{2}-x^{-1}$. Then, using the power - rule $(x^n)' = nx^{n - 1}$, we have $f'(x)=2x + x^{-2}=2x+\frac{1}{x^{2}}$.

Step2: Find the slope and point for $x = 1$

When $x = 1$, $f(1)=1^{2}-\frac{1}{1}=0$. The slope $m=f'(1)=2\times1+\frac{1}{1^{2}}=3$.

Step3: Use the point - slope form

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(1,0)$ and $m = 3$. So $y-0=3(x - 1)$, which simplifies to $y = 3x-3$.

Answer:

$y=3x - 3$