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 (correct). (b) find the equation of the tangent line at the point x = 1: y=x + 1 (incorrect). (c) find the equation of the tangent line at the point x = 2: y= (blank). 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 $\frac{d}{dx}(x^{n})=nx^{n - 1}$, we have $f^\prime(x)=2x+x^{-2}=2x+\frac{1}{x^{2}}$.

Step2: Evaluate $f(2)$ and $f^\prime(2)$

Calculate $f(2)$: $f(2)=2^{2}-\frac{1}{2}=4 - \frac{1}{2}=\frac{7}{2}$.
Calculate $f^\prime(2)$: $f^\prime(2)=2\times2+\frac{1}{2^{2}}=4+\frac{1}{4}=\frac{17}{4}$.

Step3: Use the point - slope form $y - y_{1}=m(x - x_{1})$

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(2,\frac{7}{2})$ and $m = f^\prime(2)=\frac{17}{4}$.
Substitute these values: $y-\frac{7}{2}=\frac{17}{4}(x - 2)$.
Expand the right - hand side: $y-\frac{7}{2}=\frac{17}{4}x-\frac{17}{2}$.
Add $\frac{7}{2}$ to both sides: $y=\frac{17}{4}x-\frac{17}{2}+\frac{7}{2}=\frac{17}{4}x - 5$.

Answer:

$y=\frac{17}{4}x - 5$