Tangent at a given point

Question

I have a function $\frac{x^2+2}{x-1}$. I want to find a tangent at a given point of $x=1+\sqrt{3}$. At first, I found a value of the function at given point. This is what I got: $2+\frac{6}{\sqrt{3}}$ Then, I found a derivative of this point and I think it's $0$. So, for me the tangent is $y=(2+\frac{6}{\sqrt{3}})$ but, the proper result is $-2\sqrt{3}-2$. What am I doing wrong?

Answer 1

$$f(x)=\frac{x^2+2}{x-1}\to f'(x)=\frac{x^2-2x-2}{(x-1)^2}=1-\frac{3}{(x-1)^2}\to f'(1+\sqrt{3})=0$$

So the tangent line is given by

$$y=f(1+\sqrt{3})=\frac{(1+\sqrt{3})^2+2}{\sqrt{3}}=2+\frac{6}{\sqrt{3}}=2+2\sqrt{3}$$

Answer 2

Note that $\frac{6}{\sqrt3} = \frac{(2)(3)}{\sqrt3} = 2\sqrt3$

So your $2 + \frac{6}{\sqrt3} = 2\sqrt3 + 2$

So you agree with the answer key with the exception of a sign.

Calculating mentally I agree with your derivative of zero (but yes you should show your work -- easy enough to slip up, especially when I do a mental check.)

So you have to track a sign error. Did you mis-copy, for example was it 1 - x rather than x - 1? Or was there a negative sign in front of the whole fraction? Or did you work this out from a previous problem and maybe lose a sign there?

If, after you have triple-checked everything, a sign error remains, it is possible (rare but possible) that your answer key is in error.

Answer 3

Your work seems correct.

The $\color{red}{\text{red}}$ line is your function and the $\color{blue}{\text{blue}}$ line is the tangent line you've derived. The $\color{orange}{\text{orange}}$ line is the tangent line given by your answer key.

Also, note that your tangent line can be simplified to: $$y=2+\frac{6}{\sqrt{3}}=2+\frac{2\cdot 3}{\sqrt{3}}=2+2\sqrt{3}$$

Answer 4

The derivative of a function $f$ at a point $x_0$ is the slope of the tangent line to $f$ at $x_0.$

So, let $f(x)=\frac{x^2+2}{x-1}$ and $x_0=1+\sqrt{3}.$ $$\Rightarrow f'(x)=\frac{(x-1)\cdot 2x - (x^2+2)\cdot 1}{(x-1)^2}=\frac{x^2-2x-2}{(x-1)^2}.$$ and $$f'(x_0)=f'(1+\sqrt{3})=\frac{(1+\sqrt{3})^2-2(1+\sqrt{3})-2}{(1+\sqrt{3}-1)^2}=0.$$ Now, to get the equation of the line we have $$y-f(x_0)=f'(x_0)(x-x_0)$$ but $f'(x_0)=0$ so we have $$y=f(x_0).$$ $$\Rightarrow y=f(x_0)=f(1+\sqrt{3})=\frac{(1+\sqrt{3})^2+2}{(1+\sqrt{3})-1}=\frac{6+2\sqrt{3}}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{6\sqrt{3}+6}{3}=2\sqrt{3}+2.$$ Therefore, $$y=2\sqrt{3}+2.$$