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March 20th, 2016, 02:30 AM  #1 
Senior Member Joined: Oct 2014 From: Complex Field Posts: 119 Thanks: 4  How to find the following tangent line
Hello, I am given f(x)=3sqrt(x). So I know that f'(x)=(3/(2sqrt(x))), I need to find the tangent to f(x) that goes through (3,3).. I know that y=mx+n, and I have a point and the tangent, but I seem to be missing a detail to solve it. Any ideas? Thanks 
March 20th, 2016, 08:41 AM  #2 
Senior Member Joined: Dec 2015 From: holland Posts: 149 Thanks: 36 Math Focus: tetration 
"https://www.desmos.com/calculator/lsfe9wa7a8"> y = 1.5x + 1.5
Last edited by manus; March 20th, 2016 at 09:31 AM. 
March 20th, 2016, 01:06 PM  #3 
Senior Member Joined: Oct 2014 From: Complex Field Posts: 119 Thanks: 4 
Thanks, this is the correct answer, but what is the way?

March 20th, 2016, 01:49 PM  #4 
Math Team Joined: Nov 2014 From: Australia Posts: 672 Thanks: 239 
Let $(x_0,3\sqrt{x_0})$ be the point where the tangent line touches the graph. The gradient of this line is then given by $f'(x_0) = \dfrac{3}{2\sqrt{x_0}}$. However, the gradient is also given by $\dfrac{3\sqrt{x_0} + 3}{x_0 + 3}$ (can you think why?). Now you have the equation $\dfrac{3}{2\sqrt{x_0}} = \dfrac{3\sqrt{x_0} + 3}{x_0 + 3}$ solve for $x_0$, then find the gradient. After that, you have all the information you need. 
March 23rd, 2016, 10:31 AM  #5 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,714 Thanks: 697 
The function is $\displaystyle f(x)= 3\sqrt{x}$ and the derivative is $\displaystyle f'(x)= \frac{3}{2\sqrt{x}}$ as said. So any tangent to the curve at $\displaystyle \left(x_0, 3\sqrt{x_0}\right)$ is of the form $\displaystyle y= \frac{3}{2\sqrt{x_0}}\left(x x_0\right)+ 3\sqrt{x_0}$. Since that tangent line must go through (3, 3), we must have $\displaystyle \frac{3}{2\sqrt{x_0}}\left(3 x_0\right)+ 3\sqrt{x_0}= 3$. Solve that equation for $\displaystyle x_0$.


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