009A Sample Midterm 3, Problem 4

Find the equation of the tangent line to $y=3{\sqrt {-2x+5}}$ at $(-2,9).$

Foundations:
Equation of a tangent line
The equation of the tangent line to $f(x)$ at the point $(a,b)$ is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=m(x-a)+b} where $m=f'(a).$

Solution:

Step 1:
First, we need to calculate the slope of the tangent line.
Let $f(x)=3{\sqrt {-2x+5}}.$
From Problem 3, we have
$f'(x)={\frac {-3}{\sqrt {-2x+5}}}.$
Therefore, the slope of the tangent line is
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{array}{rcl}\displaystyle {m}&=&\displaystyle {f'(-2)}\\&&\\&=&\displaystyle {\frac {-3}{\sqrt {-2(-2)+5}}}\\&&\\&=&\displaystyle {\frac {-3}{\sqrt {9}}}\\&&\\&=&\displaystyle {-1.}\end{array}}}

Step 2:
Now, the tangent line has slope $m=-1$
and passes through the point $(-2,9).$
Hence, the equation of the tangent line is
$y=-1(x+2)+9.$

Final Answer:
$y=-1(x+2)+9$

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