Find the slope of the curve at the indicated point.
-Find an equation of the tangent to the curve $f ( x ) = \sqrt { x + 4 }$ that has slope $\frac { 1 } { 4 }$.
A) $y = \frac { 1 } { 4 } x$
B) $y = \frac { 1 } { 4 } x + 2$
C) $y = \frac { 1 } { 4 } x - 2$
D) $y = - \frac { 1 } { 4 } x + 2$

Question

Quizplus · Accepted Answer

To find the slope of the curve, we need to take the derivative of the function:
$$\frac{df}{dx}=\frac{1}{2\sqrt{x+4}}$$
At the point where the tangent has a slope of $\frac{1}{4}$, we have:
$$\frac{1}{2\sqrt{x+4}}=\frac{1}{4}$$
Solving for $x$, we get $x=36$. 
Plugging this value into the function, we get $f(36)=\sqrt{40}=2\sqrt{10}$.
So the point where the tangent has a slope of $\frac{1}{4}$ is $(36,2\sqrt{10})$.
Now we can use the point-slope form of a line to find the equation of the tangent:
$$y-2\sqrt{10}=\frac{1}{4}(x-36)$$
Simplifying, we get:
$$y=\frac{1}{4}x+2-\sqrt{10}$$
So the answer is (B).

Find the Slope of the Curve at the Indicated Point f(x)=x+4f ( x ) = \sqrt { x + 4 }f(x)=x+4​

Find the Slope of the Curve at the Indicated Point $f ( x ) = \sqrt { x + 4 }$