Find the equation of the tangent line to the curve when x has the given value.
-$$f ( x ) = \sqrt { x } ; x = 25$$
A) $y = \frac { 1 } { 10 } x + 4$
B) $y = - \frac { 1 } { 10 } x + \frac { 5 } { 2 }$
C) $y = \frac { 1 } { 10 } x + \frac { 5 } { 2 }$
D) $y = \frac { 1 } { 10 } x - \frac { 5 } { 2 }$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line to the curve at $x = 25$, we first find the derivative of $f(x) = \sqrt{x}$, which is $f'(x) = \frac{1}{2\sqrt{x}}$. Evaluating this derivative at $x = 25$ gives $f'(25) = \frac{1}{2\sqrt{25}} = \frac{1}{10}$, which is the slope of the tangent line. The point on the curve at $x = 25$ is $(25, \sqrt{25}) = (25, 5)$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line, we get $y - 5 = \frac{1}{10}(x - 25)$. Simplifying this equation gives $y = \frac{1}{10}x + \frac{5}{2}$, which matches choice C.

Find the Equation of the Tangent Line to the Curve f(x)=x;x=25f ( x ) = \sqrt { x } ; x = 25f(x)=x​;x=25

Find the Equation of the Tangent Line to the Curve $f ( x ) = \sqrt { x } ; x = 25$