Find the equation of the tangent line to the graph of the given function at the point with $x = 4$ . $f ( x ) = \frac { x + 4 } { x + 1 }$
A) $y = - 0.12 x$
B) $y = 0.12 x + 1.12$
C) $y = - 1.6$
D) $y = - 0.12 x + 2.08$
E) $y = 1.6$

Question

Find the equation of the tangent line to the graph of the given function at the point with $x = 4$ .  $f ( x ) = \frac { x + 4 } { x + 1 }$  
A) $y = - 0.12 x$
B) $y = 0.12 x + 1.12$
C) $y = - 1.6$
D) $y = - 0.12 x + 2.08$
E) $y = 1.6$

Quizplus · Accepted Answer

To find the equation of the tangent line, we first need to find the derivative of the function $f(x) = \frac{x + 4}{x + 1}$, which gives us the slope of the tangent line at any point $x$. The derivative, using the quotient rule, is $f'(x) = \frac{(x + 1)(1) - (x + 4)(1)}{(x + 1)^2} = \frac{-3}{(x + 1)^2}$. Evaluating this derivative at $x = 4$ gives $f'(4) = \frac{-3}{(4 + 1)^2} = \frac{-3}{25} = -0.12$, which is the slope of the tangent line at $x = 4$. The function value at $x = 4$ is $f(4) = \frac{4 + 4}{4 + 1} = \frac{8}{5} = 1.6$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -0.12$, $x_1 = 4$, and $y_1 = 1.6$, we get $y - 1.6 = -0.12(x - 4)$. Simplifying, $y = -0.12x + 1.6 + 0.48 = -0.12x + 2.08$.

Find the Equation of the Tangent Line to the Graph x=4x = 4x=4

Find the Equation of the Tangent Line to the Graph $x = 4$