Find the equation of the tangent line to the curve when x has the given value.
-$$f ( x ) = x ^ { 2 } + 5 x ; x = 4$$
A) $y = - 39 x - 80$
B) $y = - \frac { 4 x } { 25 } + \frac { 8 } { 5 }$
C) $y = \frac { x } { 20 } + \frac { 1 } { 5 }$
D) $y = 13 x - 16$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line to the curve at $x = 4$, first, find the derivative of $f(x)$ to get the slope of the tangent line. The derivative $f'(x) = 2x + 5$. Then, plug in $x = 4$ to find the slope at that point: $f'(4) = 2(4) + 5 = 13$. The slope of the tangent line is 13. Next, use 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. Here, $x_1 = 4$ and $y_1 = f(4) = 4^2 + 5(4) = 16 + 20 = 36$. So, the equation is $y - 36 = 13(x - 4)$, which simplifies to $y = 13x - 52 + 36$, or $y = 13x - 16$.

Find the Equation of the Tangent Line to the Curve f(x)=x2+5x;x=4f ( x ) = x ^ { 2 } + 5 x ; x = 4f(x)=x2+5x;x=4

Find the Equation of the Tangent Line to the Curve $f ( x ) = x ^ { 2 } + 5 x ; x = 4$