Find an equation for the tangent to the curve at the given point.
-$$h ( x ) = t ^ { 3 } - 25 t - 1 , ( 5 , - 1 )$$
A) $y = 50 t - 251$
B) $y = 49 t - 251$
C) $y = - 1$
D) $y = 50 t - 1$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line to the curve at the given point, we first need to find the derivative of $h(t)$ with respect to $t$, which gives us the slope of the tangent line at any point $t$. The derivative of $h(t) = t^3 - 25t - 1$ is $h'(t) = 3t^2 - 25$. Substituting $t = 5$ into $h'(t)$ gives us the slope of the tangent line at $t = 5$: $h'(5) = 3(5)^2 - 25 = 75 - 25 = 50$. The slope of the tangent line is 50, and we are given a point on the line $(5, -1)$. We can use the point-slope form of a line equation, $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line. Substituting $m = 50$, $x_1 = 5$, and $y_1 = -1$ gives us:$$y - (-1) = 50(t - 5)$$$$y + 1 = 50t - 250$$$$y = 50t - 251$$Therefore, the correct equation of the tangent line is $y = 50t - 251$, which corresponds to choice A.

Find an Equation for the Tangent to the Curve at the Given