Find an equation of the tangent line to the graph of f at the given point. $$f ( t ) = ( t - 5 ) \left( t ^ { 2 } - 3 \right) , \text { at } ( 2 , - 3 )$$
A) $y = 11 t + 19$
B) $y = - 11 + 19 t$
C) $y = - 11 t + 19$
D) $y = 25 t - 11$
E) $y = - 11 t - 25$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line to the graph of $f$ at the point $(2, -3)$, we first need to find the derivative of $f(t)$, which gives us the slope of the tangent line at any point $t$. The derivative of $f(t) = (t - 5)(t^2 - 3)$ is found using the product rule:$$f'(t) = (t - 5) \cdot 2t + (t^2 - 3) \cdot 1 = 2t^2 - 10t + t^2 - 3 = 3t^2 - 10t - 3$$Evaluating this derivative at $t = 2$ gives the slope of the tangent line at the point $(2, -3)$:$$f'(2) = 3(2)^2 - 10(2) - 3 = 12 - 20 - 3 = -11$$The slope of the tangent line is $-11$. 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:$$y - (-3) = -11(t - 2)$$Simplifying this equation gives:$$y + 3 = -11t + 22$$$$y = -11t + 19$$Therefore, the correct equation of the tangent line is $y = -11t + 19$, which corresponds to choice C.

Find an Equation of the Tangent Line to the Graph f(t)=(t−5)(t2−3), at (2,−3)f ( t ) = ( t - 5 ) \left( t ^ { 2 } - 3 \right) , \text { at } ( 2 , - 3 )f(t)=(t−5)(t2−3), at (2,−3)

Find an Equation of the Tangent Line to the Graph $f ( t ) = ( t - 5 ) \left( t ^ { 2 } - 3 \right) , \text { at } ( 2 , - 3 )$