Find the equation of the tangent line to the graph of f at the given point.
-$$f ( x ) = - 4 x ^ { 2 } + 7 x \text { at } ( 5,65 )$$
A) $y = 2 x - 7$
B) $y = - 4 x + 7$
C) $y = - 33 x + 230$
D) $y = - 8 x + 7$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line, we first need to find the derivative of $f(x)$, which gives the slope of the tangent line at any point $x$. The derivative $f'(x) = -8x + 7$. Evaluating this derivative at $x = 5$ gives $f'(5) = -8(5) + 7 = -40 + 7 = -33$, which is the slope of the tangent line at the point (5, 65). To find the equation of the tangent line, we 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. Substituting $m = -33$, $x_1 = 5$, and $y_1 = 65$ gives $y - 65 = -33(x - 5)$. Simplifying, we get $y = -33x + 230$.

Find the Equation of the Tangent Line to the Graph f(x)=−4x2+7x at (5,65)f ( x ) = - 4 x ^ { 2 } + 7 x \text { at } ( 5,65 )f(x)=−4x2+7x at (5,65)

Find the Equation of the Tangent Line to the Graph $f ( x ) = - 4 x ^ { 2 } + 7 x \text { at } ( 5,65 )$