Find the equation of the tangent line to the curve when x has the given value.
-$$f ( x ) = - 7 - x ^ { 2 } ; x = 1$$
A) $y = - 2 x + - 6$
B) $y = 1 x + - 6$
C) $y = 2 x - - 6$
D) $y = - 2 x$

Question

Quizplus · Accepted Answer

To find the equation of the tangent line to the curve at $x = 1$, first find the derivative of $f(x) = -7 - x^2$, which is $f'(x) = -2x$. Then, evaluate $f'(1) = -2(1) = -2$ to get the slope of the tangent line. The y-coordinate at $x = 1$ is $f(1) = -7 - (1)^2 = -8$. Using the point-slope form of a line, $y - y_1 = m(x - x_1)$, with $m = -2$, $x_1 = 1$, and $y_1 = -8$, we get $y - (-8) = -2(x - 1)$, simplifying to $y = -2x - 6$.

Find the Equation of the Tangent Line to the Curve f(x)=−7−x2;x=1f ( x ) = - 7 - x ^ { 2 } ; x = 1f(x)=−7−x2;x=1

Find the Equation of the Tangent Line to the Curve $f ( x ) = - 7 - x ^ { 2 } ; x = 1$