The equation of the line tangent to the graph of $f ( x ) = x ^ { 2 } + 2 x$ at x = 7 is
A) y = 16x - 7
B) y = 16x - 49
C) y = 16x - 343
D) y = 16x - 686

Question

Quizplus · Accepted Answer

The slope of the tangent line to the graph of $f(x) = x^2 + 2x$ at any point x is given by the derivative of $f(x)$, which is $f'(x) = 2x + 2$. At $x = 7$, the slope is $f'(7) = 2(7) + 2 = 14 + 2 = 16$. The point on the graph at $x = 7$ is $f(7) = 7^2 + 2(7) = 49 + 14 = 63$. The equation of the tangent line can be found using the point-slope form: $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is the point on the graph. Substituting $m = 16$, $x_1 = 7$, and $y_1 = 63$ gives $y - 63 = 16(x - 7)$. Simplifying, we get $y = 16x - 112 + 63$, which simplifies further to $y = 16x - 49$.

The Equation of the Line Tangent to the Graph Of f