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

To find the equation of the tangent line to the graph of $f(x) = x^2 + 2x$ at $x = 7$, we first need to find the derivative of $f(x)$, which gives us the slope of the tangent line at any point $x$. The derivative $f'(x) = 2x + 2$. Evaluating this at $x = 7$ gives $f'(7) = 2(7) + 2 = 14 + 2 = 16$, so the slope of the tangent line is 16. The point on the graph at $x = 7$ is $f(7) = 7^2 + 2(7) = 49 + 14 = 63$. Therefore, 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