Question

Find the equation of the tangent line to the curve f(x) = 3x^3 + 2 at the point (1, 5).

Answer 1

First take the derivative of the function, and plug in the x coordinate of the point. that will tell you the slope of the line:
\[f(x) = 3x^3+2 \Rightarrow f'(x) = 9x^2 \Rightarrow f'(1) = 9(1)^2 = 9\]
So the slope of the tangent line at (1, 5) is 9.
Then use the point slope equation of a line to get the equation:
\[y-y_1 = m(x-x_1), x_1=1,y_1=5,m=9\]
this gives:
\[y-5=9(x-1)\]
If you wish to simplify, it would give:
\[y-5=9(x-1) \Rightarrow y-5=9x-9\Rightarrow y=9x-4\]