Question

write an equation of the line tangent to the graph of $g(x)=3x^{2}-15x$ at the point where $x = 3$.

Explanation:

Step1: Find the derivative of \(g(x)\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(g(x)=3x^{2}-15x\), \(g^\prime(x)=(3x^{2})^\prime-(15x)^\prime\).
\(g^\prime(x)=3\times2x-15\times1 = 6x - 15\).

Step2: Find the slope of the tangent line at \(x = 3\)

Substitute \(x = 3\) into \(g^\prime(x)\).
\(m=g^\prime(3)=6\times3-15=18 - 15=3\).

Step3: Find the \(y\) - coordinate of the point on the curve at \(x = 3\)

Substitute \(x = 3\) into \(g(x)\).
\(y=g(3)=3\times3^{2}-15\times3=3\times9-45=27 - 45=-18\).

Step4: Use the point - slope form \(y - y_1=m(x - x_1)\)

Here \(x_1 = 3\), \(y_1=-18\) and \(m = 3\).
\(y-(-18)=3(x - 3)\).
Simplify the equation:
\(y + 18=3x-9\).
\(y=3x-9 - 18\).
\(y=3x-27\).

Answer:

\(y = 3x-27\)