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

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

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

Answer

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)