Question

which of the following is an equation of the line tangent to the graph of $g(x)=2x^{3}+5x^{2}$ at the point where $x=-2$?
choose 1 answer:
a $y = 4x + 12$
b $y = 4x - 18$
c $y = 4x - 12$
d $y = -2x + 12$

Explanation:

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

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(g(x)=2x^{3}+5x^{2}\), \(g^\prime(x)=6x^{2}+10x\).

Step2: Find the slope of the tangent line at \(x = - 2\)

Substitute \(x=-2\) into \(g^\prime(x)\): \(g^\prime(-2)=6\times(-2)^{2}+10\times(-2)=6\times4 - 20=24 - 20 = 4\).

Step3: Find the \(y\) - coordinate of the point of tangency

Substitute \(x = - 2\) into \(g(x)\): \(g(-2)=2\times(-2)^{3}+5\times(-2)^{2}=2\times(-8)+5\times4=-16 + 20 = 4\). So the point of tangency is \((-2,4)\).

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

Here \(m = 4\), \(x_1=-2\), \(y_1 = 4\). \(y - 4=4(x + 2)\), expand it: \(y-4=4x + 8\), then \(y=4x+12\).

Answer:

A. \(y = 4x + 12\)