a. find the slope of the tangent line to the graph of f(x)=2x² at the point (-3,18). b. find the slope…

a. find the slope of the tangent line to the graph of f(x)=2x² at the point (-3,18). b. find the slope - intercept equation of the tangent line to the graph of f(x)=2x² at the point (-3,18). a. what is the slope of the tangent line? m_tan =□

a. find the slope of the tangent line to the graph of f(x)=2x² at the point (-3,18). b. find the slope - intercept equation of the tangent line to the graph of f(x)=2x² at the point (-3,18). a. what is the slope of the tangent line? m_tan =□

Answer

Explanation:

Step1: Find the derivative of the function

The derivative of $f(x)=2x^{2}$ using the power - rule $(x^n)^\prime=nx^{n - 1}$ is $f^\prime(x)=4x$.

Step2: Evaluate the derivative at the given x - value

We want to find the slope of the tangent line at $x=-3$. Substitute $x = - 3$ into $f^\prime(x)$. So $m_{tan}=f^\prime(-3)=4\times(-3)=-12$.

Step3: Find the slope - intercept equation of the tangent line

The slope - intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y - intercept. We know $m=-12$ and the line passes through the point $(-3,18)$. Substitute $x=-3$, $y = 18$ and $m=-12$ into $y=mx + b$: $18=-12\times(-3)+b$. $18 = 36 + b$. Solve for $b$: $b=18 - 36=-18$. The equation of the tangent line is $y=-12x-18$.

Answer:

a. $m_{tan}=-12$ b. $y=-12x - 18$