Questions: If f(x)=3x^2+3x-6, find the equation of the line tangent at x=-1. Write the equation in the form y=ax+b.

Transcript text: If $f(x)=3 x^{2}+3 x-6$, find the equation of the line tangent at $x=-1$. Write the equation in the form $y=a x+b$.

Solution

Solution Steps

To find the equation of the tangent line to the curve $ f(x) = 3x^2 + 3x - 6 $ at $ x = -1 $, we need to follow these steps:

Calculate the derivative $ f'(x) $ to find the slope of the tangent line at any point $ x $.
Evaluate $ f'(x) $ at $ x = -1 $ to get the slope of the tangent line at that point.
Find the y-coordinate of the point on the curve at $ x = -1 $ by evaluating $ f(-1) $.
Use the point-slope form of a line equation, $ y - y_1 = m(x - x_1) $, where $ m $ is the slope and $ (x_1, y_1) $ is the point on the curve, to write the equation of the tangent line.

Step 1: Find the Derivative of the Function

To find the slope of the tangent line at a given point, we first need to find the derivative of the function $ f(x) = 3x^2 + 3x - 6 $. The derivative is given by:

\[ f'(x) = 6x + 3 \]

Step 2: Evaluate the Derivative at $ x = -1 $

Next, we evaluate the derivative at $ x = -1 $ to find the slope of the tangent line at this point:

\[ f'(-1) = 6(-1) + 3 = -3 \]

Thus, the slope of the tangent line at $ x = -1 $ is $ -3 $.

Step 3: Find the y-coordinate of the Point on the Curve

We need to find the y-coordinate of the point on the curve at $ x = -1 $ by evaluating the original function:

\[ f(-1) = 3(-1)^2 + 3(-1) - 6 = -6 \]

So, the point on the curve is $(-1, -6)$.

Step 4: Write the Equation of the Tangent Line

Using the point-slope form of a line, $ y - y_1 = m(x - x_1) $, where $ m = -3 $, $ x_1 = -1 $, and $ y_1 = -6 $, we can write:

\[ y - (-6) = -3(x - (-1)) \]