Question

the slope of the tangent line to the parabola ( y = 3x^2 + 6x + 2 ) at the point ( (3, 47) ) is:
the equation of this tangent line can be written in the form ( y = mx + b ) where ( m ) is:
and where ( b ) is:

Explanation:

Step1: Find the derivative of the function

To find the slope of the tangent line to the parabola \( y = 3x^2 + 6x + 2 \), we first find its derivative. Using the power rule, the derivative \( y' \) of \( y = ax^n \) is \( y' = nax^{n - 1} \).
For \( y = 3x^2 + 6x + 2 \), the derivative \( y' \) is:
\( y' = \frac{d}{dx}(3x^2) + \frac{d}{dx}(6x) + \frac{d}{dx}(2) \)
\( y' = 3\times2x + 6 + 0 \)
\( y' = 6x + 6 \)

Step2: Evaluate the derivative at \( x = 3 \)

The slope of the tangent line at a point \( x = a \) is given by the value of the derivative at \( x = a \). Here, \( a = 3 \).
Substitute \( x = 3 \) into \( y' = 6x + 6 \):
\( y'(3) = 6\times3 + 6 \)
\( y'(3) = 18 + 6 \)
\( y'(3) = 24 \)
So the slope \( m \) of the tangent line is 24.

Step3: Find the equation of the tangent line

We use the point - slope form of a line, which is \( y - y_1 = m(x - x_1) \), where \( (x_1,y_1)=(3,47) \) and \( m = 24 \).
Substitute the values into the point - slope form:
\( y - 47 = 24(x - 3) \)
Expand the right - hand side:
\( y - 47 = 24x - 72 \)
Add 47 to both sides to get the equation in slope - intercept form \( y=mx + b \):
\( y=24x - 72 + 47 \)
\( y=24x - 25 \)
So, \( m = 24 \) and \( b=-25 \).

Answer:

The slope of the tangent line: \( 24 \)
The value of \( m \): \( 24 \)
The value of \( b \): \( - 25 \)