How do you find the slope of the curve at the given point P, and what is the equation of the tangent line at that point for the function y = 6x^2 – 19?

To find the slope of the curve at a specific point and the equation of the tangent line, follow these steps:

Step 1: Identify the function and the point

The given function is:

y = 6x² – 19

We need to find the slope at point P. Let’s assume point P is (p, y). To compute this, we first need the x value of point P.

Step 2: Find the derivative of the function

The slope of the tangent line to the curve at any point is given by the derivative of the function. Let’s calculate the derivative:

y’ = d/dx(6x² – 19)

Using basic differentiation rules, we get:

y’ = 12x

Step 3: Calculate the slope at point P

Now, substitute the x value of point P into the derivative:

m = 12p

This gives us the slope of the tangent line at point P.

Step 4: Find the y-coordinate at point P

Substitute the x value of P into the original function to find the corresponding y value:

y = 6p² – 19

So, the coordinates of point P are (p, 6p² – 19).

Step 5: Use point-slope form to find the equation of the tangent line

The equation of the tangent line can be expressed with the point-slope form:

y – y₁ = m(x – x₁)

Here, m is the slope we computed, and (x₁, y₁) is point P:

y – (6p² – 19) = (12p)(x – p)

This equation can be rearranged to the slope-intercept form:

Final Result

This tangent line equation expresses the relationship at the point P, providing both the slope and the specific point where the tangent touches the curve.

Substituting the specific values of P into these equations will yield both the slope and the tangent line equation particular to that point.