To find the slope of the curve and the equation of the tangent line at the given point (p), we need to follow a few steps.
Given the equation of the curve: y = 5 – 6x²
1. **Finding the Slope:**
First, we need to calculate the derivative of the function. This derivative represents the slope of the curve at any point x.
The derivative of the function y = 5 – 6x² is:
dy/dx = -12x
2. **Evaluating the Derivative at Point P:**
To find the slope at the specific point p where x = 2, we substitute x = 2 into the derivative:
dy/dx = -12(2) = -24
This means the slope of the curve at the point p(2, 19) is -24.
3. **Finding the Equation of the Tangent Line:**
Now that we have the slope, we can use the point-slope form of the equation of a line to find the equation of the tangent line. The point-slope form is given by:
y – y₁ = m(x – x₁)
Here, m is the slope, and (x₁, y₁) is the point of tangency, which is (2, 19). Using the slope we found:
y – 19 = -24(x – 2)
4. **Simplifying the Equation of the Tangent Line:**
Now we’ll simplify this equation:
y – 19 = -24x + 48
y = -24x + 67
Thus, the equation of the tangent line at the point p is:
y = -24x + 67
In summary, the slope of the curve at the given point p(2, 19) is -24, and the equation of the tangent line at that point is y = -24x + 67.