For the function \( P(x)=x^{3}-9 x \), at the point \( (3,0) \), find the following. (a) the slope of the tangent to the curve (b) the instantaneous rate of change of the function (b)

To find the slope of the tangent to the curve at the point \( (3,0) \) for the function \( P(x) = x^{3} - 9x \), we need to find the derivative of the function and then evaluate it at \( x = 3 \). The derivative of \( P(x) \) with respect to \( x \) is given by: \[ P'(x) = \frac{d}{dx}(x^{3} - 9x) \] Using the power rule for differentiation, we have: \[ P'(x) = 3x^{2} - 9 \] Now, we need to find the instantaneous rate of change of the function at \( x = 3 \), which is the value of \( P'(3) \). Substitute \( x = 3 \) into the derivative \( P'(x) \): \[ P'(3) = 3(3)^{2} - 9 \] Calculate the value of \( P'(3) \) to find the instantaneous rate of change of the function at \( x = 3 \). Calculate the value by following steps: - step0: Calculate: \(3\times 3^{2}-9\) - step1: Calculate: \(3^{3}-9\) - step2: Evaluate the power: \(27-9\) - step3: Subtract the numbers: \(18\) The slope of the tangent to the curve at the point \( (3,0) \) for the function \( P(x) = x^{3} - 9x \) is 18. The instantaneous rate of change of the function at \( x = 3 \) is also 18.