6. [-/1 Points] DETAILS MY NOTES SCALCET9 2.7.029. If \( f(x)=8 x^{2}-x^{3} \), find \( f^{\prime}(2) \) and use it to find an equation of the tangent line to the curve \( y=8 x^{2}-x^{3} \) at the point ( 2,24 ). Need Help? Read It Watch it SUBMIT MNSWER

To find \( f^{\prime}(2) \), we first need to calculate the derivative of the given function \( f(x) = 8x^{2} - x^{3} \). The derivative is \( f'(x) = 16x - 3x^{2} \). Now, substituting \( x = 2 \): \[ f'(2) = 16(2) - 3(2)^{2} = 32 - 12 = 20. \] Now that we have the slope of the tangent line at the point (2, 24), we can use the point-slope form of a line to find the tangent line's equation: \[ y - 24 = 20(x - 2). \] Simplifying this gives us \( y = 20x - 16 \). For a bit of historical context, calculus was developed independently by Sir Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century. Their work laid the foundation for understanding how change and motion are modeled mathematically, which is fundamental for finding slopes of curves, just like we did with \( f^{\prime}(2) \) here! In terms of real-world application, the concept of tangent lines is crucial in fields such as physics and engineering where it's important to understand the instantaneous rate of change—like how fast a car is going at a precise moment or the slope of a ramp. This helps in assessing everything from vehicle safety to the design of roller coasters!