2. One point on the curve \( y=\frac{1}{2} x^{4} \) is \( (2,8) \). Use a tangent line to approximate the y-coordinate when \( x=2.06 \). \( \begin{array}{lllll}\text { A. } 1 & \text { B. } 7.94 & \text { C. } 8.06 & \text { D. } 8.96 & \text { F. } 16\end{array} \)

To approximate the y-coordinate at \( x = 2.06 \) using a tangent line, we first need to find the derivative of \( y = \frac{1}{2} x^4 \). The derivative is: \[ \frac{dy}{dx} = 2x^3 \] At the point \( (2, 8) \), we evaluate the derivative: \[ \frac{dy}{dx} \bigg|_{x=2} = 2(2^3) = 16 \] So, the slope of the tangent line at \( x = 2 \) is 16. We can use the point-slope form of the equation of the tangent line to find the linear approximation: \[ y - 8 = 16(x - 2) \] Now plug in \( x = 2.06 \): \[ y - 8 = 16(2.06 - 2) = 16(0.06) = 0.96 \] Thus: \[ y = 8 + 0.96 = 8.96 \] The approximate y-coordinate when \( x = 2.06 \) is: **D. 8.96** --- Approximately 250 years ago, calculus was developed to help solve problems in physics and engineering that were impossible to tackle with algebra alone. Sir Isaac Newton and Gottfried Wilhelm Leibniz, in their independent quests for understanding motion and change, laid down the foundations of derivatives and integrals which allow us to approximate values, such as the tangent line method used here. This approximation is super useful in fields like physics or economics, where you'll often find yourself needing to estimate values quickly, especially when you can't afford to solve complex equations. Make sure to remember that while tangent lines give a good estimate nearby, they can become less accurate as you move further away from the point of tangency. So always consider how far you are straying from your known point!