Ify=4 \( x+3 x^{2} \), find the approximate change in \( y \) when \( x \) changes from 9 to 8.98. \( y(x)=4 \sqrt{x}+3 x^{2} \) \( y^{2} \) fion

We are given \[ y(x)=4\sqrt{x}+3x^{2}. \] We want to approximate the change in \( y \) when \( x \) changes from \( 9 \) to \( 8.98 \). Using differentials, the approximate change \( \Delta y \) is given by \[ \Delta y\approx y'(x)\,\Delta x, \] where \( \Delta x =-0.02 \) (since \( 8.98-9=-0.02 \)). **Step 1. Find the derivative \( y'(x) \).** Differentiate each term of \( y(x) \): 1. For \( 4\sqrt{x} \): \[ \frac{d}{dx}(4\sqrt{x}) = 4\cdot\frac{1}{2\sqrt{x}}=\frac{2}{\sqrt{x}}. \] 2. For \( 3x^{2} \): \[ \frac{d}{dx}(3x^{2}) = 6x. \] Thus, the derivative is \[ y'(x)=\frac{2}{\sqrt{x}}+6x. \] **Step 2. Evaluate the derivative at \( x=9 \).** Since \[ \sqrt{9}=3, \] we have \[ y'(9)=\frac{2}{3}+6(9)=\frac{2}{3}+54. \] Combine the terms: \[ y'(9)=\frac{2+162}{3}=\frac{164}{3}. \] **Step 3. Calculate the approximate change in \( y \).** Using \[ \Delta x=-0.02, \] we get \[ \Delta y\approx y'(9)\,\Delta x=\frac{164}{3}\times(-0.02). \] Performing the multiplication: \[ \Delta y\approx -\frac{164\times0.02}{3}=-\frac{3.28}{3}\approx -1.093. \] Thus, the approximate change in \( y \) is \[ \boxed{\Delta y\approx -1.09.} \]