Given the function below \[ f(x)=\sqrt[3]{-45 x^{3}+72} \] Find the equation of the tangent line to the graph of the function at \( x=1 \). Answer in \( m x+b \) form. \( L(x)= \) Use the tangent line to approximate \( f(1.1) \). \( L(1.1)= \) Compute the actual value of \( f(1.1) \). What is the error between the function value and the linear approximation? Answer as a positive value only. lerror \( 1 \approx \) ? (Approximate to at least 5 decimal places.)

To find the equation of the tangent line \( L(x) \) to the graph of the function at \( x = 1 \), we first need to find \( f(1) \) and \( f'(x) \). 1. **Calculate \( f(1) \)**: \[ f(1) = \sqrt[3]{-45(1^3) + 72} = \sqrt[3]{-45 + 72} = \sqrt[3]{27} = 3 \] 2. **Find the derivative \( f'(x) \)** using the chain rule: \[ f(x) = \sqrt[3]{-45x^3 + 72} \implies f'(x) = \frac{1}{3}(-45x^3 + 72)^{-\frac{2}{3}}(-135x^2) \] Simplifying this gives: \[ f'(x) = \frac{-135x^2}{3(-45x^3 + 72)^{\frac{2}{3}}} = \frac{-45x^2}{(-45x^3 + 72)^{\frac{2}{3}}} \] 3. **Evaluate \( f'(1) \)**: \[ f'(1) = \frac{-45(1^2)}{(-45(1^3) + 72)^{\frac{2}{3}}} = \frac{-45}{27^{\frac{2}{3}}} = \frac{-45}{9} = -5 \] 4. **Equation of the tangent line** \( L(x) \) at \( x = 1 \): Using the point-slope form \( y - f(1) = f'(1)(x - 1) \): \[ y - 3 = -5(x - 1) \implies y = -5x + 5 + 3 \implies y = -5x + 8 \] Thus, \( L(x) = -5x + 8 \). 5. **Approximation of \( f(1.1) \)**: \[ L(1.1) = -5(1.1) + 8 = -5.5 + 8 = 2.5 \] 6. **Compute the actual value of \( f(1.1) \)**: \[ f(1.1) = \sqrt[3]{-45(1.1^3) + 72} \] Calculating \( 1.1^3 = 1.331 \): \[ f(1.1) = \sqrt[3]{-45(1.331) + 72} = \sqrt[3]{-59.895 + 72} = \sqrt[3]{12.105} \approx 2.28736 \] 7. **Calculate the error**: \[ \text{error} = |f(1.1) - L(1.1)| = |2.28736 - 2.5| \approx 0.21264 \] In conclusion: - The equation of the tangent line is \( L(x) = -5x + 8 \). - The approximate value of \( f(1.1) \) using the tangent line is \( L(1.1) = 2.5 \). - The actual value of \( f(1.1) \) is approximately \( 2.28736 \), and the error is \( \text{error} \approx 0.21264 \).