Parts a,b,c ### Linear Approximation of FunctionsThis problem involves using linear approximation to estimate the value of a function at a given point \(\mathbf{a}\) and also requires computing the percent error in the approximation. Here's a step-by-step guide to solve the problem:#### 1. Problem DescriptionGiven the function: \[ f(x) = (512+x)^{\frac{1}{3}} \]And the point: \[ a = 0; \quad f(0) \]#### 2. Tasks**a. Write the equation of the line that represents the linear approximation to the following function at the given point a.****b. Use the linear approximation to estimate the given quantity.****c. Compute the percent error in the approximation** using the formula: \[ \text{Percent Error} = 100 \cdot \frac{|\text{approximation} - \text{exact}|}{\text{exact}} \]where the exact value is given by a calculator.### Step-by-Step Solution**a. Writing the Equation of the Linear Approximation**The linear approximation \( L(x) \) of a function at the point \( a \) is given by:\[ L(x) = f(a) + f'(a)(x - a) \]First, we need to calculate \( f(a) \):\[ f(a) = (512 + 0)^{\frac{1}{3}} = 512^{\frac{1}{3}} \]Next, we find the derivative \( f'(x) \):\[ f(x) = (512 + x)^{\frac{1}{3}} \]\[ f'(x) = \frac{1}{3} (512 + x)^{-\frac{2}{3}} \]Evaluating the derivative at \( x = 0 \):\[ f'(0) = \frac{1}{3} (512 + 0)^{-\frac{2}{3}} = \frac{1}{3} \cdot 512^{-\frac{2}{3}} \]Now we construct \( L(x) \):\[ L(x) = f(0) + f'(0)(x - 0) \]**b. Using the Linear Approximation**Estimate \( f(x) \) at a given value using \( L(x) \). This involves substituting the given

a. Write the equation of the line that represents the linear approximation to the following function at the given point a. b. Use the linear approximation to estimate the given quantity. c. Compute the percent error in the approximation, 100. 1 f(x) = (512+x) 3; a = 0; f(0.1) a. L(x) = approximation - exact exact where the exact value is given by a calculat ...