Question 6 Use linear approximation to estimate 8.23\sqrt[3]{8.2}. (a) Define an appropriate function f(x)f(x), and choose a point aa near 8.2. (b) Find the linear approximation L(x)L(x) of f(x)f(x) at aa. (c) Use L(x)L(x) to estimate 8.23\sqrt[3]{8.2}, and compare it to the actual value. Show solutionHide solution+Question 6 - Solution Let f(x)=x3=x1/3f(x) = \sqrt[3]{x} = x^{1/3}, and choose a=8a = 8 since 83=2\sqrt[3]{8} = 2 is exact and close to 8.2. (a) Compute derivative: f′(x)=13x−2/3=13x23,sof′(8)=13643=13⋅4=112f'(x) = \frac{1}{3}x^{-2/3} = \frac{1}{3\sqrt[3]{x^2}}, \quad \text{so} \quad f'(8) = \frac{1}{3\sqrt[3]{64}} = \frac{1}{3 \cdot 4} = \frac{1}{12} (b) Linear approximation: L(x)=f(a)+f′(a)(x−a)=2+112(x−8)L(x) = f(a) + f'(a)(x - a) = 2 + \frac{1}{12}(x - 8) (c) Estimate: L(8.2)=2+112(0.2)=2+0.212=2+160≈2.0167L(8.2) = 2 + \frac{1}{12}(0.2) = 2 + \frac{0.2}{12} = 2 + \frac{1}{60} \approx 2.0167 Actual value: 8.23≈2.0164(from calculator)\sqrt[3]{8.2} \approx 2.0164 \quad \text{(from calculator)} Conclusion: 8.23≈2.0167(Linear Approximation)\boxed{\sqrt[3]{8.2} \approx 2.0167} \quad \text{(Linear Approximation)} Actual: 8.23≈2.0164⇒Very close!\text{Actual: } \sqrt[3]{8.2} \approx 2.0164 \Rightarrow \text{Very close!}
