Linear Approximations — Question 4

Question 4 - Solution

Let $f(x) = \sqrt[3]{x} = x^{1/3}$, and choose $a = 8$, since $\sqrt[3]{8} = 2$.

(a) Linear approximation formula: $$L(x) = f(a) + f'(a)(x - a)$$

Compute: $$f'(x) = \frac{1}{3}x^{-2/3}, \quad f'(8) = \frac{1}{3} \cdot 8^{-2/3} = \frac{1}{3} \cdot \frac{1}{4} = \frac{1}{12}$$

So: $$L(x) = 2 + \frac{1}{12}(x - 8)$$

(b) Estimate: $$L(8.1) = 2 + \frac{1}{12}(0.1) = 2 + \frac{1}{120} = \frac{241}{120} \approx 2.0083$$

(c) Actual value: $$\sqrt[3]{8.1} \approx 2.00828 \quad \text{(using calculator)}$$

Conclusion: The linear approximation gives: $$\boxed{\sqrt[3]{8.1} \approx 2.0083}$$ which is very close to the actual value.