Differentials — Question 1

Question 1 - Solution

We are given: $$f(x) = x^3, \quad f'(x) = 3x^2$$

(a) Use: $$df = f'(x) \, dx$$

Let $x = 2$, $dx = 0.05$: $$f'(2) = 3(2)^2 = 12$$

$$df = 12 \cdot 0.05 = \boxed{0.6}$$

So, the approximate change in $f$ is $\boxed{0.6}$

(b) Actual change: $$f(2.05) = (2.05)^3 = 8.615125, \quad f(2) = 8$$ $$\text{Actual change} = 8.615125 - 8 = \boxed{0.615125}$$

Conclusion: The differential estimate $0.6$ is close to the actual change $0.615125$