Maths · Applications of derivatives: Rate of change of quantities, monotonic-Increasing and decreasing functions, Maxima and minima of functions of one variable
Use differentials to a approximate the values of ; (i) and (ii) {\mathbf{2 6}} \
Use differentials to a approximate the values of ; (i) \( \sqrt{\mathbf{3 6 . 6}} \) and (ii) \( \sqrt[3]{\mathbf{2 6}} \)
- A. (i) 6.02 (ii) \( \frac{80}{9} \)
- B. (i) 6.03 (ii) \( \frac{80}{18} \)
- C. \( (\text { i) } 6.25 \) (ii) \( \frac{80}{27} \)
- D. (i) 6.05 (ii) \( \frac{80}{27} \)
Step-by-step solution
Use linear approximation with differentials. For sqrt(36.6): f(x)=√x, a=36, f(36)=6, f'(36)=1/(2√36)=1/12, so √36.6≈6+(1/12)(0.6)=6+0.05=6.05. For cube root of 26: f(x)=x^(1/3), a=27, f(27)=3, f'(27)=1/3 * 27^(-2/3)=1/27, so ∛26≈3+(1/27)(-1)=3-1/27=80/27.
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