## Calculus (3rd Edition)

$$0.023,\ \ 0.02299797 ,\ \ 0.00000203$$
Given $$\sin (0.023)$$ Consider $f(x)=\sin x$, $a= 0$, $\Delta x=0.023$, since \begin{align*} f'(x) &= \cos x \\ f'(0)&=1 \end{align*} Then the linear approximation is given by \begin{align*} \Delta &f \approx f^{\prime}(a) \Delta x\\ &= (1)(0.023)\\ &= 0.023 \end{align*} and the actual change is given by \begin{align*} \Delta f&=f(a+\Delta x)-f(a)\\ &=f(0.023)-f(0)\\ & = 0.02299797 \end{align*} Hence the error is $$|0.02299797- 0.023| \approx 0.00000203$$