Science:Math Exam Resources/Courses/MATH104/December 2014/Question 01 (m)/Solution 1

We are looking at the linear approximation of $f(x)=\sin x$ about the point $0$. Hence $a=0$ and the linear approximation is given by

$\begin{array}{rl}L(x)& =f(a)+f^{\prime }(a)(x-a)\\ & =\sin (0)+\cos (0)(x-0)\\ & =0+(1)(x)\\ & =x\end{array}$

We now want to estimate the error of this approximation at the point $x=0.12$. From the formula for the error of a linear approximation we find that

$\begin{array}{rl}|\sin (0.12)-0.12|& =|\sin (0.12)-L(0.12)|\\ & =|f(x)-L(x)|\\ & \le \frac{M(x-a{)}^2}{2}\\ & =\frac{M(0.12-0{)}^2}{2}\\ & =0.0072M\end{array}$

Finally it remains to calculate $M$, which is the largest absolute value of the second derivative of $f(x)$ in the interval $[a,x]=[0,0.12]$. Doing the math we obtain $|f^{″}(x)|=|-\sin (x)|\le \sin (0.12)$ (the number 1 could also be used as an upper bound though it is not as tight of an upper bound), therefore the final answer is

$|\sin (0.12)-0.12|\le 0.0072M=0.0072\sin (0.12)$