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

MATH104 December 2014

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Question 01 (m)
Estimate $|\sin(0.12)-0.12|$ by considering the linear approximation of $\sin {x}$ at $x=0$.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Given $f(x)$, what is the linearization of $f(x)$? What is the error term associated to this approximation?

Hint 2
The linearization (or linear approximation) of $f(x)$ about a point $a$ is given by $f(x)\approx L(x)=f(a)+f'(a)(x-a)$ and it holds that $|f(x)-L(x)|\leq {\frac {M(x-a)^{2}}{2}}$ Do you remember how to obtain the constant $M$?

Hint 3
The constant $M$ in the error term for the linear approximation $L(x)$, $|f(x)-L(x)|\leq {\frac {M(x-a)^{2}}{2}}$ is the maximum of the second derivative of $f(x)$ in the interval between $x$ and $a$. What is $x$, what is $a$ in our case?

Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. 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 ${\begin{aligned}L(x)&=f(a)+f'(a)(x-a)\\&=\sin(0)+\cos(0)(x-0)\\&=0+(1)(x)\\&=x\end{aligned}}$ 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 ${\begin{aligned}|\sin(0.12)-0.12|&=|\sin(0.12)-L(0.12)|\\&=|f(x)-L(x)|\\&\leq {\frac {M(x-a)^{2}}{2}}\\&={\frac {M(0.12-0)^{2}}{2}}\\&=0.0072M\end{aligned}}$ 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 $|f''(x)|=|-\sin(x)|\leq \sin(0.12)$ (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 $|\sin(0.12)-0.12|\leq 0.0072M=0.0072\sin(0.12)$

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