Science:Math Exam Resources/Courses/MATH100 B/December 2024/Question 14

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MATH100 B December 2024

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  •  December 2023  •  December 2024  •  

Question 14

Let $f(x)$ be a function that is defined for all real numbers by $$f(x)=\{\begin{array}{ll}{x}^2& \text{if }x=\frac{1}{n},\text{where}n\text{is a positive integer}\\ 0& \text{otherwise.}\end{array}$$ A portion of the function is sketched below, to help you visualize it. For what values of x is this function differentiable? For those values, compute $f^{\prime }(x)$. Remember to fully justify your answer. Sketch of the function outlined in Question 14.

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Consider three cases separately: $x\ne \frac{1}{n},x=\frac{1}{n}$, and $x=0$.

Checking a solution serves two purposes: helping you if, after having used the hint, 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

For any nonzero $x$ that is not of the form $\frac{1}{n}$, the function is differentiable. Indeed, one can find a small interval around such an $x$ that does not contain any points of the form $\frac{1}{n}$, so the function is identically zero in that interval. Hence, $$f^{\prime }(x)=0.$$ For any $a=\frac{1}{n}$, the function is not continuous because $$\underset{x\to a}{\lim }f(x)=0\ne f(a)=\frac{1}{n^2}.$$ Therefore, $f$ is not differentiable at such points. It remains to consider $a=0$. Using the definition of the derivative, $$f^{\prime }(0)=\underset{h\to 0}{\lim }\frac{f(h)-f(0)}{h}.$$ If $h=\frac{1}{n}$, then $$\frac{f(h)-f(0)}{h}=\frac{h^2}{h}=h.$$ If $h$ is not of the form $\frac{1}{n}$, then $$\frac{f(h)-f(0)}{h}=0.$$ In either case, the limit as $h\to 0$ is 0. Thus, $$f^{\prime }(0)=0.$$ Therefore, $f$ is differentiable for all $x\ne \frac{1}{n}$, and $$f^{\prime }(x)=0\text{for all such }x.$$

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