Science:Math Exam Resources/Courses/MATH110/April 2018/Question 01 (b)

MATH110 April 2018

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Question 01 (b)

Select all that apply: When we say that ${\textstyle f}$ is not differentiable at ${\textstyle x=2}$, we mean (a) ${\textstyle f(2)}$ does not exist. (b) ${\textstyle f'(2)}$ does not exist. (c) ${\textstyle \lim _{x\to 2}f(x)\neq f(2)}$. (d) the graph of ${\textstyle f}$ may have a corner at ${\textstyle x=2}$. (e) the graph of ${\textstyle f}$ may have a vertical asymptote at ${\textstyle x=2}$.

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
A function can be continuous but not differentiable.

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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. Statement (a) is incorrect because a function can be defined at a point without being differentiable there (for example, if the function has a corner at ${\textstyle x=2}$). Statement (c) says that ${\textstyle f}$ is discontinuous at ${\textstyle x=2}$, but a function can be continuous at a point without being differentiable there (again, a function with a corner is an example of this). Statement (b) says precisely that ${\textstyle f}$ is not differentiable at ${\textstyle x=2}$, and statements (d) and (e) are each examples of cases when a function is not differentiable at a point. Answer: The correct answers are ${\textstyle {\color {blue}(b),(d),(e)}}$.

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