Science:Math Exam Resources/Courses/MATH110/December 2016/Question 05 (c)

MATH110 December 2016

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Question 05 (c)
Consider the function ${\textstyle f(x)=x^{1/3}}$. (c) Using the results from parts (a) and (b), show that $f$ has a vertical tangent line 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 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 vertical line has a slope of infinity.

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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. By part (b), the slope of the tangent line of ${\textstyle f(x)}$ at ${\textstyle x=0}$ is $$f'(0)=\lim \limits _{h\to 0}m(h).$$ By part (a), $$m(h)={\dfrac {1}{h^{\frac {2}{3}}}}.$$ Therefore, the slope of the tangent line of ${\textstyle f(x)}$ at ${\textstyle x=0}$ is $$f'(0)=\lim \limits _{h\to 0}{\dfrac {1}{h^{\frac {2}{3}}}}=\infty .$$ By the Hint, this means that the tangent line at ${\textstyle x=0}$ is vertical. Answer: ${\textstyle {\color {blue}{\mbox{The tangent line has slope }}\lim \limits _{h\to 0}m(h)=\infty {\mbox{ and is therefore vertical.}}}}$