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

MATH110 December 2016

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Question 05 (a)
Consider the function ${\textstyle f(x)=x^{1/3}}$. (a) Let ${\textstyle m}$ be the slope of the line that goes through the points ${\textstyle P(0,f(0))}$ and ${\textstyle Q(h,f(h))}$, where ${\textstyle h}$ is a nonzero real number. Find an expression for ${\textstyle m}$ in terms of ${\textstyle h}$.

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
The slope of the line passing through two different points $(x_{1},y_{1})$ and $(x_{2},y_{2})$ is ${\frac {y_{2}-y_{1}}{x_{2}-x_{1}}}$.

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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 the Hint, the slope of the line is $$m={\frac {f(h)-f(0)}{h-0}}={\frac {h^{1/3}}{h}}={\frac {1}{h^{2/3}}}.$$ The second equality follows from $f(h)=h^{1/3}$ and $f(0)=0^{1/3}=0$. Also, in the third equality, we use $h\neq 0$. Answer: $\color {blue}m={\frac {1}{h^{2/3}}}$