What is steepness, what is flatness?
I have 3 graphs rather like $$y = \frac{1}{x}$$ and I am supposed to describe which one is steepest and which one is flattest. This is for an Econ class, so I'm not sure the terminology being used is mathematically correct. This is what I want information about.
I calculated the derivatives and what not. But I am confused what "steepness" means. For a graph like $\frac{1}{x}$, the graph could be regarded as "steep" both towards the $y$ axis and towards the $x$ axis. And that's the problem. My graph 1 is steeper towards the $y$-axis ($y = \frac{1}{x^{\frac{1}{3}}}$) and my Graph 3 ($y = \frac{1}{x^3}$) is steeper towards the $x$-axis. So I'm not sure how to rank them.
My Question:
Can someone define the terms steepness and flatness in a way that allows me to rank these 3 graphs in terms of steepness?
$\endgroup$1 Answer
$\begingroup$If $f^{\prime}(x)=0$ the tangent line is horizontal, so that could be viewed as flat (at least at that point). If the derivative is very large (in absolute value) the tangent line is very "steep" at that point. So perhaps, the best definition for "flattest on an interval" is having the smallest (in absolute value) derivative on that interval. And "steepest on an interval" is having the largest (in absolute value) derivative on the interval. The comparison should be done pointwise (pick a point in the interval and compare all the derivatives at that point). The interval should be chosen so that the same ordering of functions happens for any point chosen in the interval.
For the examples you gave, the interesting interval to look at for steepness will probably be $(0, 1)$, for flatness, consider $(1,\infty)$.
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