Question

I need help with all those questions in the image!

Answer 1

The rate of change of elevation tends to a constant value.

Step-by-step explanation:

The average rate of change of e(x) on the interval [a, b] defined as

`m_{\text{avg}}=(e(b)-e(a))/(b-a)`

which explicitly we can write as

`m_{\text{avg}}=\frac{\sqrt[]{b-10}-\sqrt[]{a-10}}{b-a}`

Now, the question is, what happens to m_avg as we increase b while keeping a fixed?

As b becomes large then √b -10 becomes √b and b - a becomes b (since a is comparatively small); therefore, m_avg becomes

`m_{\text{avg}}=\frac{\sqrt[]{b-10}-\sqrt[]{a-10}}{b-a}\Rightarrow\frac{\sqrt[]{b}-\sqrt[]{a-10}}{b}\Rightarrow\frac{\sqrt[]{b}}{b}`
`\Rightarrow m_{\text{avg}}=\frac{\sqrt[]{b}}{b}`

which for any fixed value of b is a constant.

The same behaviour can be extrapolated by looking at the graph of e(x).

As can be seen from the graph, as x increases, the slope of the function becomes flatter and flatter, meaning it tends to be a constant. In other words, for large values of x, you can approximate the slope of the function by a straight line.