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4.8: Exploration- Applications of Derivatives
Opening the book…
We have a lot of practice with calculating derivatives, and we hopefully have a good idea of how we might interpret what this derivative is: a function that tells us slopes of a line tangent to a curve at a specific \(x\)-value.
Let’s think about this idea.
Exploration \(\PageIndex{1}\): A Tangent Triangle.
Let’s consider the function \(f(x)=\dfrac{1}{x}\).
Consider a point on this curve at some \(x\)-value \(x = a\) where \(a > 0\). Find the equation of the line tangent to \(f(x)\) at this point.
We can graph this line, as well as the function \(f(x)\).
Find the area of the triangle formed by the tangent line and the axes.
Hopefully you found something surprising about this area. Show that this surprising fact also occurs for the function \(g(x)=\dfrac{k}{x}\), where \(k > 0\).
Are there other functions that have this interesting property about the area of a triangle formed by the tangent line and the axes? What are some requirements that these functions would need to have?
