Exploration: Applications of Derivatives

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Section 4.8 Exploration: Applications of Derivatives

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.

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Let’s think about this idea.

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Exploration 4.8.1. A Tangent Triangle.

Let’s consider the function $f(x)=\dfrac{1}{x}\text{.}$

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(a)

Consider a point on this curve at some $x$-value $x=a$ where $a\gt 0\text{.}$ Find the equation of the line tangent to $f(x)$ at this point.

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(b)

We can graph this line, as well as the function $f(x)\text{.}$

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Find the area of the triangle formed by the tangent line and the axes.

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(c)

Hopefully you found something surprising about this area. Show that this surprising fact also occurs for the function $g(x)=\dfrac{k}{x}\text{,}$ where $k\gt 0\text{.}$

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(d)

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?

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