Skip to content

Interesting Derivative and Areas

This take-home quiz you will take a look at an interesting property in math that involves derivatives and areas.

Consider the function \(f(x) = \frac{1}{x}\). (For the duration of this quiz, just consider the function in the first quadrant only.)

  1. Find the equation of the tangent line to \(f(x)\) at the point \(x = a\). (where \(x > 0\)).
  2. Call the point of tangency \(A\), the point where the tangent line interesects the y-axis \(B\), and the point where the tangent line intersects the x-axis \(C\).
    1. What are the coordinates of point \(B\) and point \(C\)?
    2. Find the ratio of the length \(\overline{AC}\) to \(\overline{BC}\).
    3. How does the point \(x = a\) relate to what you found in 2.b?
  3. What is the area of the right triangle with vertices \(B\), \(C\), and the origin? How does the point of tangency relate to what you found?
  4. What is the area of the bottom-left vertex at the origin and a top-right vertex at \(A\)? What is the ratio of the area of the rectangle to the area of the triangle? As before, how does the point of tangency relate to what you found?
  5. A more interesting question is whether the properties that we saw in the previous question holds when the power changes. Consider the function \(f(x) = \frac{1}{x^2}\). Repeat #1-4 (call them 5a, 5b, etc.) for this function. What does the point of tangency do in 5b? What about the ratios of areas in 5d?
  6. Now generalize this by using \(f(x) = \frac{1}{x^n}\) and repeating #1-4 (call it 6a, 6b, etc.) What does the point of tangency do in 6b? What about the ratios of areas in 6d?

Solutions

Click here to view the solution