• Find a related rate.
  • Use related rates to solve real-life problems.

Assignment

  • Vocabulary and teal boxes
  • p195 2, 3, 7, 10–12, 14, 15–19 odd, 20, 23, 24, 28 35, 36, 40, 45–47

Additional Resources

Now that we’ve looked at implicit differentiation, we can start looking at how math models change with respect to time. For instance, how the area of a circle changes with respect to time.

\[\begin{align*} \frac{d}{dt}[A] &= \frac{d}{dt}[\pi r^2] \\ \frac{dA}{dt} &= 2\pi r \frac{dr}{dt} \end{align*}\]

So, the rate of change of the area is related to both the current radius and the rate of change of the radius. When you are typically given these types of problems, you’ll have values for most of the variables or rates, and be left to solve for the remaining one. In example 2 from the book, it uses our area of a circle model from above and asks for the rate of change of the area when the radius is 4 feet and the rate at which the radius changes is 1 foot per second.

\[\begin{align*} \frac{dA}{dt} &= 2\pi r \frac{dr}{dt} \\ &= 2\pi (4\text { ft})(1 \text{ft/sec}) \\ & = 8\pi \text{ ft}^2\text{/sec} \end{align*}\]

Note the use of a units. Using dimensional analysis is highly recommended so that you can a) determine the correct units, and b) verify you arrived at a correct solution so long as the units make sense in the problem. Here, square feet per second makes sense we are talking about area over time.

And that’s it. We’ll go through a number of examples, and of course look through the ones provided by the book.