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Connecting f and f’ (Lesson 9.6 Day 1)

Unit 9 - Day 11

​Learning Objectives​
  • Use the first derivative to justify whether a function is increasing, decreasing, or not changing.

  • Use rates of change to describe when a function has a relative maximum or minimum and relate this to optimization

  • Justify where a function has a relative extrema using the first derivative test

Quick Lesson Plan
Activity: How Much Should You Spend on Advertising?



Lesson Handout

Answer Key

Experience First

Today your students will dive into the world of business and marketing by deciding the best (optimal) dollar amount to spend on advertising a new product.


Students are given an equation for P(x), which gives a company’s profit as a function of money spent on advertising. This function is quadratic with a negative leading coefficient, signaling to students that the profit will have a peak at some x-value and increasing the advertising budget beyond this point will cause a loss of profit. Making sense of functions is a critical skill at every level of math, and now students have some additional calculus tools to aid in this analysis.


In question 2, students are asked to consider how P’(x), the derivative of P(x) , might give valuable insight into the optimal amount of money to spend on advertising. We want students to recognize that the derivative will tell the company the additional profit that can be earned by increasing the advertising budget by some marginal amount. If this value is negative, then profit will actually decrease.


When the company spends 80,000 on advertising (x=80), P’(x) is positive, meaning that profit is not yet at its peak. This is a critical understanding: As long as the first derivative is positive, there is still more profit to be made! An optimal advertising budget would occur when the profit has stopped getting better (increasing) and is about to get worse (start decreasing). This happens when P’(x)=0. Many students were able to explain why a zero derivative is actually the best case scenario for this company, since a horizontal tangent occurs at the relative maximum.


In questions 4 and 5, students connect this analysis of the company’s changing profit using the graph of P and the graph of P’. The main goal of today’s lesson is that students would be able to justify (and interpret) the behavior of the original function (increasing, decreasing, locations of relative extrema) using the derivative function, given either as an equation or a graph.

Formalize Later

Justifying function behavior is a critical aspect of Calculus. Today’s “Important Ideas” give students a helpful framework for connecting the behaviors of a function and its derivative, but we want students to do more than simply memorize the associated phrases. The color-coding on the front page will be helpful for seeing these relationships. Encourage students to use highlighters or colored pencils to identify how the same function feature can be seen on both graphs.

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