Lesson 1: Interpretation of the Derivative In Context PDF

Lesson 1: Interpretation of the Derivative In Context Topic 4.1: Interpreting the Meaning of the Derivative in Context In our last two units we have defined the derivative graphically, algebraically, and numerically as a slope of the tangent line and the instantaneous rate of change at a point. In t...

Lesson 1: Interpretation of the Derivative In Context Topic 4.1: Interpreting the Meaning of the Derivative in Context In our last two units we have defined the derivative graphically, algebraically, and numerically as a slope of the tangent line and the instantaneous rate of change at a point. In this lesson, we want to interpret the meaning of the derivative in a verbal manner, as it relates to the context of real-world problems. A Look at Some Mathematical History When Newton was developing his ideas of the infinitesimal calculus, namely when given a fluent (we call function) to find its fluxion (we call derivative), his thinking was concentrated on velocity. But, today, we know the derivative’s importance reaches into so many other areas involving rates of change. Let’s look at the Leibniz notation one more time in a different way. He stated that the variable 𝑦𝑦 depended on 𝑥𝑥, and if 𝑦𝑦 = 𝑓𝑓 𝑥𝑥 , then the derivative could be expressed as 𝑑𝑑𝑑𝑑/𝑑𝑑𝑑𝑑. If we allow ourselves to think of "𝑑𝑑𝑑 as representing “a small difference in some process,” we are reminded that the derivative is simply the limit of ratios in the familiar form 𝑑𝑑𝑑𝑑 ∆ 𝑖𝑖𝑖𝑖 𝑦𝑦−𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 𝑑𝑑𝑑𝑑 = 𝑓𝑓 ′ 𝑥𝑥 = ∆ 𝑖𝑖𝑖𝑖 𝑥𝑥−𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣. 𝑑𝑑 Similarly, we can read 𝑑𝑑𝑑𝑑 as a single symbol meaning “the derivative with respect to x of ….” which brings 𝑑𝑑 𝑑𝑑 us to this new place of understanding the notation 𝑑𝑑𝑑𝑑 (𝑦𝑦) or 𝑑𝑑𝑑𝑑 𝑓𝑓(𝑥𝑥) , “the derivative with respect to 𝑥𝑥 of 𝑦𝑦 or (𝑓𝑓(𝑥𝑥)). The Leibniz notation can be a little cumbersome when you need to specify the exact x-value where you need to evaluate the derivative. However, the benefit is that you can read so much about the rate of change through this expression. It’s very easy to write 𝑓𝑓 ′ 3 but with Leibniz’s notation we will write the derivative format like this: 𝑑𝑑𝑑𝑑 𝑑𝑑𝑑𝑑 𝑥𝑥=3 EX #1: Sean is cleaning out his saltwater fish tank and needs to drain some of the water in order to adjust the saline levels. If the volume of the tank 𝑉𝑉, is in gallons, is being drained over time, 𝑡𝑡, in minutes. What is a meaningful interpretation of the following statement as it relates to Sean’s fish tank? 𝑑𝑑𝑉𝑉 =−2 𝑑𝑑𝑡𝑡 𝑡𝑡=3 © 2021 Jean Adams Flamingo Math.com  Using Meaningful Variables Just like any other language course, we need to use proper names, spellings, and grammar conventions. AP Calculus is no different than AP Language. There are many real-world functions that depend on variables other than 𝑥𝑥, 𝑦𝑦, or time. Learning to read and write the correct notation will help you succeed on Free Response prompts throughout our course. EX #2: Read each scenario and write the proper notation for the derivative. A. The Alberta tar sands are one of the biggest oil reserves in the world. Yet extracting the fossil fuel costs more than the profits made. If the cost C (in dollars) to extract a barrel of oil is losing $3 on every barrel, write the derivative. B. The Deepwater Horizon oil spill is regarded as one of the largest environmental disasters in American history. The oil leak was discovered on April 22, 2010. BP originally estimated that the flow rate was about 790 cubic meters per day at the end of the first week. C. The population P of the endangered mountain gorilla has an ongoing study tracking the number of gorillas since 1992. The population increased by 114 gorillas during 2003 of this same study. EX #3: Suppose a planet is discovered in 2024; and, is named Newtonia. If the alien population 𝑃𝑃 𝑡𝑡 is recorded in millions of aliens, and t represents the number of years since the planet’s discovery. Explain the meaning of each of the statements below. A. P ′ 3 = 0.5 B. 𝑃𝑃−1 7.5 = 5 C. 𝑃𝑃−1 7.5 ′ = 0.75 © 2021 Jean Adams Flamingo Math.com  Using Units to Interpret the Derivative In the next examples, we will show how clues from the units can help you recognize interpretations of the derivative as they relate to real-world scenarios. EX #4: Suppose you launch a water balloon from a giant slingshot. Let 𝑠𝑠(𝑡𝑡) give the distance, in feet, that the water balloon traveled from its initial starting point as a function of time, t, in seconds. Explain the meaning of the derivative notation. 𝑑𝑑𝑑𝑑 𝑓𝑓𝑓𝑓. = 𝑠𝑠 ′ 2 = 38 𝑑𝑑𝑑𝑑 𝑡𝑡=2 𝑠𝑠𝑠𝑠𝑠𝑠. EX #5: The total cost C ,in dollars, of an airplane flight is a function of the number of passengers, 𝑛𝑛 can be calculated by Derivative Airlines using the function 𝐶𝐶 = 𝑓𝑓 𝑛𝑛. What does it mean to say 𝑓𝑓 ′ 182 = 58 ? EX #6: Non-potable water is being pumped into a processing center where the depth, in feet, of the water at time t in hours is given as 𝑦𝑦 = ℎ(𝑡𝑡). Interpret the following statements using the proper units. A. ℎ 7 = 5 B. ℎ′ 7 = 0.6 C. ℎ−1 7 = 9 D. ℎ−1 ′ 7 = 1.4 © 2021 Jean Adams Flamingo Math.com