Focus Session 4 Derivatives and Marginal Cost MTH 124 Goals De nitions by cid 136 Understand the concept of marginal cost and its relationship to the derivative cid 136 Use the de nition of the derivative to compute and study marginal costs cid 136 Practice using Desmos Pre activity Do before class The following calculus tools are essential for studying marginal cost a fundamental topic in economics cid 136 For a function y f x the average rate of change of f x on the interval a a h is given f x f a h f a a h a f a h f a h Note that the average rate of change of f x on the interval a a h is equal to the slope of the line connecting the points a f a and a h f a h cid 136 The instantaneous rate of change or derivative at x a is the limit of the average rate of change of C x on the interval a a h as the size h of the interval goes to zero C a h C a Instantaneous rate of change C cid 48 a lim h 0 h For the cost function C x the value C cid 48 a is called the marginal cost when x a It measures the approximate cost of producing one more item after having already produced a items For example if C cid 48 1000 4 5 then it costs approximately 4 50 to produce the 1001st jar of almond butter For the cost function C x the derivative C cid 48 x lim h 0 C x h C x h is also known as the marginal cost function The value of C cid 48 x approximates the cost of producing one more jar of almond butter after having already produced x jars The expression which appears inside this limit is called the di erence quotient C x h C x h Suppose a company s total cost function is given by C x 1000 20x 0 005x2 cid 136 Find the average rate of change on the intervals a 500 1000 b 750 1000 c 990 1000 cid 136 Use your answers from the previous question to estimate the instantaneous rate of change of C x at x 1000 1 MTH 124 Focus Session 4 Derivatives and Marginal Cost In Class Activity The Better Butter Company produces organic almond butter Their nance team calculates that their cost function in dollars is C x 5000 5x 0 00025x2 during the production of the rst 9000 jars In other words C x gives the total cost of producing x jars of organic almond butter 1 Use Desmos to graph C x on the domain 0 x 9000 Copy the graph to the right 2 Is the total cost increasing at a faster rate when 2000 jars of almond butter have been produced or when 8000 jars have been produced Use tangent lines to justify your answer 30000 20000 10000 s r a l l o d 0 0 2000 4000 6000 8000 jars of almond butter The nance team at Better Butter is interested in determining how fast the cost function is increasing when 6000 jars of almond butter have been produced 3 Calculate the average rate of change of the cost function C x on each of the indicated intervals What is the correct unit for the average rate of change Interval 6000 9000 6000 6500 6000 6010 Average rate of change of C x ZOOM POLL Q1 What are the correct units for the average rate of change in question 3 A Dollars B Dollars per jar C Dollars per month D Jars per month 2 MTH 124 Focus Session 4 Derivatives and Marginal Cost 4 In your work from question 3 what is happening to AROC as the size of the interval changes Does the AROC appear to be approaching some number What does this seem to imply about the IROC at 6000 What is the unit for this value 5 The value of C 6000 is 26000 You estimated the value of C cid 48 6000 above Using units write a sentence interpreting what those numbers tell you about the costs of producing the jars of almond butter 6 Simplify the di erence quotient as much as possible HINT Note that C x h 5000 5 x h 0 00025 x h 2 You will need to multiply everything out and simplify the numerator of the di erence quotient C x h C x h ZOOM POLL Q2 What is the correct answer for question 6 above A 5 0 0005x 0 00025h B 5 0 0005x 0 00025h C 5 0 0005x 0 00025h D 5 0 0005x 0 00025h 3 MTH 124 7 Notice that the de nition of the derivative marginal cost is given by C cid 48 x lim h 0 h Focus Session 4 Derivatives and Marginal Cost C x h C x in question 6 Substitute your answer into the de nition and nd the and we calculated C x h C x derivative by taking the limit as h 0 h 8 Use your solution to question 7 to check your estimate in question 4 In other words plug in 6000 to nd C cid 48 6000 What is the correct unit for this value 9 The nance team at Better Butter would like to know when the cost of producing one more jar i e the marginal cost is equal to 3 25 per jar Find the production quantity at which that happens 6 4 2 0 0 10 In Desmos plot the derivative function C cid 48 x on the domain 0 9000 Label the unit for the vertical axis Determine when the marginal cost is highest What is the maximum marginal cost ZOOM POLL Q3 For a cost function C x and a production quantity a the marginal cost at x a approximates the cost of producing the a 1 th item A True B False 2000 4000 6000 8000 jars of almond butter 4