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# Interpretation of the Derivative MATH127

UMass

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This 4 page Class Notes was uploaded by Elena Bassin on Friday February 13, 2015. The Class Notes belongs to MATH127 at University of Massachusetts taught by Adena Calden in Spring2015. Since its upload, it has received 48 views. For similar materials see Calculus for Life and Social Problems I in Mathematics (M) at University of Massachusetts.

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Date Created: 02/13/15

Interpretation of the Derivative A The average rate of change IS f39x for small Ax Liebniz notation reminds us of this dy This equation means the same as the one above but this is f x easier because you can see the units and where they come dx from This notation reminds us that the derivative is the limit of ratios of difference of y values the form d1 f f erence of x values We say the derivative of y with respect to x units of the dependent variable units of the independent variable Units of the derivative The derivative it tells us change in the function when the independent variable s increased by 1 unit We can use the derivative to make estimations of future and past behavior of a function Local L39near Approx39mat39on use when you want to project data of a function by using the last information given Ify fx and Ax is near 0 then Ay z f xAx Then for x near a and Ax x a fx fa f aAx OR fnewfodf39odnewod This is also sometimes called the Tangent Line Approximation f3 10 f393 2 f5 f3253 The closer you are to the points given the 14 10 22 more accurate the information Chapter 2 Page 1 Example 1 f5t is the and Find the units of the derivative of Stalso known as the velocity feet 5 t l seconds Va If the velocity vt of an object is given in feet seconds then what are the units of acceleration the derivative of velouty feet sec sec alt v lt feet sec 2 Example 2 Use the graph to the left to a Estimate g 20 using the tangent line that has been drawn in 160 4039 1395 9a gb 916 g40 15 O 15 ab 1640 4016 24 39 IE ll llama 3 m 4 32 4D wealis fierlaatmanstruatijii39ji b Find the units of g39t 0625 kg week Example 3 Chapter 2 Page 2 The time for a chemical reaction Tl in minutes is a function of the amount of catalyst present a in milliliters so T fla Interpret the following a fthO 7 If you add 100 milliliters of catalyst to the reaction it will take 7 minutes I b f 60 4 dependent variable is one that you are always plugging 4 into the function independent is already inside of the mllwtes function milliliters Example 4 Suppose the population of a town was In and the town39s population was Use this information to approximate the Given P2006125000 P392006 32000 pplyear P2013 P2006 P39200620132006 P2013 125000 3200 x 7 P2013 125000 22400 P2013 147400 Example 5 Suppose the table below gives the total sales in millions of CD39s for a music store as a function of time in years Year t 2008 2009 2010 2011 2012 Sales ft 025 606 575 47 425 Use the table above to estimate the sales in 2013 using local linear approximation Chapter 2 Page 3 ga gb g2012 g2008 425 625 2 5 million cds year a b 2012 2008 4 4 P2013 P2012 P39201220132012 P2013 425 5 P2013 375 Chapter 2 Page 4

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