University:
Glendale Community College
Course:
MAT120 | Intermediate Algebra
Academic year:

2022
Views:

361

Pages:

3
Author:

Davon Fischer

Section 3.2C Average Rate of change and applications of slope Find the Average Rate of Change of a Function 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 between the points (𝑥1 , 𝑓(𝑥1 )) (𝑥2 , 𝑓(𝑥2 )) = 𝑓(𝑥2 )−𝑓(𝑥1 ) 𝑥2 −𝑥1 (𝑢𝑠𝑒𝑑 𝑤ℎ𝑒𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑖𝑠 𝑤𝑟𝑖𝑡𝑡𝑒𝑛 𝑤𝑖𝑡ℎ 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑛𝑜𝑡𝑎𝑡𝑖𝑜𝑛) 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡𝑠 (𝑥1 , 𝑦1 ) (𝑥2 , 𝑦2 ) = 𝑦2 −𝑦1 𝑥2 −𝑥1 (used when equation is written without function notation) Both equations are symbolic ways to say the average rate of change is the slope of the line that connects two points. • Regardless of the notation, to find the average rate of change: First: find the two necessary points Second: find the slope of the line that connects the points (the slope is the average rate of change.) For example: Use the graph of 𝑓(𝑥) to find the average rate of change from 𝑥 = 1 𝑡𝑜 𝑥 = 9. The average rate of change is just the slope of the line that connects two points. First find two points. The points I need are (1, 1) and (9,3) as they have the 𝑥 − 𝑣𝑎𝑙𝑢𝑒𝑠 that are required, 1 𝑎𝑛𝑑 9. Second find the slope of the line that connects the points. 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 = 3−1 2 1 = = 9−1 8 4 Answer: Average rate of change from 𝑥 = 1 1 𝑡𝑜 𝑥 = 9 𝑖𝑠 4. Section 3.2C Average Rate of change and applications of slope For example: Find the average rate of change in the cost of a new car between 2000 and 2010. Write your answer in a sentence. Year Cost of the average new car (rounded) 1940 $600 1950 $850 1960 $1500 1970 $2600 1980 $3500 1990 $7200 2000 $17000 2010 $30000 2020 $38000 First find two points. The points needed are: (2000, 17000) (2010, 30000) Second find the slope of the line that connects the two points. 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 30000 − 17000 = 2010 − 2000 13000 𝑑𝑜𝑙𝑙𝑎𝑟𝑠 = = 1300 10 𝑦𝑒𝑎𝑟𝑠 Answer: Average rate of change is $1300 per year. The cost of a new car increased on average by $1300 per year between 2000 and 2010. Section 3.2C Average Rate of change and applications of slope For example: Find the average rate of change given a function. Find the average rate of change of 𝑓(𝑥) = 3𝑥+5 𝑥−5 on the interval: [6, 10]. I need to create two points to find the average rate of change. The numbers in the brackets are 𝑡ℎ𝑒 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 that I need. I will find the 𝑦 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒𝑠 by plugging in the values into the function. First point (6, ) y-coordinate of first point: 𝑓(6) = First point (6, 23) 3(6)+5 23 6−5 = 1 = 23 Second point (10, ) y-coordinate of second point: 𝑓(10) = 3(10)+5 10−5 = 35 5 =7 Second point (10, 7) Now find the average rate of change between the points (6, 23) 𝑎𝑛𝑑 (10, 7): 7 − 23 −16 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 = = = −4 10 − 6 4 Answer: Average rate of change on the interval [6, 10] 𝑖𝑠 − 4