web analytics

Example 5 - The total cost C(x), associated with production of - Finding rate of change

Slide18.JPG
Slide19.JPG

  1. Chapter 6 Class 12 Application of Derivatives
  2. Serial order wise
Ask Download

Transcript

Example 5 The total cost C(x) in Rupees, associated with the production of x units of an item is given by C(x) = 0.005 x3 โ€“ 0.02 x2 + 30x + 5000 Find the marginal cost when 3 units are produced, where by marginal cost we mean the instantaneous rate of change of total cost at any level of output. Marginal cost is the rate of change of total cost w.r.t output (unit produce) Let MC be marginal cost โˆด MC = ๏ท๐‘‘๐ถ๏ทฎ๐‘‘๐‘ฅ๏ทฏ It is given C(๐‘ฅ) = 0.005๐‘ฅ3 โ€“ 0.02๐‘ฅ2 + 30๐‘ฅ + 5000 We need to find marginal cost when 3 unit produced i.e. need to find MC = ๏ท๐‘‘๐ถ๏ทฎ๐‘‘๐‘ฅ๏ทฏ at ๐‘ฅ = 3 Now, MC = ๏ท๐‘‘(๐‘)๏ทฎ๐‘‘๐‘ฅ๏ทฏ MC = ๏ท๐‘‘๏ท0.05๐‘ฅ3 โˆ’ 0.02๐‘ฅ2 + 30 + 5000๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ MC = ๏ท๐‘‘๏ท0.05๐‘ฅ3๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ โ€“ ๏ท๐‘‘(0.02๐‘ฅ2)๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๏ท๐‘‘ (30๐‘ฅ)๏ทฎ๐‘‘๐‘ฅ๏ทฏ + ๏ท๐‘‘ ๏ท5000๏ทฏ๏ทฎ๐‘‘๐‘ฅ๏ทฏ MC = 3 ร— 0.05x2 โ€“ 2 ร— 0.02x + 30 + 0 MC = 0.015x2 โ€“ 0.04x + 30 We need MC at ๐‘ฅ = 3 Putting ๐‘ฅ= 3 MC = 0.015(32) โ€“ 0.04 (3) + 30 MC = 30.015 Hence, Required marginal cost is Rs. 30.02 (nearly)

About the Author

Davneet Singh's photo - Teacher, Computer Engineer, Marketer
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
Jail