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Basic calculus (i)

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Calculus for one-variable functions

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Basic calculus (i)

  1. 1. Basic Calculus (I) Recap (for MSc & PhD Business, Management & Finance Students) First Draft: Autumn 2013 Revised: Autumn 2014 Lecturer: Farzad Javidanrad One-Variable Functions
  2. 2. Exponents (Powers) โ€ข Given ๐’ a positive integer and ๐’‚ a real number, ๐’‚ ๐’ indicates that ๐’‚ is multiplied by itself ๐’ times: ๐’‚ ๐’ = ๐’‚ ร— ๐’‚ ร— โ‹ฏ ร— ๐’‚ ๐’ ๐’•๐’Š๐’Ž๐’†๐’” โ€ข According to definition: ๐’‚ ๐ŸŽ = ๐Ÿ and ๐’‚ ๐Ÿ = ๐’‚
  3. 3. Exponents Rules ๏‚ง If ๐’Ž and ๐’ are positive integers and ๐’‚ is a real number, then: ๐’‚ ๐’Ž ร— ๐’‚ ๐’ = ๐’‚ ๐’Ž+๐’ With this rule we can define the concept of negative exponent (power): ๐‘Ž0 = 1 ๐‘Ž ๐‘šโˆ’๐‘š = 1 ๐‘Ž ๐‘š+(โˆ’๐‘š) = 1 ๐‘Ž ๐‘š ร— ๐‘Žโˆ’๐‘š = 1 ๐’‚โˆ’๐’Ž = ๐Ÿ ๐’‚ ๐’Ž
  4. 4. Exponents Rules โ€ข We can also define rational power as: ๐’‚ ๐’Ž ๐’ = ๐’ ๐’‚ ๐’Ž Some other rules are: (๐’‚ and ๐’ƒ are real numbers) ๏‚ง ๐’‚ ๐’Ž ๐’‚ ๐’ = ๐’‚ ๐’Žโˆ’๐’ e.g.( 311 38 = 311โˆ’8 = 33 = 27) ๏‚ง ๐’‚ ๐’Ž ๐’ = ๐’‚ ๐’ ๐’Ž = ๐’‚ ๐’Ž.๐’ ( 23 2 = 22 3 = 26 = 64) ๏‚ง ๐’‚. ๐’ƒ ๐’Ž = ๐’‚ ๐’Ž . ๐’ƒ ๐’Ž ( 3. ๐‘ฅ 2 = 32. ๐‘ฅ2 = 9๐‘ฅ2) ๏‚ง ๐’‚ ๐’ƒ ๐’Ž = ๐’‚ ๐’Ž ๐’ƒ ๐’Ž ( 3 5 3 = 33 53 = 27 125 ) ๏‚ง ๐’‚ โˆ’๐’Ž ๐’ = ๐Ÿ ๐’‚ ๐’Ž ๐’ = ๐Ÿ ๐’ ๐’‚ ๐’Ž = ๐Ÿ ๐’ ๐’‚ ๐’Ž (๐‘ฅ โˆ’2 3 = 1 ๐‘ฅ 2 3 = 1 3 ๐‘ฅ2 )
  5. 5. Algebraic Expressions, Equations and Identities โ€ข An algebraic expression is a combination of real numbers and variables, such as: ๏ฑMonomials : 5๐‘ฅ3 , โˆ’1.75 ๐‘ฆ , 3๐‘ฅ 4๐‘ง2 = 3 4 ๐‘ฅ๐‘งโˆ’2 ๏ฑBinomials: 4๐‘ฅ3 + 3๐‘ฅ2 , 3๐‘ฅ + 1 4๐‘ง2 = 3 4 ๐‘ฅ๐‘งโˆ’2 + 1 4 ๐‘งโˆ’2 ๏ฑPolynomials: ๐‘ฅ2 โˆ’ 3๐‘ฅ โˆ’ 6 , ๐‘ฅ3 + ๐‘ฅ๐‘ฆ2 + 6๐‘ฅ๐‘ฆ๐‘ง
  6. 6. Algebraic Expressions, Equations and Identities โ€ข Equations can be made when two expressions are equal to one another or an expression is equal to a number: 3๐‘ฅ โˆ’ 1 = ๐‘ฅ 4๐‘ฅ + 3๐‘ฆ = 2 5๐‘ฅ2 โˆ’ 2๐‘ฅ๐‘ฆ = ๐‘ฅ โˆ’ 6๐‘ฆ2 ๐‘ฅ2 โˆ’ 3๐‘ฅ โˆ’ 6 = 0 The first and second equations are linear with one and two variables respectively and the third equation is a quadratic in terms of ๐’™ and ๐’š and the forth equation is a quadratic equation in terms of ๐’™ . Note: Not all equations are solvable and many of them have no unique solutions.
  7. 7. Algebraic Expressions, Equations and Identities โ€ข If two expressions are equal for all values of their variable(s), the equation is called an identity. โ€ข For example; ๐‘ฅ + 3 2 = ๐‘ฅ2 + 6๐‘ฅ + 9 ๏‚ง Some important identities are: โ€ข ๐’‚ ยฑ ๐’ƒ ๐Ÿ = ๐’‚ ๐Ÿ ยฑ ๐Ÿ๐’‚๐’ƒ + ๐’ƒ ๐Ÿ โ€ข ๐’‚ ยฑ ๐’ƒ ๐Ÿ‘ = ๐’‚ ๐Ÿ‘ ยฑ ๐Ÿ‘๐’‚ ๐Ÿ ๐’ƒ + ๐Ÿ‘๐’‚๐’ƒ ๐Ÿ ยฑ ๐’ƒ ๐Ÿ‘ โ€ข ๐’‚ โˆ’ ๐’ƒ ๐’‚ + ๐’ƒ = ๐’‚ ๐Ÿ โˆ’ ๐’ƒ ๐Ÿ โ€ข ๐’‚ ยฑ ๐’ƒ ๐’‚ ๐Ÿ โˆ“ ๐’‚๐’ƒ + ๐’ƒ ๐Ÿ = ๐’‚ ๐Ÿ‘ ยฑ ๐’ƒ ๐Ÿ‘ โ€ข ๐’™ ยฑ ๐’‚ ๐’™ ยฑ ๐’ƒ = ๐’™ ๐Ÿ ยฑ ๐’‚ + ๐’ƒ ๐’™ + ๐’‚๐’ƒ
  8. 8. Some Other Identities โ€ข ๐’™ โˆ’ ๐’š = ๐’™ โˆ’ ๐’š ๐’™ + ๐’š = ๐Ÿ‘ ๐’™ โˆ’ ๐Ÿ‘ ๐’š ๐Ÿ‘ ๐’™ ๐Ÿ + ๐Ÿ‘ ๐’™๐’š + ๐Ÿ‘ ๐’š ๐Ÿ โ‹ฎ = ๐’ ๐’™ โˆ’ ๐’ ๐’š ๐’ ๐’™ ๐’โˆ’๐Ÿ + ๐’ ๐’™ ๐’โˆ’๐Ÿ ๐’š + ๐’ ๐’™ ๐’โˆ’๐Ÿ‘ ๐’š ๐Ÿ + โ‹ฏ + ๐’ ๐’š ๐’โˆ’๐Ÿ โ€ข ๐’™ + ๐’š + ๐’› ๐Ÿ = ๐’™ ๐Ÿ + ๐’š ๐Ÿ + ๐’› ๐Ÿ + ๐Ÿ๐’™๐’š + ๐Ÿ๐’™๐’› + ๐Ÿ๐’š๐’› โ€ข ๐‘ ๐‘–๐‘›2 ๐‘ฅ + ๐‘๐‘œ๐‘ 2 ๐‘ฅ = 1 โ€ข ๐‘ ๐‘–๐‘› ๐‘ฅ ยฑ ๐‘ฆ = ๐‘ ๐‘–๐‘›๐‘ฅ. ๐‘๐‘œ๐‘ ๐‘ฆ ยฑ ๐‘๐‘œ๐‘ ๐‘ฅ. ๐‘ ๐‘–๐‘›๐‘ฆ โ€ข ๐‘๐‘œ๐‘  ๐‘ฅ ยฑ ๐‘ฆ = ๐‘๐‘œ๐‘ ๐‘ฅ. ๐‘๐‘œ๐‘ ๐‘ฆ โˆ“ ๐‘ ๐‘–๐‘›๐‘ฅ. ๐‘ ๐‘–๐‘›๐‘ฆ โ€ข ๐‘ก๐‘Ž๐‘› ๐‘ฅ ยฑ ๐‘ฆ = ๐‘ก๐‘Ž๐‘›๐‘ฅยฑ๐‘ก๐‘Ž๐‘›๐‘ฆ 1โˆ“๐‘ก๐‘Ž๐‘›๐‘ฅ.๐‘ก๐‘Ž๐‘›๐‘ฆ
  9. 9. โ€ข ๐’™ + ๐’š ๐’ = ๐ŸŽ ๐’ ๐’™ ๐’ + ๐Ÿ ๐’ ๐’™ ๐’โˆ’๐Ÿ ๐’š + โ‹ฏ + ๐’“ ๐’ ๐’™ ๐’โˆ’๐’“ ๐’š ๐’“ + โ‹ฏ + ๐’ ๐’ ๐’š ๐’ Where ๐’“ ๐’ = ๐‘ช ๐’“ ๐’ = ๐’“๐‘ช ๐’ = ๐’! ๐’“! ๐’ โˆ’ ๐’“ ! And ๐’! = ๐’ ร— ๐’ โˆ’ ๐Ÿ ร— ๐’ โˆ’ ๐Ÿ ร— โ‹ฏ ร— ๐Ÿ‘ ร— ๐Ÿ ร— ๐Ÿ ๐ŸŽ! = ๐Ÿ! = ๐Ÿ So, ๐ŸŽ ๐’ = ๐ถ 0 ๐‘› = 0๐ถ ๐‘› = ๐‘›! 0! ๐‘› โˆ’ 0 ! = ๐‘›! ๐‘›! = ๐Ÿ ๐Ÿ ๐’ = ๐ถ 1 ๐‘› = 1๐ถ ๐‘› = ๐‘›! 1! ๐‘› โˆ’ 1 ! = ๐‘›! ๐‘› โˆ’ 1 ! = ๐‘› ร— ๐‘› โˆ’ 1 ! ๐‘› โˆ’ 1 ! = ๐’ ๐Ÿ ๐’ = ๐ถ 2 ๐‘› = 2๐ถ ๐‘› = ๐‘›! 2! ๐‘› โˆ’ 2 ! = ๐‘›! ๐‘› โˆ’ 2 ! = ๐‘› ร— ๐‘› โˆ’ 1 ร— ๐‘› โˆ’ 2 ! 2! ๐‘› โˆ’ 2 ! = ๐’(๐’ โˆ’ ๐Ÿ) ๐Ÿ Some Other Identities
  10. 10. Functions โ€ข All equations represent a relationship between two or more variables, e.g.: ๐‘ฅ๐‘ฆ = 1 , ๐‘ฅ 2๐‘ฆ + ๐‘ง = 0 โ€ข Given two variables in relation, there is a functional relationship between them if for each value of one of them there is one and only one value of another. โ€ข If the relationship between ๐’š and ๐’™ can be shown by ๐’š = ๐’‡ ๐’™ and for each value of ๐’™ there is one and only one value of ๐’š , then there is a functional relationship between them or alternatively it can be said that ๐’š is a function of ๐’™ , which means ๐’š as a dependent variable follows ๐’™ as an independent variable.
  11. 11. Functions โ€ข The idea of function is close to a processing (matching) machine. It receives inputs (which are the values of ๐’™ and is called domain of the function, ๐‘ซ ๐’‡) and after the processing them the output will be values of ๐’š in correspondence with ๐’™โ€ฒ ๐’” (which is called range of the function, ๐‘น ๐’‡). โ€ข There should be no element from ๐‘ซ ๐’‡ without a match from ๐‘น ๐’‡, but it might be found some free elements in ๐‘น ๐’‡. ๐’‡ = ๐’™ ๐Ÿ, ๐’š ๐Ÿ , ๐’™ ๐Ÿ, ๐’š ๐Ÿ , โ€ฆ , ๐’™ ๐’, ๐’š ๐’ ๐’‡ ๐’™ ๐Ÿ, ๐’™ ๐Ÿ, โ€ฆ , ๐’™ ๐’ ๐’š ๐Ÿ, ๐’š ๐Ÿ, โ€ฆ , ๐’š ๐’
  12. 12. Functions โ€ข Functions can be considered as correspondence (matching) rules, which corresponds all elements of ๐’™ to some elements of ๐’š.* โ€ข For example, the correspondence rule (f), which corresponds ๐’™ to each value of ๐’™, can be written as: Or ๐‘ฆ = ๐‘ฅ xxf ๏ก: 1 2 4 15 1 ๐Ÿ 2 ๐Ÿ๐Ÿ“ 20 x y
  13. 13. Functions โ€ข The correspondence rule, which corresponds ๐’™ ๐Ÿ โˆ’ ๐Ÿ๐ŸŽ to each value of ๐’™ can be shown as: Or ๐’š = ๐’™ ๐Ÿ โˆ’ ๐Ÿ๐ŸŽ 10: 2 ๏€ญxxg ๏ก -3 -2 0 2 3 -1 -6 -10 x y
  14. 14. Functions โ€ข Some correspondence rules indicate there is a relationship between ๐’™ and ๐’š but not a functional relationship, i.e. the relationship cannot be considered as a function. โ€ข For example, ๐’š = ยฑ ๐’™ (๐’š ๐Ÿ = ๐’™) is not a function (according to the definition of function) because for each value of ๐’™ there are two symmetrical values of ๐’š . Adopted from http://www.education.com/study-help/article/trigonometry-help-inverses-circular/
  15. 15. Functions โ€ข Note that in the graphical representation of a function, any parallel line with y-axis cross the graph of a function at one and only one point. Why? Adopted from http://mrhonner.com/archives/8599
  16. 16. Some Basic Functions โ€ข Power Function : ๐’š = ๐’™ ๐’ Adoptedfromhttp://mysite.verizon.net/bnapholtz/Math/powers.html If n>0 they all pass through the origin. If n<0 the function is not defined at x=0 ๐‘ฆ = ๐‘ฅโˆ’1 ๐‘ฆ = ๐‘ฅโˆ’1
  17. 17. Some Basic Functions โ€ข Exponential Function : ๐’š = ๐’‚ ๐’™ (๐’‚ > ๐ŸŽ, โ‰  ๐Ÿ) Adopted from http://www.softmath.com/tutorials-3/relations/exponential-functions-2.html All exponential functions passing through the point (0,1)
  18. 18. Some Basic Functions โ€ข Logarithmic Function : ๐’š = ๐ฅ๐จ๐  ๐’‚ ๐’™ (๐’‚ > ๐ŸŽ, โ‰  ๐Ÿ) Adopted fromhttp://mtc.tamu.edu/9-12/index_9-12.htm?9-12M2L2.htm Adopted from http://www.cliffsnotes.com/math/calculus/precalculus/exponential-and- logarithmic-functions/logarithmic-functions All logarithmic Functions passing through the point (1,0)
  19. 19. Some Basic Functions โ€ข Trigonometric Functions: ๐’š = ๐ฌ๐ข๐ง ๐’™ , ๐’š = ๐œ๐จ๐ฌ ๐’™ , ๐’š = ๐ญ๐š๐ง ๐’™ , ๐’š = ๐œ๐จ๐ญ ๐’™ Adoptedfromhttp://www.docstoc.com/docs/41284635/Graphs-of-the-Six-Trigonometric- Functions
  20. 20. โ€ข All trigonometric functions are periodic, i.e. after adding or subtracting a constant, which is called principal periodic constant, they repeat themselves. This periodic constant is ๐Ÿ๐… for ๐’”๐’Š๐’๐’™ and ๐’„๐’๐’”๐’™ but it is ๐… for ๐’•๐’‚๐’๐’™ and ๐’„๐’๐’•๐’™ , i.e. : (k is a positive integer) ๐‘ ๐‘–๐‘›๐‘ฅ = sin ๐‘ฅ ยฑ ๐Ÿ๐… = sin ๐‘ฅ ยฑ 4๐œ‹ = โ‹ฏ = sin ๐‘ฅ ยฑ 2๐‘˜๐œ‹ ๐‘๐‘œ๐‘ ๐‘ฅ = cos ๐‘ฅ ยฑ ๐Ÿ๐… = cos ๐‘ฅ ยฑ 4๐œ‹ = โ‹ฏ = cos ๐‘ฅ ยฑ 2๐‘˜๐œ‹ ๐‘ก๐‘Ž๐‘›๐‘ฅ = tan ๐‘ฅ ยฑ ๐… = tan ๐‘ฅ ยฑ 2๐œ‹ = โ‹ฏ = tan(๐‘ฅ ยฑ ๐‘˜๐œ‹) ๐‘๐‘œ๐‘ก๐‘ฅ = cot ๐‘ฅ ยฑ ๐… = cot ๐‘ฅ ยฑ 2๐œ‹ = โ‹ฏ = cot(๐‘ฅ ยฑ ๐‘˜๐œ‹) Some Basic Functions
  21. 21. Elementary Functions โ€ข Elementary functions can be made by combining basic functions through adding, subtracting, multiplying, dividing and also composing these basic functions. โ€ข For example: ๐‘ฆ = ๐‘ฅ2 + 4๐‘ฅ โˆ’ 1 ๐‘ฆ = ๐‘ฅ. ๐‘’โˆ’๐‘ฅ = ๐‘ฅ ๐‘’ ๐‘ฅ ๐‘ฆ = ๐‘’ ๐‘ ๐‘–๐‘›๐‘ฅ ๐‘ฆ = ln ๐‘ฅ2 + 4 ๐‘ฆ = ๐‘’ ๐‘ฅ (๐‘ ๐‘–๐‘›3๐‘ฅ โˆ’ ๐‘๐‘œ๐‘ 3๐‘ฅ)
  22. 22. Behaviour of a Function โ€ข After finding the relationship between two variables ๐’™ and ๐’š in the functional form ๐’š = ๐’‡(๐’™) the first question is how this function behaves. โ€ข Here we are interested in knowing about the magnitude and the direction of the change of ๐’š (๐‘–. ๐‘’. โˆ†๐’š) when the change of ๐’™ (๐‘–. ๐‘’. โˆ†๐’™) is getting smaller and smaller around a point in its domain. The technical term for this locality around a point is neighbourhood. So, we are trying to find the magnitude and the direction of the change of ๐’š in the neighbourhood of ๐’™. โ€ข Slope of a function is the concept which helps us to have this information. The value of the slope shows the magnitude of the change and the sign of slope shows the direction of the change.
  23. 23. Slope of a Linear Function โ€ข Letโ€™s start with one of the most used functions in science , which is the linear function: ๐’š = ๐’Ž๐’™ + ๐’‰ Where ๐’Ž shows the slope of the line (the average change of ๐’š in terms of a change in ๐’™). That is; ๐’Ž = ๐šซ๐’š ๐šซ๐’™ = ๐ญ๐š๐ง ๐œถ . The value of intercept is ๐’‰ which is the distance between the intersection point of the graph and y-axis from the Origin. The slope of a liner function is constant in its whole domain. y x h ๐’š = ๐’Ž๐’™ + ๐’‰ โˆ†๐’™ โˆ†๐’š ๐œถ ๐œถ
  24. 24. Slope of a Function in its General Form โ€ข Imagine we want to find the slope of the function ๐’š = ๐’‡(๐’™) at a specific point (for e.g. at ๐’™ ๐ŸŽ) in its domain. โ€ข Given a change of ๐’™ from ๐’™ ๐ŸŽ to ๐’™ ๐ŸŽ + โˆ†๐’™ the change of ๐’š Would be from ๐’‡ ๐’™ ๐ŸŽ to ๐’‡(๐’™ ๐ŸŽ + โˆ†๐’™) . โ€ข This means a movement along the curve from A to B. Adopted from http://www.bymath.com/studyguide/ana/sec/ana3.htm
  25. 25. Slope of a Function in its General Form โ€ข The average change of ๐’š in terms of a change in ๐’™ can be calculated by ๐šซ๐’š ๐šซ๐’™ = ๐ญ๐š๐ง ๐œถ , which is the slope of the line AB. โ€ข If the change in ๐’™ gradually disappear (โˆ†๐’™ โ†’ ๐ŸŽ)*, point B moves toward point A and the slope line (secant line) AB reaches to a limiting (marginal) situation AC, which is a tangent line on the curve of ๐’š = ๐’‡(๐’™) at point ๐‘จ(๐’™ ๐ŸŽ, ๐’‡(๐’™ ๐ŸŽ)).
  26. 26. Slope of a Function in its General Form โ€ข The slope of this tangent line AC is what is called derivative of ๐’š in terms of ๐’™ at point ๐‘ฅ0 and it is shown by different symbols such as ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ ๐‘ฅ=๐‘ฅ0 , ๐‘“โ€ฒ ๐‘ฅ0 , ๐‘‘๐‘“ ๐‘‘๐‘ฅ ๐‘ฅ=๐‘ฅ0 , ๐‘ฆโ€ฒ (๐‘ฅ0) , . โ€ข The slope of the tangent line at any point of the domain of the function is denoted by: ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ , ๐‘“โ€ฒ ๐‘ฅ , ๐‘‘๐‘“ ๐‘‘๐‘ฅ , ๐‘ฆโ€ฒ, ๐‘“๐‘ฅ โ€ฒ โ€ข Definition: The process of finding a derivative of a function is called differentiation . ' 0xf
  27. 27. Slope of a Function in its General Form โ€ข Therefore, the derivative of ๐’š = ๐’‡(๐’™)at any point in its domain is: ๐’šโ€ฒ = ๐’…๐’š ๐’…๐’™ = ๐’๐’Š๐’Ž โˆ†๐’™โ†’๐ŸŽ โˆ†๐’š โˆ†๐’™ = ๐’๐’Š๐’Ž โˆ†๐’™โ†’๐ŸŽ ๐’‡ ๐’™+โˆ†๐’™ โˆ’๐’‡(๐’™) โˆ†๐’™ And the derivative of ๐’š = ๐’‡(๐’™) at the specific point ๐’™ = ๐’™ ๐ŸŽ is: ๐’‡โ€ฒ ๐’™ ๐ŸŽ = ๐ฅ๐ข๐ฆ โˆ†๐’™โ†’๐ŸŽ ๐’‡ ๐’™ ๐ŸŽ + โˆ†๐’™ โˆ’ ๐’‡(๐’™ ๐ŸŽ) โˆ†๐’™ Where ๐ฅ๐ข๐ฆ stands for โ€œlimitโ€, showing limiting (marginal) situation of the ratio ๐šซ๐’š ๐šซ๐’™ .
  28. 28. Slope of a Function in its General Form โ€ข Note: For non-linear functions, slope of the function at any point depends on the value of that point and it is not constant in the whole domain of the function. This means that the derivative of a function is a function of the same variable itself. Adopted from http://www.columbia.edu/itc/sipa/math/slope_nonlinear.html http://www.pleacher.com/mp/mlessons/calc2006/day21.html
  29. 29. Derivative of Fundamental Basic Functions โ€ข Find the derivative of ๐‘ฆ = 2๐‘ฅ โˆ’ 1 at any point in its domain. ๐‘“ ๐‘ฅ = 2๐‘ฅ โˆ’ 1 ๐‘“ ๐‘ฅ + โˆ†๐‘ฅ = 2 ๐‘ฅ + โˆ†๐‘ฅ โˆ’ 1 = 2๐‘ฅ + 2โˆ†๐‘ฅ โˆ’ 1 โˆ†๐’š = ๐’‡ ๐’™ + โˆ†๐’™ โˆ’ ๐’‡ ๐’™ = ๐Ÿโˆ†๐’™ According to definition: ๐‘ฆโ€ฒ = ๐‘‘๐‘ฆ ๐‘‘๐‘ฅ = lim โˆ†๐‘ฅโ†’0 ๐‘“ ๐‘ฅ + โˆ†๐‘ฅ โˆ’ ๐‘“(๐‘ฅ) โˆ†๐‘ฅ = lim โˆ†๐‘ฅโ†’0 2โˆ†๐‘ฅ โˆ†๐‘ฅ = 2
  30. 30. Derivative of the Fundamental Basic Functions โ€ข Applying the same method, the derivative of the fundamental basic functions can be obtained as following: ๏ฑ ๐’š = ๐’™ ๐’ โ†’ ๐’šโ€ฒ = ๐’๐’™ ๐’โˆ’๐Ÿ e.g. : ๐‘ฆ = 3 โ†’ ๐‘ฆโ€ฒ = 0 ๐‘ฆ = ๐‘ฅ3 โ†’ ๐‘ฆโ€ฒ = 3๐‘ฅ2 ๐‘ฆ = ๐‘ฅโˆ’1 โ†’ ๐‘ฆโ€ฒ = โˆ’๐‘ฅโˆ’2 ๐‘ฆ = 5 ๐‘ฅ โ†’ ๐‘ฆโ€ฒ = 1 5 ๐‘ฅ 1 5 โˆ’1 = 1 5 5 ๐‘ฅ4
  31. 31. Derivative of the Fundamental Basic Functions ๏ฑ ๐’š = ๐’‚ ๐’™ โ†’ ๐’šโ€ฒ = ๐’‚ ๐’™. ๐’๐’๐’‚ ๐‘Ž > 0, โ‰  1 e.g. : ๐‘ฆ = 2 ๐‘ฅ โ†’ ๐‘ฆโ€ฒ = 2 ๐‘ฅ. ๐‘™๐‘›2 ๐‘ฆ = ๐‘’ ๐‘ฅ โ†’ ๐‘ฆโ€ฒ = ๐‘’ ๐‘ฅ ๏ฑ ๐’š = ๐ฅ๐จ๐  ๐’‚ ๐’™ โ†’ ๐’šโ€ฒ = ๐Ÿ ๐’™.๐’๐’๐’‚ e.g. : ๐‘ฆ = log ๐‘ฅ โ†’ ๐‘ฆโ€ฒ = 1 ๐‘ฅ. ๐‘™๐‘›10 ๐‘ฆ = ln ๐‘ฅ โ†’ ๐‘ฆโ€ฒ = 1 ๐‘ฅ
  32. 32. Derivative of the Fundamental Basic Functions ๏ฑ ๐’š = ๐ฌ๐ข๐ง ๐’™ โ†’ ๐’šโ€ฒ = ๐œ๐จ๐ฌ ๐’™ ๏ฑ ๐’š = ๐œ๐จ๐ฌ ๐’™ โ†’ ๐’šโ€ฒ = โˆ’ ๐ฌ๐ข๐ง ๐’™ ๏ฑ ๐’š = ๐ญ๐š๐ง ๐’™ โ†’ ๐’šโ€ฒ = ๐Ÿ + ๐ญ๐š๐ง ๐Ÿ ๐ฑ = ๐Ÿ ๐œ๐จ๐ฌ ๐Ÿ ๐ฑ ๏ฑ ๐’š = ๐œ๐จ๐ญ ๐’™ โ†’ ๐’šโ€ฒ = โˆ’ ๐Ÿ + ๐œ๐จ๐ญ ๐Ÿ ๐ฑ = โˆ’๐Ÿ ๐ฌ๐ข๐ง ๐Ÿ ๐ฑ
  33. 33. Differentiability of a Function ๏ƒ˜A function is differentiable at a point if despite any side approach to the point in its domain (from left or right) the derivative is the same and a finite number. Sharp corner points and points of discontinuity* are not differentiable. Adopted from Ahttp://www-math.mit.edu/~djk/calculus_beginners/chapter09/section02.html
  34. 34. Rules of Differentiation โ€ข If ๐’‡(๐’™) and ๐’ˆ ๐’™ are two differentiable functions in their common domain, then: ๏ถ ๐’‡(๐’™) ยฑ ๐’ˆ(๐’™) โ€ฒ = ๐’‡โ€ฒ(๐’™) ยฑ ๐’ˆโ€ฒ(๐’™) ๏ถ ๐’‡ ๐’™ . ๐’ˆ(๐’™) โ€ฒ = ๐’‡โ€ฒ ๐’™ . ๐’ˆ ๐’™ + ๐’ˆโ€ฒ ๐’™ . ๐’‡(๐’™) ๏ถ ๐’‡(๐’™) ๐’ˆ(๐’™) โ€ฒ = ๐’‡โ€ฒ ๐’™ .๐’ˆ ๐’™ โˆ’๐’ˆโ€ฒ ๐’™ .๐’‡(๐’™) ๐’ˆ(๐’™) ๐Ÿ (Quotient Rule) ๏ถ ๐’‡(๐’ˆ ๐’™ ) โ€ฒ = ๐’ˆโ€ฒ ๐’™ . ๐’‡โ€ฒ(๐’ˆ ๐’™ ) (Chain Rule) (Summation & Sub. Rules. They can be extended to n functions) (Multiplication Rule and can be extended to n functions)
  35. 35. ๏ƒ˜ Find the derivative of the following functions: o ๐‘ฆ = ๐‘ฅ + ๐‘™๐‘›๐‘ฅ โˆถ ๐’šโ€ฒ = ๐Ÿ + ๐Ÿ ๐’™ o ๐‘ฆ = ๐‘’ ๐‘ฅ. ๐‘ ๐‘–๐‘›๐‘ฅ โˆถ ๐’šโ€ฒ = ๐’† ๐’™. ๐’”๐’Š๐’๐’™ + ๐’† ๐’™. ๐’„๐’๐’”๐’™ = ๐’† ๐’™ ๐’”๐’Š๐’๐’™ + ๐’„๐’๐’”๐’™ o ๐‘ฆ = 2๐‘ฅ ๐‘ฅ2+1 โˆถ ๐’šโ€ฒ = ๐Ÿ ๐’™ ๐Ÿ+๐Ÿ โˆ’๐Ÿ๐’™.๐Ÿ๐’™ ๐’™ ๐Ÿ+๐Ÿ ๐Ÿ = ๐Ÿโˆ’๐Ÿ๐’™ ๐Ÿ ๐’™ ๐Ÿ+๐Ÿ ๐Ÿ o ๐‘ฆ = 3 ๐‘ฅ2 + 1 โˆถ ๐’šโ€ฒ = ๐Ÿ๐’™. ๐Ÿ ๐Ÿ‘ . ๐’™ ๐Ÿ + ๐Ÿ ๐Ÿ ๐Ÿ‘ โˆ’๐Ÿ = ๐Ÿ๐’™ ๐Ÿ‘ ๐Ÿ‘ ๐’™ ๐Ÿ+๐Ÿ ๐Ÿ Rules of Differentiation
  36. 36. o ๐‘ฆ = ๐‘™๐‘›2 ๐‘ฅ โˆถ ๐’šโ€ฒ = ๐Ÿ ๐’™ . ๐Ÿ. ๐’๐’๐’™ = ๐Ÿ๐’๐’๐’™ ๐’™ o ๐‘ฆ = 5 ๐‘ฅ2 + tan 3๐‘ฅ โˆถ ๐’šโ€ฒ = ๐Ÿ“ ๐’™ ๐Ÿ . ๐ฅ๐ง๐Ÿ“. (๐Ÿ๐’™) + ๐Ÿ‘(๐Ÿ + ๐’•๐’‚๐’ ๐Ÿ ๐Ÿ‘๐’™) โ€ข The last rule(page 32) is called the chain rule which should be applied for composite functions such as the above functions, but it can be extended to include more functions. โ€ข If ๐’š = ๐’‡ ๐’– and ๐’– = ๐’ˆ ๐’› and ๐’› = ๐’‰ ๐’ and ๐’ = ๐’Œ(๐’™) then ๐’š depends on ๐’™ but through some other variables ๐’š = ๐’‡ ๐’ˆ ๐’‰ ๐’Œ ๐’™ Rules of Differentiation
  37. 37. โ€ข Under such circumstances we can extend the chain rule to cover all these functions, i.e. ๐’…๐’š ๐’…๐’™ = ๐’…๐’š ๐’…๐’– . ๐’…๐’– ๐’…๐’› . ๐’…๐’› ๐’…๐’ . ๐’…๐’ ๐’…๐’™ o ๐‘ฆ = ๐‘๐‘œ๐‘ 3 2๐‘ฅ + 1 โˆถ ๐‘ฆ = ๐‘ข3 ๐‘ข = ๐‘๐‘œ๐‘ ๐‘ง ๐‘ง = 2๐‘ฅ + 1 ๐’šโ€ฒ = ๐’…๐’š ๐’…๐’™ = ๐’…๐’š ๐’…๐’– . ๐’…๐’– ๐’…๐’› . ๐’…๐’› ๐’…๐’™ = ๐Ÿ‘๐’– ๐Ÿ. โˆ’๐’”๐’Š๐’๐’› . ๐Ÿ = โˆ’๐Ÿ”๐’„๐’๐’” ๐Ÿ ๐Ÿ๐’™ + ๐Ÿ . ๐ฌ๐ข๐ง(๐Ÿ๐’™ + ๐Ÿ) Rules of Differentiation
  38. 38. Implicit Differentiation โ€ข ๐’š = ๐’‡ ๐’™ is an explicit function because the dependent variable ๐’š is at one side and explicitly expressed by independent variable ๐’™. Implicit form of this function can be shown by ๐‘ญ ๐’™, ๐’š = ๐ŸŽ where both variables are in one side: o Explicit Functions: ๐‘ฆ = ๐‘ฅ2 โˆ’ 3๐‘ฅ , ๐‘ฆ = ๐‘’ ๐‘ฅ. ๐‘™๐‘›๐‘ฅ , ๐‘ฆ = ๐‘ ๐‘–๐‘›๐‘ฅ ๐‘ฅ o Implicit Functions: 2๐‘ฅ โˆ’ 7๐‘ฆ + 3 = 0 , 2 ๐‘ฅ๐‘ฆ โˆ’ ๐‘ฆ2 = 0 โ€ข Many implicit functions can be easily transformed to an explicit function but it cannot be done for all. In this case, differentiation with respect to ๐’™ can be done part by part and ๐’š should be treated as a function of ๐’™.
  39. 39. o Find the derivative of ๐Ÿ๐’™ โˆ’ ๐Ÿ•๐’š + ๐Ÿ‘ = ๐ŸŽ. Differentiating both sides with respect to ๐’™, we have: ๐‘‘ ๐‘‘๐‘ฅ 2๐‘ฅ โˆ’ 7๐‘ฆ + 3 = ๐‘‘ ๐‘‘๐‘ฅ 0 2 โˆ’ 7๐‘ฆโ€ฒ + 0 = 0 โ†’ ๐’šโ€ฒ = ๐Ÿ ๐Ÿ• o Find the derivative of ๐’™ ๐Ÿ โˆ’ ๐Ÿ๐’™๐’š + ๐’š ๐Ÿ‘ = ๐ŸŽ. Using the same method, we have: 2๐‘ฅ โˆ’ 2๐‘ฆ โˆ’ 2๐‘ฅ๐‘ฆโ€ฒ + 3๐‘ฆ2 ๐‘ฆโ€ฒ = 0 โ†’ ๐’šโ€ฒ = ๐Ÿ๐’š โˆ’ ๐Ÿ๐’™ ๐Ÿ‘๐’š ๐Ÿ โˆ’ ๐Ÿ๐’™ Implicit Differentiation
  40. 40. o Find the derivative of ๐Ÿ ๐’™๐’š โˆ’ ๐’š ๐Ÿ = ๐ŸŽ ๐‘ฆ + ๐‘ฅ๐‘ฆโ€ฒ 2 ๐‘ฅ๐‘ฆ โˆ’ 2๐‘ฆ๐‘ฆโ€ฒ = 0 โ†’ ๐’šโ€ฒ = ๐’š. ๐Ÿ ๐’™๐’š ๐Ÿ๐’š โˆ’ ๐’™. ๐Ÿ ๐’™๐’š o Find the derivative of ๐’”๐’Š๐’ ๐’™ ๐’š โˆ’ ๐ฅ๐ง ๐’™๐’š = ๐ŸŽ ๐‘ฆ โˆ’ ๐‘ฅ๐‘ฆโ€ฒ ๐‘ฆ2 . ๐‘๐‘œ๐‘  ๐‘ฅ ๐‘ฆ โˆ’ ๐‘ฆ + ๐‘ฅ๐‘ฆโ€ฒ ๐‘ฅ๐‘ฆ = 0 Then ๐’šโ€ฒ = ๐Ÿ ๐’š . ๐’„๐’๐’” ๐’™ ๐’š โˆ’ ๐Ÿ ๐’™ ๐’™ ๐’š ๐Ÿ . ๐’„๐’๐’” ๐’™ ๐’š + ๐Ÿ ๐’š Implicit Differentiation
  41. 41. Higher Orders Derivatives โ€ข As ๐’šโ€ฒ = ๐’‡โ€ฒ(๐’™) is itself a function of ๐’™ , in case it is differentiable, we can think of second, third or even n-th derivatives: โ€ข Second Derivative: ๐’šโ€ฒโ€ฒ , ๐’… ๐Ÿ ๐’š ๐’…๐’™ ๐Ÿ , ๐’…( ๐’…๐’š ๐’…๐’™ ) ๐’…๐’™ , ๐’… ๐’…๐’™ ๐’‡โ€ฒ , ๐’‡โ€ฒโ€ฒ ๐’™ โ€ข Third Derivative: ๐’šโ€ฒโ€ฒโ€ฒ , ๐’… ๐Ÿ‘ ๐’š ๐’…๐’™ ๐Ÿ‘ , ๐’…( ๐’… ๐Ÿ ๐’š ๐’…๐’™ ๐Ÿ) ๐’…๐’™ , ๐’… ๐’…๐’™ ๐’‡โ€ฒโ€ฒ , ๐’‡โ€ฒโ€ฒโ€ฒ ๐’™ โ€ข N-th Derivative: ๐’š(๐’) , ๐’… ๐’ ๐’š ๐’…๐’™ ๐’ , ๐’…( ๐’…(๐’โˆ’๐Ÿ) ๐’š ๐’…๐’™(๐’โˆ’๐Ÿ)) ๐’…๐’™ , ๐’… ๐’…๐’™ ๐’‡(๐’โˆ’๐Ÿ) , ๐’‡(๐’) ๐’™
  42. 42. o Find the second and third derivatives of ๐’š = ๐’†โˆ’๐’™. ๐‘ฆโ€ฒ = โˆ’๐‘’โˆ’๐‘ฅ ๐‘ฆโ€ฒโ€ฒ = ๐‘’โˆ’๐‘ฅ ๐‘ฆโ€ฒโ€ฒโ€ฒ = โˆ’๐‘’โˆ’๐‘ฅ o If ๐’š = ๐’† ๐œฝ๐’™ show that the equation ๐’šโ€ฒโ€ฒโ€ฒ โˆ’ ๐’šโ€ฒโ€ฒ = ๐ŸŽ has two roots. ๐‘ฆโ€ฒ = ๐œƒ๐‘’ ๐œƒ๐‘ฅ ๐‘ฆโ€ฒโ€ฒ = ๐œƒ2 ๐‘’ ๐œƒ๐‘ฅ ๐‘ฆโ€ฒโ€ฒโ€ฒ = ๐œƒ3 ๐‘’ ๐œƒ๐‘ฅ ๐‘ฆโ€ฒโ€ฒโ€ฒ โˆ’ ๐‘ฆโ€ฒโ€ฒ = ๐œƒ3 ๐‘’ ๐œƒ๐‘ฅ โˆ’ ๐œƒ2 ๐‘’ ๐œƒ๐‘ฅ = 0 ๐œƒ2 ๐‘’ ๐œƒ๐‘ฅ ๐œƒ โˆ’ 1 = 0 ๐‘’ ๐œƒ๐‘ฅ โ‰  0 โ†’ ๐œƒ = 0, ๐œƒ = 1 Higher Orders Derivatives
  43. 43. First & Second Order Differentials โ€ข If ๐’š = ๐’‡(๐’™) is differentiable on an interval then at any point of that interval the derivative of ๐’‡ can be defined as: ๐’šโ€ฒ = ๐’‡โ€ฒ ๐’™ = ๐’…๐’š ๐’…๐’™ = ๐ฅ๐ข๐ฆ โˆ†๐’™โ†’๐ŸŽ ๐šซ๐’š ๐šซ๐’™ โ€ข This means when ๐šซ๐’™ becomes โ€œinfinitesimalโ€ (getting smaller infinitely; โˆ†๐’™ โ†’ ๐ŸŽ), the ratio ๐šซ๐’š ๐šซ๐’™ approaches to the derivative of the function, i.e. the difference between ๐šซ๐’š ๐šซ๐’™ and ๐’‡โ€ฒ ๐’™ is infinitesimal itself and ignorable: ๐šซ๐’š ๐šซ๐’™ โ‰ˆ ๐’‡โ€ฒ ๐’™ ๐’๐’“ โˆ†๐’š โ‰ˆ ๐’‡โ€ฒ ๐’™ . โˆ†๐’™ โ€ข ๐’‡โ€ฒ ๐’™ . โˆ†๐’™ is called โ€œ differential of ๐’š โ€ and is shown by ๐’…๐’š, so: โˆ†๐’š โ‰ˆ ๐’‡โ€ฒ ๐’™ . โˆ†๐’™ = ๐’…๐’š As โˆ†๐’™ is an independent increment of ๐’™ we can always assume that ๐’…๐’™ = โˆ†๐’™; so we can re-write the above as โˆ†๐’š โ‰ˆ ๐’‡โ€ฒ ๐’™ . ๐’…๐’™ = ๐’…๐’š
  44. 44. โ€ข The geometric interpretation of ๐’…๐’š and โˆ†๐’š : โˆ†๐’š represents the change in height of the curve and ๐’…๐’š represents the change in height of the tangent line when โˆ†๐’™ changes (see the graph) Adopted fromhttp://www.cliffsnotes.com/math/calculus/calculus/applications-of-the- derivative/differentials So: ๐’…๐’š = ๐’šโ€ฒ. ๐’…๐’™ Some rules: If ๐’– and ๐’— are differentiable functions, then: i. ๐’… ๐’„๐’– = ๐’„. ๐’…๐’– (c is constant) ii. ๐’… ๐’– ยฑ ๐’— = ๐’…๐’– ยฑ ๐’…๐’— (can be extended to more than two functions) iii. ๐’… ๐’–. ๐’— = ๐’—. ๐’…๐’– + ๐’–. ๐’…๐’— (extendable) iv. ๐’… ๐’– ๐’— = ๐’—.๐’…๐’–โˆ’๐’–.๐’…๐’— ๐’— ๐Ÿ First & Second Order Differentials
  45. 45. โ€ข Using the third rule of differentials, the second order differential of ๐’š can be calculated, i.e. : ๐’… ๐Ÿ ๐’š = ๐’… ๐’…๐’š = ๐’… ๐’šโ€ฒ . ๐’…๐’™ = ๐’…๐’šโ€ฒ. ๐’…๐’™ + ๐’šโ€ฒ. ๐’… ๐’…๐’™ = ๐’šโ€ฒโ€ฒ. ๐’…๐’™. ๐’…๐’™ + ๐’šโ€ฒ. ๐’… ๐Ÿ ๐’™ = ๐’šโ€ฒโ€ฒ. ๐’…๐’™ ๐Ÿ + ๐’šโ€ฒ. ๐’… ๐Ÿ ๐’™ As ๐’™ is not dependent on another variable and ๐’…๐’™ is a constant : ๐’… ๐Ÿ ๐’™ = ๐’… ๐’…๐’™ = ๐ŸŽ So, ๐’… ๐Ÿ ๐’š = ๐’šโ€ฒโ€ฒ. ๐’…๐’™ ๐Ÿ = ๐’šโ€ฒโ€ฒ. ๐’…๐’™ ๐Ÿ (or in the familiar form ๐’šโ€ฒโ€ฒ = ๐’… ๐Ÿ ๐’š ๐’…๐’™ ๐Ÿ ) Where ๐’…๐’™ ๐Ÿ = ๐’…๐’™ ๐Ÿ is always positive and the sign of ๐’… ๐Ÿ ๐’š depends on the sign of ๐’šโ€ฒโ€ฒ. โ€ข Applying the same method we have ๐’… ๐’ ๐’š = ๐’š(๐’). ๐’…๐’™ ๐’ . First & Second Order Differentials
  46. 46. Derivative and Optimisation of Functions โ€ข Function ๐’š = ๐’‡ ๐’™ is said to be an increasing function at ๐’™ = ๐’‚ if at any small neighbourhood (โˆ†๐’™) of that point: ๐‘Ž + โˆ†๐‘ฅ > ๐‘Ž โ†” ๐‘“ ๐‘Ž + โˆ†๐‘ฅ > ๐‘“ ๐‘Ž From the above inequality we can conclude that: ๐‘“ ๐‘Ž+โˆ†๐‘ฅ โˆ’๐‘“(๐‘Ž) โˆ†๐‘ฅ โ‰ˆ ๐‘“โ€ฒ(๐‘Ž) > 0 So, the function is increasing at ๐’™=๐’‚ if ๐’‡โ€ฒ(๐’‚)>๐ŸŽ , and decreasing if ๐’‡โ€ฒ(๐’‚)<๐ŸŽ . Adopted from http://portal.tpu.ru/SHARED/k/KONVAL/Sites/English_sites/calculus/3_Geometric_f.htm a a
  47. 47. โ€ข More generally, the function ๐’š = ๐’‡(๐’™) is increasing (decreasing) in an interval if at any point in that interval ๐’‡โ€ฒ ๐’™ > ๐ŸŽ ( ๐’‡โ€ฒ ๐’™ < ๐ŸŽ ). Derivative and Optimisation of Functions Adopted from http://www.webgraphing.com/polynomialdefs.jsp
  48. 48. Derivative and Optimisation of Functions โ€ข If the sign of ๐’‡โ€ฒ (๐’™) is changing when passing a point such as ๐’™ = ๐’‚ (from negative to positive or vice versa) and ๐’š = ๐’‡(๐’™) is differentiable at that point, It is very logical to think that ๐’‡โ€ฒ (๐’™) at that point should be zero, i.e. : ๐’‡โ€ฒ ๐’‚ = ๐ŸŽ. (in this case the tangent line is horizontal) โ€ข This point is called local (relative) maximum or local (relative) minimum. In some books it is called critical point or extremum point. http://www-rohan.sdsu.edu/~jmahaffy/courses/s00a/math121/lectures/graph_deriv/diffgraph.html Not an extremum or critical point
  49. 49. โ€ข If ๐’‡โ€ฒ ๐’‚ = ๐ŸŽ but the sign of ๐’‡โ€ฒ(๐’™) does not change when passing the point ๐’™ = ๐’‚, the point (๐’‚, ๐’‡ ๐’‚ ) is not a extremum or critical point (point C in the previous slide). โ€ข For a function which is differentiable in its domain(or part of that), a sign change of ๐’‡โ€ฒ when passing a point is a sufficient evidence of the point being a extremum point. Therefore, at that point ๐’‡โ€ฒ(๐’™) will be necessarily zero. Necessary and Sufficient Conditions ๐’‡โ€ฒ ๐’™ > ๐ŸŽ ๐’‡โ€ฒ ๐’™ < ๐ŸŽ ๐‘“โ€ฒ ๐‘Ž = 0 Adopted and altered from http://homepage.tinet.ie/~phabfys/maxim.htm/ ๐’‡โ€ฒ (๐’™) > ๐ŸŽ ๐‘“โ€ฒ ๐‘ = 0 ๐’‡โ€ฒ ๐’™ < ๐ŸŽ a b
  50. 50. โ€ข If a function is not differentiable at a point (see the graph, point x=c) but the sign of ๐’‡โ€ฒ changes, it is sufficient to say the point is a extremum point despite non-existence of ๐’‡โ€ฒ(๐’™) . Necessary and Sufficient Conditions Adopted from http://www.nabla.hr/Z_IntermediateAlgebraIntroductionToFunctCont_3.htm ๐’‡โ€ฒ (๐’„) is not defined as it goes to infinity These types of critical points cannot be obtained through solving the equation ๐’‡โ€ฒ ๐’™ = ๐ŸŽ as they are not differentiable at these points.
  51. 51. Second Derivative Test โ€ข Apart from the sign change of ๐’‡โ€ฒ (๐’™) there is another test to distinguish between extremums. This test is suitable for those functions which are differentiable at least twice at the critical points. โ€ข Assume that ๐’‡โ€ฒ ๐’‚ = ๐ŸŽ; so, the point (๐’‚, ๐’‡ ๐’‚ ) is suspicious to be a maximum or minimum. If ๐’‡โ€ฒโ€ฒ ๐’‚ > ๐ŸŽ, the point is a minimum point and if ๐’‡โ€ฒโ€ฒ ๐’‚ < ๐ŸŽ, the point is a maximum point. Adopted and altered from http://www.webgraphing.com/polynomialdefs.jsp Inflection point Concave Down Concave up ๐‘“โ€ฒ ๐‘ฅ = 0 ๐‘“โ€ฒ ๐‘ฅ = 0 ๐‘“โ€ฒโ€ฒ ๐‘ฅ = 0
  52. 52. Inflection Point & Concavity of Function โ€ข If ๐’‡โ€ฒ ๐’‚ = ๐ŸŽ and at the same time ๐’‡โ€ฒโ€ฒ ๐’‚ = ๐ŸŽ, we need other tests to find out the nature of the point. It could be a extremum point [e.g. ๐’š = ๐’™ ๐Ÿ’ , which has minimum at ๐’™ = ๐ŸŽ]or just an inflection point (where the tangent line crosses the graph of the function and separate that to two parts; concave up and concave down) Adopted and altered from http://www.ltcconline.net/greenl/courses/105/curvesketching/SECTST.HTM Adopted from http://www.sparkle.pro.br/tutorial/geometry ๐‘“โ€ฒโ€ฒ ๐‘ฅ = 0 ๐‘“โ€ฒ ๐‘ฅ > 0 Concave Down Concave up
  53. 53. Some Examples o Find extremums of ๐’š = ๐’™ ๐Ÿ‘ โˆ’ ๐Ÿ‘๐’™ ๐Ÿ + ๐Ÿ, if any. To find the points which could be our extremums (critical points) we need to find the roots of this equation: ๐’‡โ€ฒ ๐’™ = ๐ŸŽ, So, ๐’‡โ€ฒ ๐’™ = ๐Ÿ‘๐’™ ๐Ÿ โˆ’ ๐Ÿ”๐’™ = ๐ŸŽ โ†’ ๐Ÿ‘๐’™ ๐’™ โˆ’ ๐Ÿ = ๐ŸŽ โ†’ ๐’™ = ๐ŸŽ, ๐’™ = ๐Ÿ Two points ๐‘จ(๐ŸŽ, ๐Ÿ) and ๐‘ฉ(๐Ÿ, โˆ’๐Ÿ) are possible extremums. Sufficient condition(1st method): As the sign of ๐’šโ€ฒ = ๐’‡โ€ฒ(๐’™) changes while passing through the points there is a maximum and a minimum. ๐’™ โˆ’โˆž +โˆž ๐‘ฆโ€ฒ + โˆ’ + ๐‘ฆ 0 2 2 -2 Max Min
  54. 54. Some Examples โ€ข Sufficient condition (2nd method): we need to find the sign of ๐’‡โ€ฒโ€ฒ(๐’™) at those critical points: ๐’‡โ€ฒโ€ฒ ๐’™ = ๐Ÿ”๐’™ โˆ’ ๐Ÿ” ๐’‡โ€ฒโ€ฒ ๐’™ = ๐ŸŽ = โˆ’๐Ÿ” โ†’ ๐‘จ ๐ŸŽ, ๐Ÿ ๐’Š๐’” ๐’Ž๐’‚๐’™๐’Š๐’Ž๐’–๐’Ž ๐’‡โ€ฒโ€ฒ ๐’™ = ๐Ÿ = ๐Ÿ” โ†’ ๐‘ฉ ๐Ÿ, โˆ’๐Ÿ ๐’Š๐’” ๐’Ž๐’Š๐’๐’Š๐’Ž๐’–๐’Ž o Find the extremum(s) of ๐’š = ๐Ÿ โˆ’ ๐Ÿ‘ ๐’™ ๐Ÿ, if any. ๐’šโ€ฒ = โˆ’๐Ÿ ๐Ÿ‘ ๐Ÿ‘ ๐’™ Although ๐’šโ€ฒ cannot be zero but its sign changes when passing through ๐’™ = ๐ŸŽ, so the function has a maximum at point ๐‘จ(๐ŸŽ, ๐Ÿ). The second method of the sufficient condition cannot be used here. Why? ๐’™ โˆ’โˆž +โˆž ๐‘ฆโ€ฒ + ๐‘ฆ 0 1 Max

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