Math for Bus. and Eco. Chapter 2

• 1. Chapter TwoDifferentiation:Basic Concepts

2. Definition For the function y f(x), the derivative of f with respect to x isThe Derivativef (x x) f (x)f x limx 0 x if the limit exists. 2 3. The Power RuleTechniques of DifferentiationFor any number n, d n n 1x nxdxExample 1Differentiate (find the derivative of) eachof the following functions: 2711 y xy 27 yx yx x3 4. The Derivative of a constantTechniques of DifferentiationFor any constant C,d C0dx 4 5. The Constant Multiple RuleTechniques of DifferentiationFor any constant C,d df Cf C Cf xdxdxExample 2 5Differentiate the function y3x d 5 d 54 43x 3x3 5x15 x dxdx5 6. The Sum RuleTechniques of Differentiationd df dg f gf x g xExample 3 dxdx dx d25d2d 5x 3x x 3x dxdx dx4 2 x 15 x6 7. The Product RuleTechniques of Differentiationd df dg fg gfdxdx dxf x g x g x f xExample 4dd 2 d x 2 3x 1 3x 1 x 2 x3x 1dxdx dx3 x 1 2x x 2 3 9x 2 2x 7 8. The Derivative of a QuotientTechniques of Differentiation df dg gf d f dx dx2 dx g gExample 5Differentiate the rational function2x2 x 21yx 38 9. The Derivative of a QuotientTechniques of DifferentiationExample 5d dx 3x22 x 21 x 2 2 x 21x 3dydxdx2dxx 3x 3 2x 2 x22 x 21 2 x 3 2x 2 4 x 6 x 2 2 x 21x26 x 15 2 2 x 3 x 3 9 10. Forthe function y f(x), the change in xAverage Rate of Changeis x, the change in y is y and y f(x x) f(x) The average rate of change (ARC) of ywith respect to x is ARC y/ x Or f x xf xARCx10 11. Instantaneous Rate of Change When x approaches zero, the average rate of change becomes instantaneous rate of change (IRC). It is the derivative of the function f at any point x. IRC=f(x)=dy/dx 11 12. Instantaneous Rate of Change Example1 It is estimated that x months from now, the population of a certain community will beP(x) x2 20x 8,000 a) At what rate will the population be changing with respect to time 15 months from now? b) By how much will the population actually change during the 16th month? 12 13. Instantaneous Rate of Change Example1 a) The rate of change of the population withrespect to time is the derivative of thepopulation function. That is,Rate of Change P(x) 2x 20The rate of change of the population 15months from now will be Rate of Change 2 15 20Rate of Change 50 people /month 13 14. Instantaneous Rate of Change Example1 b) The actual change in the populationduring the 16th month is the differencebetween the population at the end of16 month and the population at theend of 15 months. That is,DP P(16) P(15)DP 8576 8525 51 people/month14 15. Percentage Rate of ChangeFor y f(x), the percentage rate ofchange of y with respect to x is definedbyf x PRC100f x 15 16. Percentage Rate of Change Example 2The gross national product (GNP) of acertain country was N(t)=t2+5t+106 billiondollars years after 1980.a) At what rate was the GNP changing with respect to time in 1988?b) At what percentage rate was theGNP changing with respect to time in1988? 16 17. Percentage Rate of Change Example 2a) The rate of change of GNP is the derivative of N(t) when t=8 (in 1988)N(t)=2t+5N(8)=2(8)+5=21 billion $/yearb) The percentage rate of change of the GNP in 1988 was PRC = 100 [N(8)/N(8)] PRC = 100 (21/210)=10%/year 17 18. Approximation by Differentialsrate of change Changechange of y within yin x respect to x18 19. Approximation by Differentials If y=f(x), and x a small change in x than the corresponding change in y is y=(dy/ dx) x In functional notation the corresponding change in f isf=f(x+x)-f(x) f (x) x19 20. Approximation by Differentials Example 1 Suppose the total cost in dollars of manufacturing q units of a certain commodity is C(q)=3q2+5q+10. If the current level of production is 40 units, estimate how the total cost will change if 40.5 units are produced. 20 21. Approximation by Differentials Example 1 The current value of the variable is q=40 and the change in variable q=0.5. By the approximation formula, the corresponding change in cost is C=C(40.5)-C(40 C(40) q =C(40) 0.5 Since C(q)=6q+5 and C(40)=6 40+5=245 It follows that C C(40) 0.5=245 0.5=$122.50 21 22. Approximation by Differentials Example 2 The daily output at a certain factory is Q(L)=900L^(1/3) where L denotes the size of the labor force measured in worker- hours. Currently, 1,000 worker-hours of labor are used each day. Use calculus to estimate the number of additional worker- hours of labor that will be needed to increase daily output by 15 units. (Answer: L= 5 worker-hours) 22 23. Approximation of Percentage change inPercentage quantity change 100 change size of quantity 23 24. Approximation of PercentageIf x is a small change in x, thecorresponding percentage change in thefunction f(x) is change f% f100 f xf x x 100 f x 24 25. Example 2Approximation of PercentageThe GNP of a certain country wasN(t)=t2+5t+200, billion dollars t years after1990. Use calculus to estimate the changepercentage change in the GNP during thefirst quarter of 1998.25 26. Example 3Approximation of Percentage N tt % N 100N t changeN(t)=t2+5t+200; N(t)=2t+5with t=8, N(8)=82+5 8+200=304N(8)=2 8+5=21; t=0.25Then%N 100(21)(0.25)/304=1.73 %26 27. Approximation of Percentage Example 4At a certain factory, the daily output isQ(K)=4,000K^(1/2) units, where Kdenotes the Firms capital investment. changeUse calculus to estimate the percentageincrease in output that will result from a1 percent increase in capital investment.(Answer: 0.5 %) 27 28. If C(x) is the total production costincurred by a manufacturer when x unitsare produced then C(x) is called theMarginal costmarginal cost.If production is increased by 1 unit, thenx=1 and the approximation formula: C=C(x+ x)-C(x) C(x) x becomes C=C(x+1)-C(x) C(x) 28 29. If R(x) is the total revenue derived fromsale of x units, then R(x) is called themarginal revenue.Marginal costIf sale is increased by 1 unit, then x 1and the approximation formula:R R(x x) R(x) R(x)x becomes R R(x 1) R(x) R(x)29 30. Marginal Cost and RevenueThe marginal cost C(x) is anapproximation to the costC(x 1) C(x) of producing the (x 1)st unit.The marginal revenue R(x) is anapproximation to the revenueR(x 1) R(x) derived from the sale of the(x 1)st unit. 30 31. Marginal Cost and Revenue Example 5A manufacture estimates that when x unitsof a particular commodity areproduced, the total cost will beC(x) (1/8)x2 3x 98 dollars, and thatP(x) (1/3)(75 x) dollars per unit is theprice at which all x units will be sold.a) Find the marginal cost and themarginal revenue.31 32. Marginal Cost and Revenue Example 5 (cont.)b) Use marginal cost to estimate the cost of producing the 9th unit.c) What is the actual cost of producing the 9th unit?d) Use the marginal revenue to estimate the revenue derived from the sale of the 9th unit.e) What is the actual revenue derived from the sale of the 9th unit?32 33. Marginal Cost and Revenue Example 5 (cont.)a) The marginal cost is C(x) x/4 3. Since x units of the commodity are sold at a price ofP(x) (75 x)/3 dollar per unit the total revenue is R(x) x P(x)=x(75 x)/3 25x x2/3 The marginal revenue isR(x) 25 2x/3 33 34. Marginal Cost and Revenue Example 5 (cont.)b) The cost of producing the 9th unit is the change in cost as x increase from 8 to 9 and can be estimated by the marginal cost C(8) 8/4 3 $5c) The actual cost of producing the 9th unit is C C(9) C(8) $5.13 34 35. Marginal Cost and Revenue Example 5 (cont.)d) The revenue obtained from the sale of the 9th unit is approximated by the marginal revenue: R(8) 25 (2/3)8 $19.67e) The actual revenue obtained from the sale of the 9th unit is R=R(9) R(8) $19.67 35 36. yy=f(x)DifferentialsPDQyD dyxxx x+Dx 36 37. From approximation formula: f f(x) x or y f(x) xwhen x approaches zero, we can writeDifferentialsdy ydx, which is called differential of y.37 38. Suppose y is a differentiable function of u and u is a differentiable function of x. Then y can be regarded as a function xThe Chain Rule and dydy du dxdu dx38 39. Example 1 Suppose that y uuand u x 3 17 Use the Chain Rule to find dy/dx andThe Chain Rule evaluate it at x 2. 39 40. Example 2, 3 o Find dy/dx if y u/(u 1) and u 3x2 1The Chain Rule when x 1. o Compute the derivatives of the following functions 324f x x 3x 2f x 2x x1f x 5 2x 3 40 41. Example 4 An environmental study of a certain suburban community suggests that the average dailyThe Chain Rule level of carbon monoxide in the air will be C(p) (0.5p2 17) parts per million when the population is p thousand. It is estimated that t years from now, the population of the community will be p(t) 3.1 0.1t2 thousand. At what rate will the carbon monoxide level be changing with respect to time 3 years from now?41 42. Example 4 The goal is to find dC/dt when t 3.1 dC1The Chain Rule 20.5 p 1720.5 2.p dp21 1 2 p 0.5 p 17 2 2 anddp 0.2tdt42 43. Example 4 It follows from the chain rule that1dc 1The Chain Rule2 p 0.5 p 17 20.2tdt 20.1pt0.5 p 2 17 when t 3, p p(3)3.1 0.1 32 4 . So, dc0.1 4 31.2 0.24 dt0.5 4 2 17 2543 44. The second derivative of a function is theThe Second Derivativederivative of its derivative. If y f(x), thesecond derivative is denoted by: 2dy 2 or f xdxThe second derivative gives the rate ofchange of the rate of change of theoriginal function. 44 45. Example 1The Second DerivativeFind both the first and second derivativesof the functions:f xx 3 12x 142f x5x 3x 3x 73x 2f x 2 x 1 45 46. Example 2An efficiency study of the morning shift atThe Second Derivativea certain factory indicates that an averageworker who arrives on the job at 8:00AM.Will have produced Q(t) t3 6t2 24t unitst hours later.a) Compute the workers rate of production at 11:00A.M46 47. Example 2b) At what rate is the workers rate ofThe Second Derivative production changing with respect to time at 11:00A.M?c) Use calculus to estimate the change in the workers rate of production between 11:00 and 11:10A.M.d) Compute the actual change in the workers rate of production between 11:00 and 11:10A.M. 47 48. Example 2a) The workers rate of production is the firstThe Second Derivative derivativeQ(t) 3t2 12t 24At 11:00 A.M.t 3 and the rate ofproduction isQ(3) 3 32 12 3 24 33 units per hour. 48 49. Example 2The Second Derivativeb) The rate of change of the rate of production is the second derivativeQ(t) 6t 12 At 11:00 A.M., the rate is Q(3) 6 3 12 6 units /hour /hour. 49 50. Example 2The Second Derivativec) Note that 10 minutes is 1/6 hours, and hence t 1/6 hour. Change in rate of production is Q Q(t) t Q6(1/6) 1 unit per hour.50 51. Example 2d) The actual change in the workers rateThe Second Derivative of production between 11:00 and 11:10 A.M. is the difference between the values of the rate Q(t) when t 3 and when t 19/6. That is Q(t) Q(19/6) Q(3) Q(t) 1.08 units per hour51 52. Suppose that f is differentiable on theinterval (a,b).The Concavity a) If f is increasing on (a,b),then the graphof f is concave upward on (a,b). b) If f is decreasing on (a,b), then thegraph of f is concave downward on(a,b). 52 53. Concave upwardThe Concavity (holds water)Concave downward (spills water)53 54. A critical point of a function is a point onits graph where either:Critical Points The derivative is zero, or The derivative is undefinedThe relative maxima and minima of thefunction can occur only at critical points. 54 55. yConcavityIncreasing, Concave upward xf (x) >0, f (x)>0 55 56. Concavity yConcavity x Increasing, concave downf x 0, f x 056 57. ConcavityyConcavityxdecreasing, concave upf x 0, f x057 58. ConcavityyConcavityxdecreasing, concave downf x 0, f x 0 58 59. Second-Derivative TestSuppose f(a)=0. If f(a)>0, then f has a relative minimum at x=a. If f(a)1, demand is said to be elasticwith respect to price.If | |