Then f 2006 4 00 and f 2006 1 75 also δ x 1 3 so f

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Basic Blueprint Reading and Sketching
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Chapter 26 / Exercise 13
Basic Blueprint Reading and Sketching
Olivo
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Thenf(2006) = 4.00andf0(2006)1.75.Also,Δx=-13.Sof(2005.67)4.00 + 1.75·(-13) = 3.42.55
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Basic Blueprint Reading and Sketching
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Chapter 26 / Exercise 13
Basic Blueprint Reading and Sketching
Olivo
Expert Verified
2.3Section 4: Second Derivatives2.3.1411Recall the functionf(x) =x2.We calculated thatf0(x) = 2x.Sof0is a linear function with slope 2.In other words: the derivative of the derivative offis 2.We say that thesecond derivativeoffis 2.f00(x) = 2Since the derivative off0is positive, that means the slope offis increasing.f00>0 =slope offincreasing =concave up2.3.2ConcavityIn general,f00>0=f0increasing=concave upf00<0=f0decreasing=concave downBut note thatf00isnotusually constant!So concavity can change within the same function.-2-1.5-1-0.500.511.52-2-1012yxff’f’’-4-3-2-101234-4-3-2-101234yxff’f’’56
2.3.3NotationIfy=f(x)then:f00(x) =ddxdydx=d2ydx2andf00(a) =d2ydx2x=a.2.4Section 5: Marginalia2.4.1Cost and RevenueSupposeC(q)is the cost of producingqgoods.We know thatC0is always+.Because producing more goods costs more money.The value ofC0reflectshowexpensive it is to produce goods.This can change depending on the production levelq.For smallq:C0is probably high.For moderateq:C0is probably low.For highq:C0is probably high.What aboutC00?For smallq:C00is probably-.For largeq:C00is probably+.-50510152025012345qff’f’’Note aroundq= 2: economies of scale change to diseconomies.Suppose thatR(q)is the revenue fromqgoods.We know thatR0is always+.Because selling more goods earns more revenue.The value ofR0is revenue/unit: the price.This can change depending on the production levelq:as more goods are produced price tends to fall.SoR0gets smaller over time:R00is-.57
-1012345012345yqff’f’’2.4.2Marginal AnalysisMarginalF=F0.Example:Suppose that a company is currently producing 1000 squeaky dog toys per month, andthatC0(1000) = 0.5andR0(1000) = 2.0. Should it increase production?Solution:Since marginal revenue exceeds marginal cost, Yes.Example:05101520012345qCostRevenueMarginal CostMarginal Revenue3Chapter 3: Differentiation Formulas3.1Polynomials3.1.1Old RulesIff(x) =mx+bthenf0(x) =m.Iff(x) =x2thenf0(x) = 2x.What about other functions?Example:Supposeg(x) =x2+ 5x-7.What isg0(3)?Solution:Think ofg(x)=(x2)+(5x-7)ddx(x2)x=3ddx(5x-7)2x|x=3= 65Sog0(3) = 11,30,1,65, or?g0(3) = 1158
3.1.2Addition RuleOne way to think about it is via linear approximation atx= 3:x29 + 6Δx5x-78 + 5Δxthereforex2+ 5x-717 + 11Δx.So evidently atx= 3the slope ofgis 11.In general,(f1+f2)0=f10+f20.So taking derivatives iseasierthan taking roots:9 + 166=9 +16.Example:Find the derivative ofx2-3x+ 4Solution:ddx(x2-3x+4)=ddxx2-ddx3x+ddx4=2x-3+0Answer:2x-3.3.1.3Constant Multiple RuleExample:Supposeg(x) = 7x2. What isg0(x)?The constant factor7produces a vertical stretch.All slopes get7times steeper.Sog0(x) = 7·2x= 14x.

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