““I feel like I’m diagonally I feel like I’m diagonally parked in a parallel parked in a parallel

universe”universe”

A)A) IntroductionIntroductiona.a. SymbolsSymbolsb.b. OperOperatioationsnsc.c. Central TendenciesCentral Tendencies

B)B) Linear AlgebraLinear AlgebraC)C)Correlation/Regression AnalysisCorrelation/Regression Analysis D)D) Applied CalculusApplied Calculus

Math ReviewMath ReviewMonday June 7 2003Monday June 7 2003

B)B)System of equationsSystem of equationsBasic Math ReviewBasic Math Review

a)a) 33xx - - yy = -7 = -755yy + 5 = -5 + 5 = -5xx

b)b) 3x + 4y = 23x + 4y = 22y = 4 - 3/2x 2y = 4 - 3/2x

D)D) Applied CalculusApplied CalculusBasic Math ReviewBasic Math Review

Rate of change (slope): Rate of change (slope): y/y/x or (yx or (y22-y-y11)/(x)/(x22-x-x11))HereHere

y/y/x is constant regardless of the “limit”x is constant regardless of the “limit”

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Broadway Address

Street Number

D)D) Applied CalculusApplied CalculusBasic Math ReviewBasic Math Review

How long does it take to fill one beaker (1L)?How long does it take to fill one beaker (1L)?

V(t)

t

⎥⎦

⎤⎢⎣

⎡=dt

dVmlt /1000

€

dV

dt

⎡ ⎣ ⎢

⎤ ⎦ ⎥=1000ml / t

D)D) Differential equationsDifferential equationsBasic Math ReviewBasic Math Review

A differential equation is an equation in which A differential equation is an equation in which one or more unknowns depend on its/their one or more unknowns depend on its/their rate rate of changeof change (or that of other variables included in (or that of other variables included in the equations)the equations)

2

2 )()(

dt

txd

dt

tdva

where

maF

==

=

Newton’s principiaNewton’s principia

D)D) Definition of a derivativeDefinition of a derivativeBasic Math ReviewBasic Math Review

x

xfxxfxf

x −+

=→

)()()( lim

00

The derivative of a function The derivative of a function ((xx)) at a point at a point “a” is the slope of the straight line tangent “a” is the slope of the straight line tangent to to ((xx)) at “a” at “a” instantaneous rate of instantaneous rate of change!change!One is pushing to limit to “0”: the slope is One is pushing to limit to “0”: the slope is close to real as close to real as ((xx)) approaches 0 approaches 0

D)D) Definition of a derivativeDefinition of a derivativeBasic Math ReviewBasic Math Review

x

xfxxfxf

x −+

=→

)()()( lim

00

The derivative of a function The derivative of a function ((xx)) = = f’f’((xx))Important derivatives:Important derivatives:ff((xx) = C ) = C f’f’((xx) = 0 ) = 0 ff((xx) = ) = xxn n f’f’((xx) = n) = nxxn-1n-1 ff((xx) = ) = eexx f’f’((xx) = ) = eexx ff((xx) = ln) = lnxx f’f’((xx) = 1/) = 1/xx

D)D) Maxima, MinimaMaxima, MinimaBasic Math ReviewBasic Math Review

One of the great applications of calculus One of the great applications of calculus (particularly in economics) is to determine the (particularly in economics) is to determine the “maxima” and “minima” of functions.“maxima” and “minima” of functions.The derivatives of the maxima and minima = 0The derivatives of the maxima and minima = 0

The function neither increase nor decreases!The function neither increase nor decreases!

ff((xx) = ) = xx33 – 3 – 3xx22 – 24 – 24x + x + 55

f’f’((xx) = 3) = 3xx22 – 6 – 6xx – 24 – 243 (3 (xx22 – 2 – 2xx – 8) – 8)3 (3 (xx + 2)( + 2)(xx –4) –4)

f’f’((xx)) vanishes (reaches a critical point) only when vanishes (reaches a critical point) only when f’f’((xx) = 0) = 0

D)D) Maxima, MinimaMaxima, Minima

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f”f”((xx) < 0 ) < 0 maximum maximumf”f”((xx) > 0 ) > 0 minimum minimum

D)D) Application: DerivativesApplication: DerivativesJi = - DS ([Ci ]/z)

DS = D0 2

J Cz=0 = -3 D0 (8 o

C) ([Ci ]/z)z=0

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CO2

Depth (cm)

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CH4

CO2 (uM)CH4 (uM)

D)D) Application: DerivativesApplication: DerivativesJi = - DS ([Ci ]/z)

DS = D0 2

J Cz=0 = -3 D0 (8 o

C) ([Ci ]/z)z=0

y = -0.0002x6 + 0.017x5 - 0.682x4 + 13.528x3 - 137.88x2 + 666.46x + 531.18

R2 = 0.9798

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Depth (cm)

CO2 (uM)

Poly.(CO2(uM))

““A River (used to) Run Through It”: Part IA River (used to) Run Through It”: Part I

Reservoirs and soil erosionReservoirs and soil erosion

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(C/N)a

Fo

C/N

1200

D)D) Application: IntegralApplication: Integral

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1200

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Jo (gC/m2.yr)

D)D) Application: IntegralApplication: Integral(x) = Polynomial (6th degree)

∫ (xn) = [n(n+1)/(n+1)] + C

y = -2E-10x 6 + 4E-08x 5 - 3E-06x 4 + 0.0001x3 - 0.0013x 2 + 0.0136x + 0.3082

R2 = 0.9795

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Jo (gC/m2.yr)

Integral between two limits (0 and 85 yrs)∫ (xn)100 - ∫ (xn)0

C - C