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Econ 300_Summer 2009_Slides 4- Differential Calculus[1]

# Econ 300_Summer 2009_Slides 4- Differential Calculus[1] -...

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Unformatted text preview: Why differential calculus? • Economic models assume rational optimizers – Consumers maximize utility – Producers maximize profits – NBA owners maximize combination of wins and profits • Optimization uses calculus to evaluate tradeoffs – How much to consume? • Consume until marginal utility = price – How much to produce? • Produce until marginal revenue = marginal cost – Which free agents to go for? Average rate of change over [x0,x1] 'y f ( x1 )  f ( x0 ) { , where 'x x1  x 0 'x { x1  x0 and 'y { y1  y0 Average rate of change examples 'y f ( x1 )  f ( x0 ) { 'x x1  x0 y 'y a  bx : 'x x12  x0 2 x1  x0 bx1  bx0 x1  x0 b y 'y 2 x: 'x ( x1  x0 )( x1  x0 ) x1  x0 x1  x0 y x 2 'y 'x 6 'y 'x 3 Average rate of change and difference quotient 'y f ( x1 )  f ( x) f ( x  'x)  f ( x) { { 'x 'x x1  x 'y b( x  'x)  bx y a  bx : b 'x 'x y 'y 2 x: 'x ( x  'x) 2  x 2 'x 2 x'x  'x 2 'x 2 x  'x Some properties • Rate of change of sum = sum of rates of change – y, w, z are functions of x and y = w + z – Then 'y ' ( w  z ) 'w 'x • Scaling: 'x '(ay ) 'x 'z  'x 'x 'y a 'x Application: quadratic '( x 2 ) 'x ( x  'x) 2  x 2 'x 2 x'x  'x 2 'x 2 x  'x y 'y 'x ax  bx  c '( x ) 'x '1 a b c 'x 'x 'x a (2 x  'x)  b 2 2 Application: cubic '( x ) 'x 3 ( x  'x)( x  2 x'x  'x )  x 'x 3 2 2 3 3 ( x  3 x 'x  3 x'x  'x )  x 'x 2 2 3 x  3 x'x  'x 3 2 3 2 2 2 3 y 'y 'x gx  ax  bx  c '( x ) '( x ) 'x '1 g a b c 'x 'x 'x 'x 2 2 g (3 x  3 x'x  'x )  a (2 x  'x)  b Exercise 6.2 1 • Find difference quotient for each function y 5x y 30  15 x y 2 'y 5 'x 'y 15 'x 6(2 x  'x)  2 (2 x  'x) 6 x  2 x  9 'y 'x 2 y 1 x 'y 'x Exercise 6.3 7 • Total revenue: TR = P Q • Price: P = 10  .5Q 10 • Difference quotient? TR (10  .5Q)Q .5Q 2  10Q 'TR .5(2Q  'Q)  10 'Q • If Q = 5, what is impact of 1 unit increase in Q? 'TR 'Q .5(2(5)  1)  10 4.5 Derivative is difference quotient as 'xo0 dy dx y 2 f ( x  'x)  f ( x) lim 'x o0 'x b( x  'x)  bx lim 'x o0 'x 2 2 'x o0 dy d a  bx : dx b y dy x: dx y 3 ( x  'x)  x lim 'x o0 'x 2 'x o0 lim 2 x  'x 2 2x dy x: dx lim 3 x  3 x'x  'x 3x 2 Some properties • Derivative of sum = sum of derivatives – y, w, z are functions of x and y = w + z – Then dy d ( w  z ) dw dz  dx • Scaling: • Application dx d (ay ) dx y dy dx 3 3 dx dy a dx 2 2 dx gx  ax  bx  c d (x ) d (x ) dx d1 g a b c dx dx dx dx 2 3 gx  2ax  b Derivative is difference quotient as 'xo0 average rate of change { difference quotient o derivative 'y 'x dy dx f ( x1 )  f ( x) x1  x f ( x  'x)  f ( x) 'x f ( x  'x)  f ( x) lim 'x o0 'x Derivative is rate of change as 'xo0 Derivative is instantaneous rate of change Tangent line is limit of secant line Derivative is slope of tangent line Total tax revenue and marginal tax revenue Exercise 6.3 3 • Cigarette tax yields revenue R(t) = 50 + 25t – 75t2 • What is marginal revenue? MR MR t * dR dt 25  75(2t ) 25  150t 0 • What tax rate maximizes revenues? 25  150t 25 / 150 1 / 6 • Why is this a maximum? concave Functions not everywhere differentiable Differentiable  Continuous Demand and cost functions Average vs. marginal Difference quotient of a polynomial '(1) '( x) 0 1 'x 'x 2 '( x ) 2 x  'x 'x 3 '( x ) 2 2 3 x  3 x'x  ('x) 'x '( x ) 'x 4 4 x  6 x 'x  4 x('x)  ('x) 3 2 2 3 Exercise 6.3 9: y = 4x2 – 8x + 3 • Find roots (x such that y = 0). y b r b  4ac 2a 2 8 r 8  4(4)(3) 2(4) 2 • • 2r 43 1 r .5 2 Derivative dy 8x  8 dx Extreme value dy * 8x  8 0  x 1 dx Exercise 6.3 9: y = 4x2 – 8x + 3 3 2 1 0.5 1.0 1.5 2.0 1 Exercise 6.3 9: y = 4x2 – 8x + 3 5 y’ = 8x - 8 y 0.5 1.0 1.5 2.0 5 Differential vs. derivative • Derivative is rate of change as 'x o 0 'y 'y o lim 'x o0 'x 'x dy f '( x)dx dy dx f '( x) • Differential is change in y along tangent line Differential approximation and actual change dy f '( x0 )dx Exercise 6.4 3c: y = 16 – 4x + x3 'y • What is rate of change? 'x • What is derivative? • What is differential? 4  3 x  3 x'x  'x 2 2 2 dy dx 4  3 x 2 dy (3 x  4)dx 'y | (3(22 )  4)(8) 64 • Let x0 = 2; 'x = 8 'y (3(22 )  4  3(2)(8)  82 )(8) 960 • Let x0 = 2; 'x = .2 'y | (3(22 )  4)(.2) 1.6 'y (3(22 )  4  3(2)(.2)  .22 )(.2) 1.848 Exercise 6.4 6 investment investment (I) and cost of borrowing (r) 2 I f (r ) 600  150r  400r Compute: 'I f c(r )  'r r0 2%; 'r1 .5%; 'r2 1% 'I1 | (800 r0  150) 'r1 'I 2 | (800 r0  150) 'r2 (800(.02)  150)(.005) (800(.02)  150)(.01) 2 .67 1.34 I 0 600  150(.02)  400(.02 ) 597.16 2 I1 600  150(.025)  400(.025 ) 596.5 2 I 2 600  150(.03)  400(.03 ) 595.86 'I1 'I 2 596.5  597.16 .66 595.86  597.16 1.3 ...
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