course Mth 272 £˥ӦふӅassignment #010
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22:36:49 5.5.23 (was 5.5.28 area in region defined by y=8/x, y = x^2, x = 1, x = 4
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RESPONSE --> to find where the functions intersect, set them equal to each other 8/x = x^2 multiply by x 8 = x^3 so x= 2 since 8/x > x^2 when x<2 and x^2> 8/x when x>2 from 1 to 2 8/x - 2^x from 2 to 4 x^2- 8/x then integrate 8lnx - x^3/3 and the opposite for the other one 8ln(2) - 8/3 - (8ln1- 1/3) = 3.212 64/3 - 8ln(4) - (8ln(2)-8/3) = 13.121 the sum is 16.333 which is the region defined confidence assessment: 3
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22:36:55 These graphs intersect when 8/x = x^2, which we solve to obtain x = 2. For x < 2 we have 8/x > x^2; for x > 2 the inequality is the reverse. So we integrate 8/x - x^2 from x = 1 to x = 2, and x^2 - 8 / x from x = 2 to x = 4. Antiderivative are 8 ln x - x^3 / 3 and x^3 / 3 - 8 ln x. We obtain 8 ln 2 - 8/3 - (8 ln 1 - 1/3) = 8 ln 2 - 7/3 and 64/3 - 8 ln 4 - (8 ln 2 - 8/3) = 56/3 - 8 ln 2. Adding the two results we obtain 49/3. **
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RESPONSE --> got it self critique assessment: 3
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00:07:30 5.5.44 (was 5.5.40 demand p1 = 1000-.4x^2, supply p2=42x
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RESPONSE --> first make it a function that can be factored -.4x^2 -42x+1000 = 0 it looks like the quadratic formula has to be used x= -(-.4)+- sqrt(-.4^2- 4(-.4*1000)/ 2(-.4) x= 20 plug this into the original eqn. 1000-.4(20)^2 = 42(20) = 840 so they intersect at (20, 840) 1000-.4x^2-840 = 160 - .4x^2 anti deriv = 160x-.4/3(x^3) consumer surplus = 2133.34 840- (42x) anti deriv = 840x- (42x^2/2) = 8400 = producer surplus confidence assessment: 3
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00:07:37 1000-.4x^2 = 42x is a quadratic equation. Rearrange to form -.4 x^2 - 42 x + 1000 = 0 and use the quadratic formula. You get x = 20 At x = 20 demand is 1000 - .4 * 20^2 = 840, supply is 42 * 20 = 840. The demand and supply curves meet at (20, 840). The area of the demand function above the equilibrium line y = 840 is the integral of 1000 - .4 x^2 - 840 = 160 - .4 x^2, from x = 0 to the equlibrium point at x = 20. This is the consumer surplus. The area of the supply function below the equilibrium line is the integral from x = 0 to x = 20 of the function 840 - 42 x. This is the producer surplus. The consumer surplus is therefore integral ( 160 - .4 x^2 , x from 0 to 20) = 2133.33 (antiderivative is 160 x - .4 / 3 * x^3). *&*&
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RESPONSE --> go it self critique assessment: 3
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