# How you would decide if they were linear or if the

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how you would decide if they were linear, or if the one were more closely linear than the other). (c) Give the local linearization of f ( x , y ) = e - y sin ( x ) at ( 2 , 1 ) : (1 pt) The gas equation for one mole of oxygen relates its pressure, P (in atmospheres), its temperature, T (in K), and its volume, V (in cubic decimeters, dm 3 ): T = 16 . 574 · 1 V - 0 . 52754 · 1 V 2 - 0 . 3879 P + 12 . 187 V P . (a) Find the temperature T and differential dT if the volume is 20 dm 3 and the pressure is 2 atmosphere. T = dT = (b) Use your answer to part (a) to estimate how much the volume would have to change if the pressure increased by 0.1 Using the second of your tables: (Hint: For the derivatives, use the difference quotient with values taken from the second col- umn and row from the second table. That is, do not calculate the actual partial derivatives of the function.) f ( x , y ) Now, use the partial derivatives: f x ( x , y ) = e - y cos ( x ) and f y ( x , y ) = - e - y ln ( e ) sin ( x ) : f ( x , y ) 0.3394 0.3379 0.3363 0.336 0.3345 0.333 0.3327 0.3312 0.3297 0.3345 + (0.333-0.336)*(x-2)/0.02 + (0.3312-0.3379)*(y-1)/0 0.334512 + -0.153092*(x - 2) + -0.334512*(y - 1) 6. (1 pt) Find the differential of f ( x , y ) = p x 3 + y 2 at the point ( 2 , 1 ) . d f = Then use the differential to estimate f ( 2 . 04 , 0 . 9 ) . f ( 2 . 04 , 0 . 9 ) Correct Answers: 2*dx + 0.333333*dy 3.04667 7. (1 pt) The gas equation for one mole of oxygen relates its pressure, P (in atmospheres), its temperature, T (in K), and its volume, V (in cubic decimeters, dm 3 ): T = 16 . 574 · 1 V - 0 . 52754 · 1 V 2 - 0 . 3879 P + 12 . 187 V P . (a) Find the temperature T and differential dT if the volume is 20 dm 3 and the pressure is 2 atmosphere. T = dT = (b) Use your answer to part (a) to estimate how much the volume would have to change if the pressure increased by 0.1