Quiz 4: Linearization and the first Taylor polynomial.

Problem 1) Suppose f(x,y)=(1/2)x⁴-x²+xy -(1/2)y². Click on the function below which you think is the local linearization of f at the point (1,2)

A) (2x²-2x+y)(x-1)+(x-y)(y-2)

B) 2(x-1)-(y-2)

C) -(1/2)+2(x-1)-(y-2)

Complete solution

Linearization: A Review

Problem 2) The following table gives the Volume (in cubic feet) at various temperatures and pressures of one pound of steam. Pressure, p, is measured in (lb/in²), while temperature, T is measured in degrees Farenheit.

	p= 20	p= 22	p= 24	p= 26
T=480	27.85	25.31	23.19	21.39
T=500	28.46	25.86	23.69	21.86
T=520	29.06	26.41	24.20	22.33

a) Based on what you know about the chemistry of gases, and the meaning of the partial derivative, click on the answer you agree with.

i) Both partial derivatives of V are always positive

ii) Both partials are always negative

iii) The partials have different signs,and the partial derivative of V with respect to p is negative.

iv) None of the above.

b) Use the table to approximate

when

T=480 and p=22. Check to make sure that your answers agree with your conclusions about the signs of the partial derivatives that you reached in 2a).

Click here for a hint

Complete solution

c) Using the table and your answers to a) give the local linearization for V when T=480 and p=22.

Click here for a hint

Complete solution

d) Use the local linearization to approximate V when T=490 and p=21.

Click here for a hint

Complete solution

Problem 3) If the length and width of a box are increased by 1%, and the height is decreased by 2%, use the differential to approximate the percent change in volume.

The Differential: A Review

Click here for a hint

Complete solution

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