Multivariate Calculus (58 problems)

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Problem #1: Application Problems

A farmer wishes to fence off a rectangular pasture along the bank of a river. The area of the pasture is to be 6,400 yd^2 and no fencing is needed along the river bank. Find the dimensions of the pasture that will require the least amount of fencing.

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Problem #2: Calculation of Partial Derivatives

Compute all first order derivatives

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Problem #3: Calculation of Partial Derivatives

For, compute all the first order derivatives.

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Problem #4: Calculation of Partial Derivatives

A grocer’s daily profit from the sale of two brands of apple juice is P(x,y) = (x-30)(70-5x+4y)+(y-40)(80+6x-7y) cents, where x is the price per can of the first brand and y is the price per can of the second. Currently the first brand sells for 50 cents per can and the second for 52 cents per can. Use marginal analysis to estimate the change in the daily profit that will result if the grocer raises the price of the second brand by 1 cent per can but keeps the price of the first brand unchanged.

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Problem #5: Calculation of Partial Derivatives

Let

Find

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Problem #6: Calculation of Partial Derivatives

Find the partial derivative of z with respect to x given

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Problem #7: Calculation of Partial Derivatives

Compute all the second order derivatives of

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Problem #8: Calculation of Partial Derivatives

Compute all the first order derivatives at P = (1, 1) for

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Problem #9: Calculation of Partial Derivatives

Compute the partial derivative for

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Problem #10: Calculation of Partial Derivatives

1. Let

a) Find

b) Find

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Problem #11: Calculation of Surface Areas

Find the area of the surface of the portion of the conebetween the cylinderand the plane.

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Problem #12: Computation of Critical Points

Find the critical points of

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Problem #13: Cylindrical Coordinates

Use cylindrical coordinates to compute the integral

where Q is the cylindrical solid given by

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Problem #14: Domain

Describe the domain of the function

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Problem #15: Double Integration

A lamina in the shape of the region bounded byand the line x = 0. The areas density varies as the distance from the x-axis. Find the center of mass. Please provide an accurate sketch of the bounded region.

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Problem #16: Double Integration

Evaluate the following double integral:

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Problem #17: Functions of Several Variables

The output at a certain factoryunits, where K is the capital investment measured in units of $1000 and L the size of the labor force measured in worker hours.