多变量微积分代写-Calculus代写-MATH2720代写

MATH2720 Multi-variable Calculus:

Assignment 5

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List of problems 多变量微积分代写

In single-variable calculus we differentiated equations that described y implicitly as a function of x via implicit differentiation. We can apply the same ideas in the multivariable situation. The equation

16x² + 25y² + 4z² = 144

describes z implicitly as a function of x and y.

(a)Take the partial derivative of both sides of this equation with respect to x and solve the resulting equation to show that^∂z = − 4x .

Solution. Taking the partial derivative of both sides of this equation with respect to x while treating z as a function of x gives

(b)Take the partial derivative of both sides of this equation with respect to y and solve the resulting equation to show that^∂z = − 25y . 多变量微积分代写

Solution. Taking the partial derivative of both sides of this equation with respect to y while treating z as a function of y gives

(c)Use the results of parts (a) and (b) to find the equation of the tangent plane to this surface at the point 1, ⁸, 4 . Use this tangent plane to approximate the value of z at the input (1.1, 1.4).

Solution. The equation of the plane tangent to a function f = f (x, y) at a point (x₀, y₀) is

z = f (x₀, y₀) + f_x(x₀, y₀)(x − x₀) + f_y(x₀, y₀)(y − y₀).

For our function we have x₀ = 1, y₀ = ⁸ = 1.6, and z₀ = 4. From parts (a) and (b) we also have

So the equation of the plane tangent to the surface at the point 1, ⁸ , 4 is

z = 4 − (x − 1) − 2.5(y − 1.6).

Thus, the value of z at the input (1.1, 1.4) is approximately 多变量微积分代写

4 − (1.1 − 1) − 2.5(1.4 − 1.6) = 4.4.

2.Thepressure P , temperature T , and volume V of a gas are related by an equation of state of the form f (P, V, T ) = 0 for some function f . For an ideal gas, the function f is f (P, V, T ) = P V nRT , where n is the number of moles of the gas (a constant) and R > 0 is the ideal gas constant. Show that, for an ideal gas,