Determine the partial derivative values of function z. with respect to x and wit

Question

Determine the partial derivative values of function z. with respect to x and with respect to y for the following function:. z(x, y) = x^3 + x^2y^3 - 2y^2 The charge q of a capacitor in a circuit containing a capacitor of capacitance C, a resistor R, and a source of voltage E is given by: q = CE(1 - e^-t/RC) where t is time Determine the value of R dq/dt + q/C The adiabatic law of the expansion of air is PV^1.4 = C At a given instant the volume is 12.5 cubic feet and the pressure is 40 pounds per square inch At what rate is the pressure changing if the volume is decreasing at a rate of 1 cubic foot per second? Water flows from a faucet into a hemispherical basin 14 inches in diameter at a rate of 2 cubic inches per second. How fast does the water rise when the water is exactly halfway to the top? just as it runs over? (The volume of a spherical segment is given by pi rh^2 - 1/3 pi h^3 where r is the radius of the sphere and h is the depth of the segment) The diameter and height of a right circular cylinder are found at a certain instant to be 10 centimeters and 20 centimeters respectively. If the diameter is increasing at the rate of I centimeters per second, what change m height will keep the volume constant?

Explanation / Answer

(1) f(x, y) = x^3 + x^2 y^3 - 2y^2

fx (x, y) = 3x^2 + (y^3)(2x) = 3x^2 + 2xy^3

fy (x, y) = (x^2)(3y^2) - 4y = 3x^2 y^2 - 4y

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