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The solutions to the math 200 exam covering topics such as resistors in parallel, gradient and directional derivatives, saddle points, lagrange multipliers, partial derivatives, double integrals, and volumes of regions.
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[15] 1. If two resistors of resistance R 1 and R 2 are wired in parallel, then the resulting resistance R satisfies the equation (^) R^1 = (^) R^11 + (^) R^12. Use the linear or differential approximation to estimate the change in R if R 1 decreases from 2 to 1.9 ohms and R 2 increases from 8 to 8.1 ohms.
[10] 2. (^) in the direction of the vector 2Assume that the directional derivative ofi − j + k, and the value of the directional derivative in w = f (x, y, z) at a point P is a maximum that direction is 3√6. (a) Find the gradient vector of w = f (x, y, z) at P. [5%] (b) Find the directional derivative of i + j. [5%] w = f (x, y, z) at P in the direction of the vector
[15] 4. (^) to find the radius of the base and the height of a right circular cylinder of maximumUse the Method of Lagrange Multipliers (no credit will be given for any other method) volume which can be fit inside the unit sphere x^2 + y^2 + z^2 = 1.
[10] 5. (^) andLet cz such that= f (x, y) where x = 2s + t and y = s − t. Find the values of the constants a, b
a ∂∂x^2 z 2 + b ∂x∂y∂^2 z + c ∂∂y^2 z 2 = ∂∂s^2 z 2 + ∂∂t^2 z 2. You may assume that z = f (x, y) is a smooth function so that the Chain Rule and Clairaut’s Theorem on the equality of the mixed partial derivatives apply.
[15] 7. Evaluate the iterated double integral
∫ (^) x= x=
∫ (^) y=√ 4 −x 2 y=0^ (x
(^2) + y (^2) ) 32 dy dx.
[15] 8. 1 Consider the region≤ ρ ≤ 1 + cos ϕ. E in 3-dimensions specified by the spherical inequalities
(a) Draw a reasonably accurate picture ofunits on the coordinates axes. [5%] E in 3-dimensions. Be sure to show the (b) Find the volume of E. [10%]
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