Each of Exercises 61 through 68 involves either the chain rule for partial derivatives or the incremental approximation formula for functions of two variables.

Ethan is an investor who derives U(x, y) units of satisfaction from owning x stock units and y bond units, where

He currently owns x = 27 stock units and y = 12 bond units.

a. Find the marginal utilities U_x and U_y.

b. Evaluate U_x and U_y for the current values of x and y.

c. Use calculus to estimate how Ethan’s satisfaction changes if he adds 3 stock units and removes 2 bond units from his portfolio.

d. Estimate how many bond units Ethan could substitute for 1 stock unit without affecting his total satisfaction with his portfolio.

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U(x, y) = (2x + 3)(y + 5)