Tutorial Week 8

Partial Derivatives

Q1: Let \(f(x, y, z) = x^3 + y^2 + sin(z)\). Compute \(f_x, f_y, f_z\). Compute \(\frac{\partial x}{\partial z} |_{(6, 8, \pi)}\).

Show Solution

Q2: Let \(g(x, y) = 2^{xy} + x\sqrt{y}\). Find \(\frac{\partial g}{\partial y}\).

Show Solution

Q3: Let \(6^{xyzw} = \sqrt{x^3 + xyz - z^3 + tan(xy)}\) and \(w\) be an implicitly defined function in terms of \(x\), \(y\), and \(z\). Find \(w_z\).

Show Solution

Marginal Costs

Q4: Let \(C(x, y, z) = 5x^2 + 3y^x - z^3\) denote the cost of producing x number of product A, y number of product B, and z number of product C. If \(x = 10\), \(y = 5\), \(z = 8\), what is the marginal cost with respect to x, marginal cost with respect to y, and the marginal cost with respect to z?

Show Solution