Multivariable calculus
Practice problems on partial derivatives, multiple integrals and optimization in several variables.
Partial derivative practice problems
f_x = \frac{\partial f}{\partial x}, \quad \nabla f = \langle f_x, f_y\rangle, \quad D_{\mathbf{u}}f = \nabla f\cdot\mathbf{u}
12 partial derivative practice problems with answers: first and second partials, gradients, directional derivatives, tangent planes and critical points.
Double integral practice problems
\iint_R f(x, y)\,dA = \int_a^b\!\int_{g_1(x)}^{g_2(x)} f(x, y)\,dy\,dx
12 double integral practice problems with answers: iterated integrals, general regions, switching the order, polar coordinates, volume and average value.
Lagrange multiplier practice problems
\nabla f = \lambda\,\nabla g, \qquad g(x, y) = k
12 Lagrange multiplier practice problems with answers: max and min on lines and circles, closest points, three variables and the meaning of lambda.
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