6. Further Pure Mathematics

Advanced Calculus — Quiz

Test your understanding of advanced calculus with 5 practice questions.

Practice Questions

Question 1

Which expression gives the total differential $df$ of a differentiable function $f(x,y)$?

Question 2

Given $f(x,y,z)=e^{xy}z+xz^2$, what is the mixed partial derivative $f_{xz}$?

Question 3

Evaluate the double integral $\iint_D(x-y)\,dA$ where $D$ is the triangle with vertices $(0,0),(1,0),(1,1)$.

Question 4

Evaluate the triple integral $\displaystyle\iiint_E z\,dV$ over the region $E$ in the first octant bounded by the plane $z=1-x-y$.

Question 5

Using spherical coordinates, evaluate $\displaystyle\iiint_G(x^2+y^2+z^2)\,dV$ over the hemisphere $G$ defined by $x^2+y^2+z^2\le a^2$, $z\ge0$.