Quant Academy Math · Probability · Quant Finance
Phase 3 Exam

Multivariable Calculus - Phase Exam

15 interleaved questions drawn from every lesson in this phase. Interleaving mixes topics on purpose - that difficulty is what builds durable, transferable understanding. Aim for 70%+ before advancing; below that, revisit the flagged lessons.

How this exam teaches
Questions are shuffled across lessons (not blocked by topic) so you practice choosing the right idea. Missed questions are added to your review queue automatically.
Q1 Easy
The level set \(\{(x,y): x^2+4y^2=9\}\) is:
A circle of radius 3
An ellipse with semi-axes 3 and 3/2
A parabola
Two straight lines
Q2 Easy
At a point where \(\nabla f=(3,4)\), the maximum rate of increase of \(f\) per unit step is:
7
5
12
1
Q3 Easy
Fubini’s theorem guarantees that for an integrable \(f\) on a rectangle:
The two orders of iterated integration give the same value
The integral factors as a product
f must be continuous
The region must be a disc
Q4 Easy
A critical point has Hessian with eigenvalues \(+3\) and \(-2\). It is:
A local minimum
A local maximum
A saddle point
Inconclusive
Q5 Easy
The Lagrange condition for minimizing \(f\) on \(g=c\) states that at the optimum:
\(\nabla f=0\)
\(\nabla f=\lambda\nabla g\)
\(f=g\)
\(\nabla g=0\)
Q6 Easy
Two nonzero vectors satisfy \(x\cdot y=0\). This means:
They point the same way
They are orthogonal (angle 90°)
One is the zero vector
Their norms are equal
Q7 Medium
For \(f(x)=\tfrac12 x^\top A x\) with symmetric \(A\), the Hessian is:
\(2A\)
\(A\)
\(A^\top A\)
the zero matrix
Q8 Easy
Under the polar change of variables, the area element \(dx\,dy\) becomes:
\(dr\,d\theta\)
\(r\,dr\,d\theta\)
\(r^2\,dr\,d\theta\)
\(\tfrac1r\,dr\,d\theta\)
Q9 Medium
In 2-D with \(D=f_{xx}f_{yy}-f_{xy}^2\gt 0\) and \(f_{xx}\lt 0\), the critical point is a:
Local minimum
Local maximum
Saddle
Inflection
Q10 Medium
The Lagrange multiplier \(\lambda\) at the optimum equals:
The constraint value c
The sensitivity \(df^*/dc\) of the optimal value to the constraint level
The objective value
Always 1
Q11 Medium
To show \(\lim_{(x,y)\to(0,0)} f(x,y)\) does NOT exist, the cleanest strategy is:
Check the limit along the x-axis only
Find two approach paths giving different limits
Show f is unbounded
Evaluate f at the origin
Q12 Medium
The gradient of a scalar field at a point is always:
Tangent to the level set through that point
Orthogonal to the level set through that point
Zero at every point
Parallel to the x-axis
Q13 Medium
The value of \(\int_{-\infty}^\infty e^{-x^2}\,dx\) is:
\(\pi\)
\(\sqrt{\pi}\)
\(2\pi\)
\(1\)
Q14 Medium
For a convex function, a point with \(\nabla f=0\) is:
Only a local minimum
A global minimum
Possibly a saddle
Never an extremum
Q15 Hard
For \(\min\ \tfrac12 w^\top\Sigma w\) s.t. \(\mathbf 1^\top w=1\) with \(\Sigma\succ0\), the solution is:
\(w=\Sigma^{-1}\mathbf 1\)
\(w=\Sigma^{-1}\mathbf 1/(\mathbf 1^\top\Sigma^{-1}\mathbf 1)\)
\(w=\mathbf 1/n\)
\(w=\Sigma\mathbf 1\)
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