Level 1 — RecognitionCalculus & Optimization Basics

Calculus & Optimization Basics

⏱ 20 minutes30 marksprintable — key stays hidden on paper

Level 1 Examination — Recognition

Time Limit: 20 minutes Total Marks: 30

Instructions: Answer all questions. For True/False items, a one-line justification is required for full marks. Use $...$ for any math you write.

Part A — Multiple Choice (1 mark each, 10 marks)

Q1. The gradient descent update rule for parameter $\theta$ with learning rate $\eta$ is: (a) $\theta \leftarrow \theta + \eta \nabla f(\theta)$ (b) $\theta \leftarrow \theta - \eta \nabla f(\theta)$ (c) $\theta \leftarrow \eta \theta - \nabla f(\theta)$ (d) $\theta \leftarrow \theta - \nabla f(\theta)/\eta$

Q2. For $f(x,y) = x^2 y$ , the partial derivative $\dfrac{\partial f}{\partial y}$ is: (a) $2xy$ (b) $x^2$ (c) $2x$ (d) $2xy + x^2$

Q3. The Hessian matrix of a scalar function $f:\mathbb{R}^n\to\mathbb{R}$ contains: (a) first-order partial derivatives (b) second-order partial derivatives (c) directional derivatives only (d) the function values

Q4. A function is convex if its Hessian is everywhere: (a) negative definite (b) positive semidefinite (c) singular (d) skew-symmetric

Q5. The derivative of $f(x) = \sin(x^2)$ using the chain rule is: (a) $\cos(x^2)$ (b) $2x\cos(x^2)$ (c) $2x\sin(x^2)$ (d) $\cos(2x)$

Q6. A saddle point of $f$ is a critical point where the Hessian is: (a) positive definite (b) negative definite (c) indefinite (d) zero matrix

Q7. If the learning rate $\eta$ is chosen too large, gradient descent typically: (a) converges faster with no downside (b) oscillates or diverges (c) always finds the global minimum (d) stops immediately

Q8. The Jacobian of a vector function $\mathbf{f}:\mathbb{R}^n\to\mathbb{R}^m$ has dimensions: (a) $n\times n$ (b) $m\times m$ (c) $m\times n$ (d) $n\times m$

Q9. The first-order Taylor approximation of $f$ about $a$ is: (a) $f(a)$ (b) $f(a) + f'(a)(x-a)$ (c) $f(a) + \tfrac12 f''(a)(x-a)^2$ (d) $f'(a)(x-a)$

Q10. The directional derivative of $f$ in the direction of unit vector $\mathbf{u}$ equals: (a) $\nabla f$ (b) $\nabla f \cdot \mathbf{u}$ (c) $\|\nabla f\|$ (d) $\mathbf{u}\cdot\mathbf{u}$

Part B — Matching (5 marks)

Q11. Match each term (Column X) to its definition (Column Y). Write pairs like 1-c.

Column X	Column Y
1. Local minimum	a. Point where every partial derivative is zero
2. Global minimum	b. Lowest value in a small neighborhood
3. Critical point	c. Method for constrained optimization using $\nabla f = \lambda \nabla g$
4. Lagrange multiplier	d. Lowest value over the entire domain
5. Continuity	e. $\lim_{x\to a} f(x) = f(a)$

Part C — True/False with justification (3 marks each, 15 marks)

Q12. A differentiable function is always continuous. (True/False + one-line reason.)

Q13. At any local minimum of a differentiable function, the gradient is the zero vector.

Q14. For a convex function, every local minimum is also a global minimum.

Q15. The gradient $\nabla f$ points in the direction of steepest decrease of $f$ .

Q16. By Clairaut's theorem, for a function with continuous second partials, $\dfrac{\partial^2 f}{\partial x\,\partial y} = \dfrac{\partial^2 f}{\partial y\,\partial x}$ .

Answer keyMark scheme & solutions

Part A — MCQ (1 mark each)

Q1. (b) — Gradient descent moves against the gradient to reduce the loss. (1)

Q2. (b) $x^2$ — Treat $x$ as constant; $\partial(x^2 y)/\partial y = x^2$ . (1)

Q3. (b) — The Hessian is the matrix of second-order partials $H_{ij}=\partial^2 f/\partial x_i\partial x_j$ . (1)

Q4. (b) — Convexity ⇔ Hessian positive semidefinite everywhere. (1)

Q5. (b) $2x\cos(x^2)$ — chain rule: outer $\cos(u)$ times inner $u'=2x$ , $u=x^2$ . (1)

Q6. (c) — A saddle has indefinite Hessian (mixed positive/negative curvature). (1)

Q7. (b) — Too large $\eta$ overshoots the minimum → oscillation/divergence. (1)

Q8. (c) $m\times n$ — rows = outputs ( $m$ ), columns = inputs ( $n$ ). (1)

Q9. (b) — First-order Taylor: $f(a)+f'(a)(x-a)$ . (1)

Q10. (b) $\nabla f\cdot\mathbf{u}$ — directional derivative = dot product of gradient with unit direction. (1)

Part B — Matching (1 mark each, 5 total)

Q11. 1-b, 2-d, 3-a, 4-c, 5-e

Local min = lowest in neighborhood (b); Global min = lowest over domain (d); Critical point = zero partials (a); Lagrange multiplier = constrained method (c); Continuity = limit equals value (e). (5)

Part C — True/False + justification (3 each)

Q12. True. (1) Differentiability at a point implies the limit defining the derivative exists, which forces the function to be continuous there. (2) (Converse is false: $|x|$ is continuous but not differentiable at 0.)

Q13. True. (1) A differentiable interior local minimum is a stationary point, so $\nabla f = \mathbf{0}$ (first-order necessary condition). (2)

Q14. True. (1) Convexity guarantees no separate lower valley elsewhere; any local min is global. (2) (Non-convex functions can have multiple distinct local minima.)

Q15. False. (1) The gradient points in the direction of steepest increase; steepest decrease is $-\nabla f$ . (2)

Q16. True. (1) Clairaut/Schwarz theorem: with continuous second partials, mixed partials are equal (symmetric Hessian). (2)

[
  {"claim":"Q2: d/dy of x^2*y equals x^2","code":"x,y=symbols('x y'); result = diff(x**2*y,y)==x**2"},
  {"claim":"Q5: d/dx of sin(x^2) equals 2x*cos(x^2)","code":"x=symbols('x'); result = simplify(diff(sin(x**2),x)-2*x*cos(x**2))==0"},
  {"claim":"Q16: mixed partials of a smooth function are equal","code":"x,y=symbols('x y'); f=x**3*y**2+sin(x*y); result = simplify(diff(f,x,y)-diff(f,y,x))==0"},
  {"claim":"Q9: first-order Taylor of exp about 0 is 1+x","code":"x=symbols('x'); result = series(exp(x),x,0,2).removeO()==1+x"}
]