MATH2010 Mock A

6 questions · 100 marks · suggested time: 90 minutes. Spend about 80 minutes solving and 10 minutes checking. Answer all six questions, show the reasoning behind each conclusion, and keep exact expressions where useful. The time beside each question is a planning guide.

Revision route · Step-by-step solutions and marking

Q1. A plane, a projection and a volume — 16 marks, 14 minutes

Let $P=(1,-1,2)$, $u=(1,2,0)$ and $v=(2,-1,1)$. The plane $\Pi$ passes through $P$ and has direction vectors $u,v$.

(a) [6] Find a nonzero normal vector and a Cartesian equation of $\Pi$.

(b) [6] Find the distance from the origin to $\Pi$ and the foot of the perpendicular from the origin.

(c) [4] Find the volume of the parallelepiped with edges $u,v,w$, where $w=(0,1,1)$. Explain the role of the sign in your calculation.

Q2. A set and its edges — 16 marks, 12 minutes

Let

$$S=\{(x,y)\in\mathbb R^2:1\le x^2+y^2\le4,\ y>0\}.$$

(a) [8] Find the interior, closure and boundary of $S$ in $\mathbb R^2$. Include all boundary pieces.

(b) [6] Decide whether $S$ is open, closed, bounded and path-connected. Justify each decision; give a continuous path construction for the last one.

(c) [2] Convert the polar coordinates $r=3/2$, $\theta=2\pi/3$ to Cartesian coordinates and decide whether the point belongs to $S$.

Q3. Which approaches prove a limit? — 18 marks, 15 minutes

For $(x,y)\ne(0,0)$ define

$$F(x,y)=\frac{x^2y^2}{x^2+y^2},\qquad G(x,y)=\frac{x^2(y-x)}{x^4+(y-x)^2},$$

and set $F(0,0)=G(0,0)=0$.

(a) [6] Prove that $F$ is continuous at the origin using a bound valid for every approach. Give an explicit $\delta$ for a prescribed $\varepsilon>0$.

(b) [4] Compute the limit of $G$ along every line through the origin, including the vertical line.

(c) [4] Determine whether the joint limit of $G$ at the origin exists. Supply an approach that settles the question.

(d) [4] Compute both iterated limits of $G$ at the origin. A student argues that agreement of these two limits proves existence of the joint limit. Assess that argument and explain what evidence is missing.

Q4. Differentiating in the correct variables — 16 marks, 12 minutes

On $D=\{(x,y):y>-1\}$ let

$$H(x,y)=e^{xy}+x^2\log(1+y).$$

(a) [6] Find $H_x,H_y$ on $D$ and evaluate $\nabla H(1,0)$.

(b) [6] Compute $H_{xy}$ and $H_{yx}$ separately and evaluate them at $(1,0)$.

(c) [4] State a sufficient mixed-partial theorem and verify its hypotheses for this function on $D$.

Q5. Linearization with a controlled error — 18 marks, 15 minutes

Let $f(x,y)=\sqrt{9+x+2y}$ on $x+2y>-9$ and consider the origin. Write $\rho=\sqrt{x^2+y^2}$ and $t=x+2y$.

(a) [4] Find the affine linearization $L$ at the origin.

(b) [6] Derive the exact identity

$$f(x,y)-L(x,y)=-\frac{t^2}{6(\sqrt{9+t}+3)^2}.$$

(c) [4] Use the identity to prove differentiability at the origin from the small-remainder definition. Specify a neighbourhood on which your estimate holds.

(d) [4] Use $L$ to approximate $f(0.06,-0.015)$ and give an absolute-error bound from the identity.

Q6. Direction, speed and differentiability — 16 marks, 12 minutes

Let $g(x,y,z)=x^2+2xy+3y^2+z^2-2z$ and $A=(1,-1,2)$.

(a) [4] Compute $\nabla g(A)$ and the unit directional derivative at $A$ in the direction $v=(2,-1,2)$.

(b) [6] Find the directions and rates of fastest increase and fastest decrease among unit directions. Justify the optimal rates.

(c) [6] Define

$$q(x,y)= \begin{cases} \dfrac{x^2y}{x^2+3y^2},&(x,y)\ne(0,0),\\ 0,&(x,y)=(0,0). \end{cases}$$

Find every unit directional derivative of $q$ at the origin. A student concludes that $q$ is differentiable because these directional derivatives all exist. Decide whether that conclusion is valid, identify the gap in the argument, and give a proof of your decision.

Check before marking

Check the normal-vector sign, missing straight boundary segments, uniformity of limit estimates, the variable held fixed in each partial derivative, the denominator in the differentiability remainder, and normalization of a direction vector. Then use the solutions to award marks and choose the next note to revisit.