Mathematical Sciences Institute
EXAMINATION: Semester 1
MATH1014_Semester 1Mathematics and Applications 2
This practise questions for calculus

### 微积分/Calculus part

Mathematical Sciences Institute
EXAMINATION: Semester 1
MATH1014_Semester 1Mathematics and Applications 2
This practise questions for calculus
Exam Duration: 90 minutes.
Reading Time: 0 minutes.
Materials Permitted In The Exam Venue:
• One A4 page with hand written notes on both sides.
(This A4 page is to cover both Linear Algebra and Calculus.)
• Unmarked English-to-foreign-language dictionary (no approval required).

Problem 1.

Check the continuity and the differentiability of $f(x, y)= \begin{cases}x y \sin \frac{1}{x^{2}+y^{2}} & \text { if } \quad(x, y) \neq(0,0) \ 0 & \text { if } \quad(x, y)=(0,0)\end{cases}$ at $(0,0)$.

Problem 2.

Let
$$f(x, y)=x y e^{-x-y} .$$
(a) Calculate the gradient of $f$.
(b) Find the equation of the plane tangent to the graph of $z=f(x, y)$ passing through $(0,1, f(0,1))$.
(c) In which direction does the function $f$ increase fastest at $(1,-1)$.
(d) Find the rate of change of $f$ at the point $(1,-1)$ in the direction from the point $(1,-1)$ to the origin $(0,0)$.

Problem 3.

(a) Find the critical points of $f(x, y)=x y e^{-x-y}$ and classify each as a local minimum, local maximum, or saddle point.
(b) Find the maximum and minimum values of $f(x, y)=4 x^{2}+10 y^{2}$ on the disk $x^{2}+y^{2} \leq 4$.

Problem 4.

Find the volume of the solid under the surface $z=\sqrt{4-x^{2}}$ over the domain$D=\left{(x, y) \mid x^{2}+y^{2} \leq 4\right}$

Problem 5.

Evaluate the double integral
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-\left(2 x^{2}-2 x y+13 y^{2}\right) / 2} d x d y$$

### 线性代数/Linear Algebra part

Mathematical Sciences Institute
EXAMINATION: Practise Questions – Linear Algebra
MATH1014: Mathematics and Applications 2, Semester 1
Exam Duration: 120 minutes.
Reading Time: 15 minutes.
Materials Permitted In The Exam Venue:
• Unmarked English-to-foreign-language dictionary (no approval required).
• One A4 page with hand written notes on both sides.

Problem 6.

Consider the matrix
$$A=\left[\begin{array}{ccc} 1 & 3 & 3 \ -3 & -5 & -3 \ 3 & 3 & 1 \end{array}\right] \text {, }$$
with characteristic polynomial $-(\lambda-1)(\lambda+2)^{2}=0$. Find a diagonal matrix $D$ and an invertible matrix $P$ that satisfy $A=P D P^{-1}$. You must justify that $P$ is invertible.

Problem 7.

Consider the matrix $A=\left[\begin{array}{cc}0.5 & 0.6 \ -0.3 & 1.4\end{array}\right]$, which is associated with the dynamical system $A \mathrm{x}{k}=\mathrm{x}{k+1}$. Classify the origin as an attractor, repeller, saddle, or spiral point. Determine the directions of greatest attraction and/or repulsion.

Problem 8.

The matrix $A$ is given by
$$A=\left[\begin{array}{ccc} 3 & -5 & 1 \ 1 & 1 & 1 \ -1 & 5 & -2 \ 3 & -7 & 8 \end{array}\right]$$
Find an orthogonal basis for the column space of $A$.

Problem 9.

Let
$$\mathbf{y}=\left[\begin{array}{c} -1 \ 4 \ 3 \end{array}\right], \quad \mathbf{u}{1}=\left[\begin{array}{l} 1 \ 1 \ 0 \end{array}\right], \quad \mathbf{u}{2}=\left[\begin{array}{c} -1 \ 1 \ 0 \end{array}\right]$$
and let $W=\operatorname{span}\left{\mathbf{u}{1}, \mathbf{u}{2}\right}$.
a) Verify that $\left{\mathbf{u}{1}, \mathbf{u}{2}\right}$ is an orthogonal set.
b) Find the orthogonal projection of $y$ onto the subspace $W$.

Problem 10.

Let
$$A=\left[\begin{array}{cr} 1 & -1 \ -1 & -2 \ 1 & -1 \end{array}\right], \quad \mathbf{b}=\left[\begin{array}{c} -2 \ 0 \ -4 \end{array}\right]$$
Find all least-squares solutions to the system $A \mathrm{x}=\mathrm{b}$. What is the least-squares error?

BS equation代写

# Mathematics and Applications 2

An undergraduate course offered by the Mathematical Sciences Institute.

This course continues on from MATH1013. It emphasises an understanding of the fundamental results from calculus and linear algebra which both can be applied across a range of fields including the physical and biological sciences, engineering and information technologies, economics and commerce, and can also serve as a base for future mathematics courses. Many applications and connections with other fields will be discussed although not developed in detail.

Topics to be covered include:

Calculus – Integration and techniques of integration, including multiple and iterated integrals. Sequences and series. Functions of several variables – visualisation, continuity, partial derivatives, and directional derivatives. Lagrange multipliers.

Linear Algebra – theory and application of Euclidean vector spaces. Vector spaces: linear independence, bases and dimension; eigenvalues and eigenvectors; orthogonality and least squares.

## Learning Outcomes

Upon successful completion, students will have the knowledge and skills to:

1. Explain the fundamental concepts of calculus and linear algebra and their role in modern mathematics and applied contexts. These concepts include vector spaces, eigenvalues and eigenvectors, orthogonality and least squares in linear algebra; and integration, sequences and series, functions of several variables, and partial differential equations in calculus.
2. Demonstrate accurate and efficient use of calculus and linear algebra techniques as they relate to the concepts listed above.
3. Demonstrate capacity for mathematical reasoning through explaining concepts from calculus and linear algebra.
4. Apply problem-solving using calculus and linear algebra techniques applied to diverse situations in physics, engineering and other mathematical contexts.