这是份THE UNIVERSITY OF QUEENSLAND昆士兰大学的MATH7000 Calculus & Linear Algebra II期末考试辅导案例，学生在我们的帮助下获得了满意的成绩。
(10 marks) Determine the general solution to
$$
y^{\prime \prime}-y=\frac{1}{\cosh (x)}
$$
Show all working.
Consider the inner product space $M_{2,2}(\mathbb{R})$ with inner product
$$
\left\langle\left(\begin{array}{ll}
a & b \
c & d
\end{array}\right),\left(\begin{array}{ll}
e & f \
g & h
\end{array}\right)\right\rangle=a e+b f+c g+d h .
$$
Let $W$ be the set of all symmetric matrices in $M_{2,2}(\mathbb{R})$. Find an orthonormal basis for $W^{\perp}$. Show all working.
(10 marks) Consider the inner product space $C[0,1]$ with inner product
$$
\langle f, g\rangle=\int_{0}^{1} f(x) g(x) d x .
$$
Determine the least squares approximation of $\sqrt{x}$ from the subspace $P_{2}(\mathbb{R})$ of $C[0,1] .$ Show all working.
Hint: The set $\left{1, \sqrt{3}(1-2 x), \sqrt{5}\left(1-6 x+6 x^{2}\right)\right}$ is an orthonormal basis of $P_{2}(\mathbb{R})$.
(10 marks) Let
$$
A=\left(\begin{array}{ccc}
0 & 0 & 0 \
0 & 0 & 0 \
1 & -1 & -1
\end{array}\right)
$$
Find an orthogonal basis for the eigenspace of $A$ corresponding to eigenvalue $0 .$ Show all working.
(10 marks) Let $D$ be the region in the first quadrant $(x, y \geq 0)$ of the $x-y$ plane bounded by the curves $y=x^{2}, x+y=2$ and $y=0$ (the $x$-axis). Calculate the volume below the surface $z=x y$ and above $D$. Show all working.
x
Consider the iterated integral
$$
I=\int_{0}^{2 \sqrt{2}} \int_{y}^{\sqrt{16-y^{2}}} x d x d y
$$
(i) (6 marks) Express $I$ as an iterated integral in terms of polar coordinates.
(ii) (4 marks) Evaluate $I$ using either rectangular or polar coordinates (i.e. you choose).
Show all working.
(10 marks) Consider a solid of mass density $\rho(x, y, z)=\rho_{0} z$ (constant $\rho_{0}>0$ ) occupying the region in $\mathbb{R}^{3}$ given by
$$
V={(x, y, z) \mid 0 \leq x \leq 1,0 \leq y \leq 1,0 \leq z \leq 1}
$$
Show that the moment of inertia of the solid about the $z$-axis, $I_{z}$, is given by the formula
$$
I_{z}=\frac{2}{3} M
$$
where $M$ is the mass of the solid.
(10 marks) Let $D$ be the region in the $x-y$ plane bounded by the curves $y=\frac{1}{2} x^{2}$ and $y=1-\frac{1}{2} x^{2}$. Calculate the net outward flux of the vector field $F(x, y)=x^{4} y i+x^{3} y^{2} j$ across the boundary of D. Show all working.
(10 marks) Find the surface area of the part of paraboloid $y=x^{2}+z^{2}$ that lies inside the cylinder $x^{2}+z^{2}=1$. Show all working.
(10 marks) Calculate the net outward flux of the vector field $\boldsymbol{F}(x, y, z)=x^{3} \boldsymbol{i}+y^{3} \boldsymbol{j}+z^{3} \boldsymbol{k}$ across the surface of the solid bounded by the cylinder $x^{2}+y^{2}=1$ and the planes $z=0$ and $z=2$.
(10 marks) Evaluate $\oint_{C} \boldsymbol{F} \cdot \boldsymbol{d} \boldsymbol{r}$, where $\boldsymbol{F}(x, y, z)=z^{2} \boldsymbol{i}+\frac{3}{2} x \boldsymbol{j}$ and $C$ is the boundary of the square ${(x, y, z) \mid 0 \leq x \leq 1,0 \leq y \leq 1, z=1}$ traversed in an anticlockwise direction when viewed from above.
(i) (5 marks) Let $C$ be a positively oriented piecewise-smooth simple closed curve in the $x-y$ plane. Show that the area enclosed by $C$ is given by the line integral
$$
-\oint_{C} y d x
$$
(ii) (5 marks) Use the result of part (i) (or otherwise) to find the area between the $x$-axis and the curve parameterised by $\boldsymbol{r}(t)=(t-\sin t) \boldsymbol{i}+(1-\cos t) \boldsymbol{j}, 0 \leq t \leq 2 \pi$.
Exam information
Course code and title
MATH7000
Calculus & Linear Algebra II
Semester Summer Semester, 2021
Exam type Online, non-invigilated, end-of-semester examination
Exam technology File upload to Blackboard Assignment
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