MATH115 GEOMETRY AND CALCULUS

Quiz 4

• Most and least What are the greatest and least values of $xy$ on the ellipse $\mathrm{\hspace{0.33em}4}{x}^2-2xy+y^2=12$?

  A) $6$ and $-2$,$$ B) 0 and 0,$$ C) $2\sqrt{3}$ and $-2\sqrt{3}$,$$ D) $\frac{1}{2}$ and $-\frac{1}{6}$,$$ E) $\sqrt{3}$ and $-1$.

• Just the most Find the maximum value of $xyz$ subject to $x^2+2y^2+4z^2=9$.

  A) $0$,$$ B) $\frac{3}{2}\sqrt{\frac{3}{2}}$,$$ C) $\frac{1}{2}\sqrt{\frac{3}{2}}$,$$ D) $\sqrt{3}$,$$ E) 3.

• Not all true Which of the following statements is not true?

  A) ${\int }_d^c{\int }_b^{a}f(x,y)𝑑x𝑑y={\int }_c^d{\int }_a^{b}f(x,y)𝑑x𝑑y$,

  B) ${\int }_c^d{\int }_a^{b}f(x)g(y)𝑑x𝑑y=({\int }_a^{b}f(x)𝑑x)({\int }_c^{d}g(y)𝑑y)$,

  C) ${\int }_c^d{\int }_a^{b}f(x,y)𝑑x𝑑y={\int }_c^d{\int }_a^{b}f(y,x)𝑑y𝑑x$,

  D) ${\int }_c^d{\int }_a^{b}f(x,y)𝑑y𝑑x={\int }_a^b{\int }_c^{d}f(x,y)𝑑x𝑑y$,

  E) ${\int }_c^d{\int }_a^b\frac{{\partial }^{2}f}{\partial x\partial y}𝑑x𝑑y=f(b,d)-f(a,c)$,

• A disc Let $R$ be the subset of ${ℝ}^2$ given by: $0\le x^2+y^2\le 4$, $x\ge 0$, $y\ge 0$. What is $\int {\int }_R\frac{1}{\sqrt{16-x^2-y^2}}𝑑x𝑑y$?

  A) $(2-\sqrt{3})\pi$,$$ B) $(4-\sqrt{12})\pi$,$$ C) $2\pi$,$$ D) ${\pi }^2/6$,$$ E) ${\pi }^2/3$.

• A slipped disc Let $R$ be the circular disc in ${ℝ}^2$ of radius $a$ and centre $(a,0)$. What is $\int {\int }_R\frac{1}{x}𝑑x𝑑y$?

  A) $\pi a^2/2$,$$ B) $4a^2$,$$ C) $2\pi a$,$$ D) $0$,$$ E) The integral doesn’t converge.

 

MATH113 CALCULUS AND GEOMETRY

Assessed Exercises 4

• (3 marks) What is the minimum value of $x^{3}y$ on the circle $x^2+y^2=4$?

• (4 marks) Show that the greatest and least values of $xy-xz$ on the sphere $x^2+y^2+z^2=1$ are $\pm 1/\sqrt{2}$, and find the points where these values occur.

• (3 marks) Find the area in the positive quadrant bounded by the curves $xy=a,xy=b,$ $y=\frac{1}{2}x,y=2x$, where .