MATH113 Calculus and Geometry

Test 11 February 2016

Please write name and tutor on each answer sheet. Total marks: 50

• i) Let $𝐚=(1,0,2)$ and $𝐛=(-2,2,1)$. Compute $𝐚\cdot 𝐛$ and $𝐚\times 𝐛$. [4]

  ii) Consider the three points $O=(0,0,0)$, $A=(2,1,0)$ and $B=(3,1,1)$. If $OACB$ is a parallelogram, what is the point $C$? [2]

  iii) Compute the area of the parallelogram $OACB$ in (ii). [4]

  iv) If $𝐮\cdot 𝐯=\sqrt{3}$ and $𝐮\times 𝐯=(1,2,2)$, find $\tan \theta$, where $\theta$ is the angle between $𝐮$ and $𝐯$. [4]

  
  

• Let $\gamma :ℝ\to {ℝ}^2$, $\gamma (t)=(e^t+e^{-t},5-2t)$.

  i) Find the length of the curve $\gamma$ between $\gamma (0)$ and $\gamma (3)$. [5]

  ii) Determine an equation of the tangent line to the image of $\gamma$ at the point $\gamma (0)$. [3]

  
  

• Determine equations of (a) the normal line and (b) the tangent plane to the surface $x^4=y^2{z}^2$ at the point $(2,2,2)$. [6]

  
  

• Determine whether either of the following vector-valued functions can be expressed as $\nabla \varphi$ for some function $\varphi (x,y,z)$, and find such a $\varphi$ when it exists. [8]

  i) $f(x,y,z)=(x^2+y^2+z^2,2xy+z^2,2xz+y^2)$.

  ii) $g(x,y,z)=(2xy-2z^2-yz,x^2+6yz-xz,\mathrm{\hspace{0.33em}3}{y}^2-4xz-xy)$.

  
  

• Find the greatest and least values of $x^2+2y^2$ on the circle $x^2+y^2=1$. [8]

  
  

• Use polar coordinates to evaluate

  $$I=\int {\int }_A\frac{1}{x^2+y^2+1}𝑑x𝑑y,$$

  where $A$ is the region defined by $0\le x^2+y^2\le 9$,   $x\ge 0$,  $y\ge 0$. [6]