MATH115 GEOMETRY AND CALCULUS

Workshop Exercises 2

• Calculate the length of the parametrized curve $\gamma (t)=(13\cos t+5t,12\sin t)$ between $t=0$ and $t=T$.

• A metal bar of length $R$, with one end fixed at the origin, rotates anti-clockwise at a constant angular velocity of $\omega$ radians per second. The free end of the bar is attached to the centre of a disc of radius $r$, which rotates anti-clockwise about its centre at a constant angular velocity of $\rho$ radians per second. A light is attached to the outer edge of the disc. At time $t=0$ the metal bar points in the direction of the positive $x$-axis and the light is at position $(r+R,0)$.

  a) Determine $\gamma (t)=(x(t),y(t))$.

  (Hint: Find an expression for the position of the centre $C$ of the disc at time $t$. Then add to this a vector for the difference between the centre of the disc and the position of the light.)

  b) Suppose $r=\omega =1$ and $R=\rho =2$. Show that ${|{\gamma }^{\prime }(t)|}^2=8(1+\cos t)$.

  (Hint: you will have to use the difference formula $\cos (a-b)=\cos a\cos b+\sin a\sin b$.)

  c) Determine the distance traversed by the light from time $t=0$ to $t=\pi$.

  (Hint: use $\cos 2a=1-2{\sin }^{2}a$.)

• Let $\gamma :ℝ\to {ℝ}^2$, $t\mapsto (t^2+2,(t-1)e^t)$; let $C$ be the image of the curve. Determine the equation of the tangent line to $C$ at $\gamma (1)$.

• Let $C$ be the set of points $P$ in ${ℝ}^2$ for which the distance from $P$ to the origin is half the distance from $P$ to the line $x=3$:

  $$\sqrt{x^2+y^2}=\frac{1}{2}|3-x|$$

  Show that $C$ is an ellipse, and determine the centre of $C$.

• True or false?

  i) There is exactly one curve between any pair of points in ${ℝ}^2$.

  ii) For any vector $𝐮$ in ${ℝ}^3$ and any $c\in ℝ$, the equation $(xyz)\cdot 𝐮=c$ defines a plane passing through the origin.

  iii) If $S$ is a surface in ${ℝ}^3$ and $L$ is a line in ${ℝ}^3$ then $S$ and $L$ intersect in exactly one point.

  iv) The line $y=1$ in ${ℝ}^2$ is a tangent line to the unit circle $C$ with centre the origin.