MATH115 GEOMETRY AND CALCULUS

Quiz 2

• Lines and planes Which of the following statements is TRUE?

  A) There is a unique curve passing through any two points in ${ℝ}^2$,

  B) There is a unique plane passing through any two points in ${ℝ}^3$,

  C) Any plane in ${ℝ}^3$ has a unique normal vector,

  D) For each pair of non-zero vectors $𝐮,𝐯$ in ${ℝ}^3$, there exists $\lambda \in ℝ$ such that $𝐮-\lambda 𝐯$ is orthogonal to $𝐯$,

  E) Vectors $𝐮,𝐯$ in ${ℝ}^3$ are orthogonal if and only if $𝐮=\lambda 𝐯$ for some $\lambda \in ℝ$.

• Thinking about gradients The following is a sketch of the derivative $f^{\prime }(x)$ of a function $f(x)$ (not of $f$ itself!).

  $\text{ curve (10,100,20,81,30,64,40,49,50,36,60,25,70,16,80,9,90,4,100,1, 110,0,120,1,130,4,140,9,150,16,160,25,170,36,180,49,190,64,200,81,210,100) }\text{ curve (-10,0,230,0) }\text{ curve (0,-10,0,120) }\text{ curve (0 ,0,-4,-2) }\text{ curve (0,0,-4,2) }\text{ curve (0,0,2,-4) }\text{ curve (0,0,-2,-4) }\text{ y }\text{ x }\text{ α }$

  Which of the following best describes the point $\alpha$?

  A) A local minimum,$$ B) A local maximum,$$ C) A saddle point,

  D) It’s not a stationary point,$$ E) It’s impossible to tell.

• Turning points How many turning points does the polynomial $f(x)=x^5-10x^3+25x+4$ have?

  A) 1,$$ B) 2,$$ C) 3,$$ D) 4,$$ E) 5.

• Arc length Let $\gamma :ℝ\to {ℝ}^2$, $t\mapsto (3t^2+1,3t^3-t-1)$. What is the length of the curve between the points $t=0$ and $t=1$?

  A) $\sqrt{13}$,$$ B) 4,$$ C) $3/4$,$$ D) $2$,$$ E) 1.

• Tangent line What is the equation of the tangent line to the parametrized curve $\gamma :ℝ\to {ℝ}^2$, $\gamma (t)=(t^3+3,e^t)$ at the point $\gamma (1)$?

  A) $3x+ey=e^2+12$,$$ B) $\frac{y-1}{x-3}=e-1$,$$ C) $ex-3y=e$,$$ D) $\frac{y-e}{x-2}=\frac{e}{2}$,$$ E) $y=ex+e+1$.

Assessed Exercises

• (2 marks) Let $\gamma :ℝ\to {ℝ}^2$ be the plane curve given by $\gamma (t)=(3\tanh t,2\text{sech}t)$. Show that the image of $\gamma$ is contained in an ellipse, and determine the equation of the ellipse. Is the image the whole ellipse?

  
  

• A metal bar has one end fixed at the origin, and rotates anti-clockwise at a constant rate of $\omega$ radians per second; at time $t=0$ it points along the $x$-axis. A beetle, starting at the origin, walks along the bar at a constant speed of $v$ metres per second.

  a) (1 mark) What is its position at time $t$?

  b) (2 marks) What is its acceleration at time $t$?

  c) (2 marks) The beetle will stop walking when the magnitude of its acceleration is $4v\omega$ m s${}^{-2}$. At what distance along the bar will it stop?

• (3 marks) Calculate the length of the curve $\gamma :ℝ\to {ℝ}^2$, $\gamma (t)=(4\cosh t,3\sinh t+5t)$ between the points $\gamma (0)$ and $\gamma (1)$.

Bonus question. Let $S$ be the surface in ${ℝ}^3$ defined by $z=1/(x^2+y^2)$. By introducing the auxiliary variable $u=x^2+y^2$, or otherwise, show that the minimum distance of a point on $S$ from the origin is $\sqrt{3}/\sqrt[3]{2}$.