#import "@local/evan:1.0.0":*

#show: evan.with(
  title: [18.02 Mock Midterm 2],
  author: "<NAME>",
  date: [10 October 2024],
  maketitle: false,
)


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= 18.02 Mock Midterm 2 Instructions

- Don't turn the page until the signal to start is given (3:05 PM in 4-370 on October 21, 2024).

- You have 50 minutes to answer five questions.
  We're not grading anything, so write your solutions anywhere
  (the space below, other loose paper, notebook, iPad, etc.).

- Like the real exam, I suggest not referring to any notes/calculators/etc.

- Solutions are posted in Section 29 of my LAMV book at #url("https://web.evanchen.cc/1802.html").
]

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= 18.02 Mock Midterm 2 Questions

/ Question 1.: A butterfly is fluttering in the $x y$ plane with position given by
  $bf(r)(t) = angle.l cos(t), cos(t) angle.r$, starting from
  time $t = 0$ at $bf(r)(0) = angle.l 1,1 angle.r$.
  - Compute the speed of the butterfly at $t = pi/3$.
  - Compute the arc length of the butterfly's trajectory from $t = 0$ to $t = 2 pi$.
  - Sketch the butterfly's trajectory from $t = 0$ to $t = 2pi$ in the $x y$ plane.

/ Question 2.: Let $k > 0$ be a fixed real number and let $f(x,y) = x^3 + k y^2$.
  Assume that the level curve of $f$ for the value $21$ passes through the point $P = (1,2)$.
  Compute the equation of the tangent line to this level curve at the point $P$.

/ Question 3.: Let $f(x,y) = x^(5y)$ for $x,y > 0$.
  Use linear approximation to estimate $f(1.001, 3.001)$ starting from the point $(1,3)$.

/ Question 4.:
  Consider the function $f : RR^2 -> RR$ defined by
  $ f(x,y) = cos(pi x) + y^4/4 - y^3/3 - y^2. $
  - Compute all the critical points and classify them as saddle point, local minimum, or local maximum.
  - Find the global minimums and global maximums of $f$, if they exist.

/ Question 5.:
  Compute the minimum and maximum possible value of $x + 2 y + 2 z$
  over real numbers $x$, $y$, $z$ satisfying $x^2 + y^2 + z^2 <= 100$.

/ Question 6.:
  Consider the level surface of $f(x,y,z) = (x-1)^2 + (y-1)^3 + (z-1)^4$
  that passes through the origin $O = (0,0,0)$.
  Let $cal(H)$ denote the tangent plane to this surface at $O$.
  Give an example of two nonzero tangent vectors to this surface at $O$ whose span is $cal(H)$.


#v(3em)

The solutions to all the problems are now posted in Section 29 of my LAMV book:
#align(center)[
  #url("https://web.evanchen.cc/upload/1802/lamv.pdf").
]
]
