MATH3503 Online Assignment 203

• Make sure to use the link on the Learning Hub, which contains a user name and a user ID unique to you, otherwise your mark would not be recorded. You may use the URL without '?uname=...&uid=...' if you just want to practice. You will get different numbers every time you open the page.
• Numerical answers are evaluated as correct if the difference from the correct value is less than `0.001 times` (correct value) unless otherwise specified. That means, you need to keep at least 4 significant digits (e.g., 12.34, 0.01234, etc.) in your final answer, and a couple of more digits in the intermediate steps.
• Arithmetic operators:

  Multiplication	* (You cannot omit it!)
  division	/

• Available functions
  • Trigonometric functions: sin(x), cos(x), tan(x), sec(x), asin(x), acos(x), atan(x)
  • Exponential and logarithmic functions with base e: exp(x) (`=e^x`), ln(x) (or log(x))
  • Square root: sqrt(x)
  • Absolute value: abs(x)

Final Examination (December 8, 2020)

1. This is an closed-book exam. You may use two double-sided cheat sheets(You must make your own).
2. Please sign the "assessment declaration" form in "Final Exam" folder under Activities > Assignments.
3. Your hand-written work should be written legibly and in a well-organized manner on blank sheets of paper (or iPad, etc.). You need to show steps leading to the solution.
4. Submit your hand-written work (single PDF if possible)and cheat sheets to the folder "Final Exam".
5. Note that a correct answer online without hand-written work will be given zero marks.

1. Compute the following integrals.
   1.   \(\displaystyle\int(4x^{-8} +10x^{1/7}-5\sqrt[9]{x^{6}})dx\)
      `=``+C`
       * \(x^{a/b}\to\) x^(a/b)
   2.   `int(9/x +3/x^{1.08}) dx = ``+C`
      
   3.   `int x e^{-4x}dx= ``+C`
             * `e^a \to` exp(a)
      
   4.   \(\displaystyle\int\frac{3}{x^2+6x+8}\,dx= ~\)\( + C\)
      
   5.   \(\displaystyle\int_{0}^{\pi/3} \cos^3(7x)\,dx=~\)* Decimal number with more than 4 significant digits
      
   6.   \(\displaystyle\int_{0}^{2} \frac{x^3\,dx}{\sqrt{4-x^2\,}}\,=~\) * Decimal number with more than 4 significant digits
      
2.  
   Waste water is flowing into a tailings pond at the rate of \(\displaystyle r(t) = \frac{7.8}{(1+t)^{1.4}}\) \(\text{m}^3/\text{sec}\).

   1.  Find the total volume `V(t)` of the water in the tailings pond at time `t` [sec], given that `V(0)= 0`.
      ` V(t) = `\(\text{m}^3\)
   2.  After a long time what will be the total volume of water? I.e., `V(\infty) = `
      
   
   
3.  
   There is a mound of dirt whose cross-section at height `y` [m] is a circle of radius \(\displaystyle r(y) = \frac{63}{1+y/2}\)[m]. The mound has a flat circular top at `y=6` [m], find the total volume of dirt in this mound.
   Volume of dirt = [m³]
   
   
4.  
   An open pit has a paraboloid shape. Its depth is 22 metres and the diameter of the opening is 144 metres.

   1.   The paraboloid can be created by rotating a parabola about the `y`-axis. Let `y=0` be the bottom of the pit, then the equation of the parabola is `y=``x^2`.
   2.   Compute the volume of dirt removed to make this pit: m³
   3.   Compute the surface area of this pit: m²
   
   
5.  

   A horizontally placed trough has two semi-circular ends (See Fig. 1) of radius 0.2 m. Find the force on each end plate if the trough is filled with water. Use \(\rho=1000\text{ kg/m}^3,~~ g=9.81\text{ kg m/s}^2\). `F = `Newtons

   Fig. 1

   
   
6.  
   Let `rho(t)` be the density of a certain pollutant in soil. `rho(t)` decreases in time by diffusing into the surrounding area. The dynamics of `rho(t)` is described by the differential equation \[ \frac{d\rho}{dt}=-k(\rho-\rho_\text{env}), ~~\text{where }\rho\text{ in ppm, and }t\text{ in days} \] Suppose `k=2.7` \(\text{day}^{-1}\) and \(\rho_\text{env}=11\) ppm. Find the density at time `t` if the initial density is \(\rho(0)=190\) ppm. `\rho(t) = `
   
   
   
7.  
   Taylor-expand \(f(t) = e^{-2t}\sin(4t)\) about `t=0` to the 2nd order* in `t`. \(\displaystyle e^{-2t}\sin(4t) \approx~ \)
   * It is also called the quadratic approximation, i.e., \( f(x)\approx f(x_0) + f'(x_0)(x-x_0) + \frac{1}{2}f''(x_0)(x-x_0)^2 \).
   
   

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