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  1. Most people are aware that cars emit CO$_2$ while running. Because fuel efficiency worsens as a car ages, it is tempting to think that, in order to minimize overall emissions, it best to buy a new car as soon as your old car shows signs of lower fuel efficiency. However, an important but often overlooked fact is that the process of building a car emits a surprisingly large amount of CO$_2$. This complicates the question of when to buy a new car if your goal is to minimize overall emissions.

    Consider that production of a typical mid-sized car releases $E_0=17,000$ kg of CO$_2$ into the atmosphere. These emissions can be counted at the moment the car is purchased. A new car burns 7 litres of fuel for every 100 km travelled or $r_0=0.07$ L/km. The fuel consumption rate of the car gradually increases with distance travelled at a rate of $2\cdot 10^{-5}$ litres per 100 km for every km driven or $a=2\cdot 10^{-7}$ L/km$^2$. Gasoline produces roughly $w=2.36$ kg of CO$_2$ per litre burned.

    To minimize overall CO$_2$ emissions, we want to choose a distance $x_{retire}$ at which to retire our old car and buy a new one. Throughout the questions below, do not substitute any numbers until step (e). All your answers until that point should depend on $E_0$, $r_0$, $a$ and $w$.

    1. What is the fuel consumption rate, $r(x)$, measured in L/km, of a car that has travelled $x$ km?
    2. What is the emission rate, $E'(x)$, measured in kg of CO$_2$ per km, of a car that has travelled $x$ km?
    3. What is the total amount of CO$_2$ emitted, $E(x)$, measured in kg, including both building the car and driving it $x$ km?
    4. What is the average emission rate over the lifetime of a car, $R(x)$, measured in kg/km, including both building the car and driving it $x$ km?
    5. The distance $x_{retire}$ should minimize the average emission rate. What is $x_{retire}$? Your answer should depend on $E_0$, $r_0$, $w$ and $a$ (or possibly a subset of those). For the values of these parameters given above, what is $x_{retire}$? Assuming an annual travel distance on 12,000 km, how many years after buying the car does the model indicate you should get a new one so as to minimize overall emissions?
    6. What is the equation of the tangent line to $E(x)$ that goes through the origin (the one whose point of tangency has a positive $x$ coordinate)? Sketch $E(x)$ along with its tangent line that goes through the origin. Explain the connection between this tangent line and $x_{retire}$.
    7. On a separate set of axes, sketch $R(x)$. Label $x_{retire}$ clearly.

  2. This question will deal with the aftermath of the midterm. It is extremely important for you to identify and address your mistakes on the midterm before they hinder your further performance in the course. Download the solutions from the Midterm information page and carefully check that your exam was graded according to the rubric. Next, for each question you got wrong, carefully go through it and characterize each, and every, mistake you made as:
    A: Algebra or Arithmetic error
    B: not knowing how to Begin the problem, not understanding what the question is asking for
    C: not understanding a Concept or how to apply it in this particular context (specify symbolic, graphic, or geometric)
    D: Distracted or careless mistake (rushed)
    E: other reasons (please specify)
    Create a list of errors you've made for every question. What types of errors are you prone to? What is your strategy to improve in these areas? You do not need to hand in a reworked and corrected version of your answers. You should just hand in a summary of your errors/issues and describe your strategy for avoiding them in the future.