Group HW01

Directions

  1. All answers have to be handwritten using pen and physical paper. Scan your answers and then upload as a pdf file to Animospace. Complete solutions with a sufficient level of explanation are preferred over incomplete ones.
  2. Only one person from the group submits the answers. Make sure that the group agrees to who should submit the answers.
  3. The deadline is May 13, 2024 before 10:00 am.
  4. You are already part of a group and the collaboration is only allowed within your group. That is the extent of collaboration that is allowed.
  5. Each member of the group should submit an evaluation form for the group homework.
  6. Read and consult the late policy in the syllabus if you have a tendency to be late with respect to submissions.
  7. Every group member has a responsibility to oneself and to the other members of their group. Hold each other accountable and give credit where it is due.

Exercises

  1. Let \(h(x)=\dfrac{1}{1+x^4}\). Assume that \(x\in [-1,1]\). Complete the non-calculus argument below to find maximum or minimum points of \(h\). You either will be filling in the blanks or supplying reasons or citing properties or theorems from EMEA. For citing properties or theorems or results from EMEA, give the page number and write down what you have cited as support.

    1. (2 points) Since \(-1\leq x\leq 1\), we must have \(0\leq x^4\leq 1\). We must have ___ \(\leq 1+x^4 \leq\) ___. Why?
    2. (2 points) Then, ___ \(\leq 1/(1+x^4) \leq\) ___. Why?
    3. (2 points) Therefore, the minimum point is \(x=\) ___ and the maximum point is \(x=\) ___. Why?

  2. Let \(h(x)=\dfrac{1}{1+x^4}\). Assume that \(x\in (-1,1)\). You are going to use calculus to find maximum or minimum points of \(h\):

    1. (1 point) Why did we assume \(x\in (-1,1)\) instead of \(x\in [-1,1]\)?
    2. (1 point) Find stationary points of \(h\).
    3. (2 points) Use the first derivative test to determine to classify the stationary points as maxima or minima or neither.

  3. (1 point) What happens when you try finding stationary points for the function in Section 8.1, Example 1(b)?

  4. A firm wants to estimate their cost function. Their cost function is a function of quantity produced \(q\geq 0\). It has fixed costs of forty thousand pesos in rent and overhead regardless of the level of \(q\). The firm believes that when \(q = 1000\), marginal cost equals 200 pesos and that when \(q = 5000\), marginal cost is 680 pesos. The firm believes that marginal cost can be approximated by a straight line.

    1. (1 point) Determine the marginal cost function given the information so far.
    2. (1 point) Using your knowledge of integration, determine what is the total cost function given your preceding answer.
    3. (2 points) At what level of \(q\) is average cost at a minimum? Show that you have indeed found a minimum.

  5. So far, you have been exposed to finding extreme points of functions of one variable. What happens if you are asked to find extreme points of functions of two variables, but those two variables are related somewhat by a constraint? Consider the following geometry problem: Find the rectangle with a perimeter of 8 cm which has the maximum area. Define \(x\) and \(y\) to be the length and width of the rectangle, respectively.

    1. (1 point) Write an equation which relates \(x\), \(y\), and the perimeter.
    2. (1 point) What would be the domains for \(x\) and \(y\) suitable for this context?
    3. (1 point) Write down an expression for the area of a rectangle in terms of \(x\) and \(y\).
    4. (1 point) Define an area function \(A(x)\) only in terms of \(x\) using what you obtained from the previous items.
    5. (1 point) What equation should be satisfied by the value of \(x\) which will maximize \(A(x)\)?

  6. Consider the worked out examples in Section 8.2 Example 1 and Section 8.3 Example 1(b).

    1. (3 points) Assume that \(t\geq 0\). Show that \[c^{\prime\prime}(t)=\frac{2t(t^2-12)}{(t^2+4)^3}.\] Determine if \(c(t)\) concave or convex over \(t\geq 0\).
    2. (2 points) Assume that \(N\geq 0\). Find \(\pi^{\prime\prime}(N)\) and determine if \(\pi(N)\) is concave or convex over \(N\geq 0\).