**UNAUTHORIZED LATE WORK MAY BE PENALIZED.

GROUND RULES: 1. You may consult a)your classmates, b)your notes, c)calculus textbooks, and d) myself.

2.You may not consult any other persons (student or faculty) than those allowed in rule 1.

3. All collaborations and consultations should be acknowledged.

4. Team papers must be signed by all partners and affirm that the work represents the collaboration of the partners on the problems. Each partner should give an estimate of the time spent working on the assignment. [Keeping a log for yourself is helpful.]

* 5. Submitted work should reflect your own understanding.

1. (10 points) Suppose f(x)= x ³ + Ax ² + Bx +C. Find A, B, and C so that f(0)=3, x=2 is a local minimum point, and x = 1 is a point of inflection. [Hint: Find A first.] Sketch the graph of f(x) labeling clearly with coordinates any local extrema and points of inflection. Justify briefly the features of your graph.
2. (14 points) Suppose y is a function of x that satisfies the equation
y' = dy/dx = (x-1)(4-y) .
1. Sketch a tangent field for the differential equation in a window with -2<x<2 and -2<y<6 showing all four quadrants with at least three integral curves.
2. DELETED: Suppose y(0) = 1. Explain briefly why each of the following statements should be true.
1. DELETED
2. DELETED
3. DELETED
3. Use implicit differentiation to show that y ''= (y-4)(x ² - 2x).
4. Suppose 0< y < 4. Discuss the graphical features of y, i.e. when is the graph of y increasing, decreasing, concave up, concave down, etc.
3. (6 points) Assume P(x) is a solution to the differential equation P'(x)=1/(1+x²) with P(0)=0.
1. Estimate P(1) using Euler's method with n = 10.
2. Is your answer in part a) an overestimate or an underestimate for the exact value? Discuss briefly your reasoning. [Hint: Consider the graphical interpretation of Euler's Method.]
4. (6 points) Assume P(x) is a solution to the differential equation P'(x) = 1/(1+x ²) with P(0) = 0.
1. Sketch a tangent field for this differential equation with the graph of an integral curve representing the function P.
2. Let Q(x) / P(tan(x)). Use the chain rule and trigonometry to show that Q'(x) = 1 for all x where -p/2 < x < p/2 .
3. Use B to explain why Q(x)= x for -p/2 < x < p/2 .[ Why is Q(0)= 0?]
4. Explain why P(1) = Q(p/4) = p/4.
5. (4 points) Use "substitution" to solve the following indefinite integrals in terms of the function P of problem 4.

[For example: ; let u = 2x.]
 
1. 
2. [Hint: Express 5+4x+x ² as 1+[Q(x)] ².]
6. (10 points)
1. Fly-by-night Airlines has just announced a special summer charter flight fare for H.S.U. students from Arcata to Hawaii and return. A minimum of 80 students must sign up for each flight at a round trip fare of $210.00 per person. However, the airline has offered to reduce each student's fare by $1.00 for each additional student who joins the flight. Under these arrangements, what number of passengers will provide the airline with the greatest revenue per flight? What kind of assumptions have you made abou the functions you used in solving this problem?
2. Suppose you are given constants A, B, C, and D.

Let . Using calculus, for what value(s) of x does g(x) assume a minimum value? Justify your answer briefly. Generalize your result (if possible) for more constants.