Math 1A: Discussion ExercisesGSI: Theo Johnson-Freydhttp://math.berkeley.edu/∼theojf/09Spring1A/Find two or three classmates and a few feet of chalkboard. As a group, try your hand at thefollowing exercises. Be sure to discuss how to solve the exercises — how you get the solution ismuch more important than whether you get the solution. If as a group you agree that you allunderstand a certain type of exercise, move on to later problems. You are not expected to solve allthe exercises: in particular, the last few exercises may be very hard.Many of the exercises are from Single Variable Calculus: Early Transcendentals for UC Berkeleyby James Stewart; these are marked with an §. Others are my own, or are independently marked.Optimization Problems: Applications to Business and Economics1. § If C(x) is the cost of producing x units of a commodity, then the average cost per unit isC(x)/x, and the marginal cost per unit is C0(x). Prove that if the average cost is a minimum,then the average cost equals the marginal cost.2. § Let R(x) be the revenue brought in by selling x units of a commodity, and C(x) the cost ofproducing those x units. Then the profit from selling x units is P (x) = R(x) − C(x). Definethe marginal cost per unit to be C0(x) and the marginal revenue per unit to by R0(x). Provethat if the company sells a number of units so as to maximize its profits, then the marginalcost is equal to the marginal revenue.3. If a company charges more for a commodity, then the number of units of the commoditythe company can sell generally decreases. Define the price function p(x) to be the unit pricefor which the company will sell x many units of the commodity. Why is p(x) a decreasingfunction? Why does R(x) = x p(x)?(a) Explain how the marginal revenue per unit depends on the demand. In particular, showthat when x = 0, the marginal revenue per unit is exactly the price.(b) If p(x) = b − mx, where m, b > 0, for what value of x is the marginal revenue 0? Forwhat values is it positive? For what values is it negative?(c) If p(x) = Ce−kxfor C, k > 0, does there ever come a time when the marginal revenue isnegative? If so, when? What about for p(x) = ae−kx+ b where a, b, k > 0?4. Economists are not known for their mathematics skills. Thus, they generally assume thatp(x) and C(x) are linear functions. Let’s say that p(x) = b − mx and C(x) = a + lx, whereb, m, a, l > 0. In terms of a, b, m, l, how many units of a commodity should the companyproduce, and what price should the company charge, so as to maximize their profits? Whatis the maximum profit the company can make?5. In your group, discuss how this shows that sometimes a company can increase its profits byraising its prices, and sometimes it can do so by lowering its prices. Explain how the companycan determine whether it should raise or lower its prices in terms of the marginal cost andmarginal revenue.6. In your group, discuss the problems with these models. For example, if p(x) is linear, then forlarge enough x, p(x) will be negative. Is this reasonable? If C(x) is linear, then the marginal1cost is constant. Is this reasonable, even as x → ∞? What happens to a company that raisesits prices if there is a competing company making a similar product?Other Optimization Problems7. § A fence 8 ft tall runs parallel to a tall building at a distance of 4 ft from the building. Whatis the length of the shortest ladder that will reach from the ground over the fence to the wallof the building?8. § An object with weight W is dragged along a horizontal plane by a force acting along a ropeattached to the object. If the rope makes an angle θ with the plane, then the magnitude ofthe force isF =µWµ sin θ + cos θwhere µ is a constant called the coefficient of friction. For what value of θ is F smallest?9. § Let A = (a, a2) and B = (b, b2) be two fixed points on the parabola y = x2, with a ≤ b.Find the point P = (x, x2) on the arc between A and B (i.e. a ≤ x ≤ b) so that the triangleAP B has the largest possible area.10. X§ Let v1be the velocity of light in air and v2the velocity of lightin water. According to Fermat’s Principle, a ray of light willtravel between a point A in the air to a point B in the waterby a path ACB that minimizes the time taken. Show thatsin θ1sin θ2=v1v2where θ1(the angle of incidence) and θ2(the angle of refraction)are as shown. This equation is known as Snell’s Law.θ1Aθ2BC11. (a) Let C be a fixed positive number. Prove that the minimum sum of two positive numberswhose product is C occurs when the two numbers are equal.(b) Prove that the minimum sum of three positive numbers whose product is C occurs whenthe three numbers are equal. Hint: Call the numbers x, y, and z. Then the product ofy and z is C/x; if we pretend that x is fixed, then how we can minimum the sum of theother two? What is this minimum sum, as a function of x? So what is the minimumpossibility for x plus this sum?(c) Generalize to the sum of n positive numbers with a fixed