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 Problems1. § Find the points on the ellips 4x2+ y2= 4 that are farthest away from the point (1, 0).2. § Find the area of the largest rectangle that can be inscribed in the ellipse x2/a2+ y2/b2= 1.You may assume that the sides of the rectangle are parallel to the axes.3. (a) § Find the area of the largest rectangle that can be inscribed in a right triangle withlegs of lengths 3 cm and 4 cm if two of the sides of the rectangle lie along with legs.(b) Find the area of the largest rectangle that can be inscribed in a right triangle with legsof lengths 3 cm and 4 cm if one of the sides of the rectangle lies along the hypotenuse.4. § Find the dimensions of the isosceles triangle of largest area that can be inscribed in a circleof radius r.5. § A right circular cylinder is inscribed in a cone with height h and base radius r. Find thelargest possible volume of such a cylinder.6. § A cylindrical can without a top is made to contain V cm3of liquid. Find the dimensionsthat will minimize the cost of the metal to make the can.7. § 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. § A cone-shaped paper drinking cup is to be made to hold 27 cm3of water. Find the heightand radius of the cup that will use the smallest amount of paper.9. § 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?10. § 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.111. § If P = (a, a2) is any point on the parabola y = x2, except for the origin, let Q by the pointwhere the normal line intersects the parabola gain. Show that the line segment P Q has theshortest possible length when a = ±1/√2.12. § 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.13. 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.θ1Aθ2BC14. (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
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