Math 1A Discussion Exercises GSI Theo Johnson Freyd http math berkeley edu theojf 09Spring1A Find two or three classmates and a few feet of chalkboard As a group try your hand at the following exercises Be sure to discuss how to solve the exercises how you get the solution is much more important than whether you get the solution If as a group you agree that you all understand a certain type of exercise move on to later problems You are not expected to solve all the exercises in particular the last few exercises may be very hard Many of the exercises are from Single Variable Calculus Early Transcendentals for UC Berkeley by James Stewart these are marked with an Others are my own or are independently marked Optimization Problems 1 Find the points on the ellips 4x2 y 2 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 y 2 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 with legs 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 legs of 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 circle of radius r 5 A right circular cylinder is inscribed in a cone with height h and base radius r Find the largest possible volume of such a cylinder 6 A cylindrical can without a top is made to contain V cm3 of liquid Find the dimensions that 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 What is the length of the shortest ladder that will reach from the ground over the fence to the wall of the building 8 A cone shaped paper drinking cup is to be made to hold 27 cm3 of water Find the height and 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 rope attached to the object If the rope makes an angle with the plane then the magnitude of the force is W F sin cos where is a constant called the coefficient of friction For what value of is F smallest 10 If C x is the cost of producing x units of a commodity then the average cost per unit is C x x and the marginal cost per unit is C 0 x Prove that if the average cost is a minimum then the average cost equals the marginal cost 1 11 If P a a2 is any point on the parabola y x2 except for the origin let Q by the point where the normal line intersects the parabola gain Show that the line segment P Q has the shortest 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 triangle AP B has the largest possible area 13 Let v1 be the velocity of light in air and v2 the velocity of light X in water According to Fermat s Principle a ray of light will travel between a point A in the air to a point B in the water by a path ACB that minimizes the time taken Show that sin 1 v1 sin 2 v2 where 1 the angle of incidence and 2 the angle of refraction are as shown This equation is known as Snell s Law A 1 C 2 B 14 a Let C be a fixed positive number Prove that the minimum sum of two positive numbers whose 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 when the three numbers are equal Hint Call the numbers x y and z Then the product of y and z is C x if we pretend that x is fixed then how we can minimum the sum of the other two What is this minimum sum as a function of x So what is the minimum possibility for x plus this sum c Generalize to the sum of n positive numbers with a fixed product 2
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