1) For a given curve, the slope of the tangent at each point xy, on the curve is equal to x

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1 Word Problems Word Problems ) For a given curve, the slope of the tangent at each point, on the curve is equal to. Find the equation of the curve. ) Given a curve, in the first quadrant, which goes through the point,3, and that the slope of its tangent at the point, equals. Find the equation of the curve. 3) Find the equation of the curve, whose normal at each point passes through the origin. 4) Find the equation of the curve, the slope of whose tangent at each point is equal to half the slope of the segment from the origin to the point. 5) Find the equation of the curve which passes through the point, and, for each point, on it, the slope of the normal is. 6) Given a curve in the first quadrant, passing through the point,4. Also given that for ach point A, on it, the difference between the slope of the tangent to the curve at A and between the slope of the line connecting A with the origin, is equal to the -oordinate of A. Find the equation of the curve. 7) Find the equation of the curve that passes through the origin and which is perpendicular to each line connecting a point on the curve to the point 3,4. 8) The area S is bounded the curve, the -ais and the lines a, (variable); see diagram below. It is known that the area S is proportional to the arc length between the points and, a, a. Find the equation of the curve. 9) Find the famil of curves orthogonal to the famil { c } Eas Education LLC All Rights Reserved

2 0) Find the famil of curves orthogonal to the famil { c }. ) Answer the following questions: a. Find the famil of curves orthogonal to the famil { c }. b. Find the curve orthogonal to the curve 9 at the point, on it. ) Find the famil of curves orthogonal to the famil { c }. 3) Find the famil of curves which form a 45 angle with the famil { c }. 4) At each point on a curve the segment of the normal between the point and the -ais is bisected b the -ais. Find the equation of the curve. 5) Find the equation of the curve passing through the point 0, such that the triangle bounded b the -ais, the tangent to the curve at an point M, on it, and the segment OM from the origin O to M, is an isosceles triangle whose base is the segment MN, where N is the intersection of the tangent with the -ais. Illustrate the problem with a sketch in the first quadrant. 6) The area S is bounded the curve the ais, (variable) ; See diagram below. It is known that and the lines Does such a curve eist such that the area of S equals? ( ) 7) The area S is bounded the curve., the -ais and the lines, (variable) ; See diagram below. It is known that. Does such a curve eist such that the area of S equals ( )? 8) The area S is bounded the curve See diagram below. It is known that S equals ( )?, the -ais and the lines, (variable) ;. Does such a curve eist such that the area of 9) Given a curve passing through the point B 0,. At each point A on the curve, the slope is equal to the area of the trapezoid ABCD as shown in the picture. What is the equation of the curve? Eas Education LLC All Rights Reserved

3 0) If a quantit t grows [decas] eponentiall; i.e. at each instant the rate of growth [deca] is proportional to its value. Suppose that at the start time t 0 the quantit is 0 and that the constant of proportionalit is k. Find a formula for the quantit at an time t. ) The population of the earth is increasing at a rate of % per ear. It was found to be 4 billion in 980. a. What will the population of the earth be in 00? b. What was the population of the earth in 974? c. When will a population of 50 billion be reached? (Assume that the population is growing eponentiall; i.e. at each instant the rate of growth is proportional to its value). ) The population in a certain cit grows eponentiall. In a certain ear there were 400 housand residents and 4 ears later there were 440 thousand. a. Find the annual growth rate (as a %). b. After how man ears (from that certain ear) were there 550 thousand residents? 3) A man deposited mone in the bank at an interest rate of 4% compounded annuall. After 5 ears he had accumulated $5000. a. How much did he initiall deposit? b. After how man ears will he have accumulated $7000? 4) The number of wild animals at a nature reserve grows eponentiall. There were 000 animals at the initial count. At a second count, 0 months later, there were 400 wild animals. How man months after the initial count will the reserve have 000 animals? 5) The radioactive isotope carbon-4 has a half-life of 5750 ears. At an given moment its rate of deca is proportion to the amount present. a. How man grams of this isotope will survive after 000 ears, if there were 00 grams initiall? b. After how man ears will there remain just 0 grams of the initial 00 grams? Eas Education LLC All Rights Reserved 3

4 6) In a certain pool there are 40 tons of fish, and the quantit of fish in it increases b 4% each week. In a second pool there are 00 tons of fish, and the quantit of fish in it increases b 0% each week. a. After how man weeks will both pools have the same quantit of fish? b. After how man weeks will the second pool have twice the quantit of fish as the first pool? 7) At time t 0 a tank contains 4 kg of salt dissolved in 00 liters of water. Salt water, at a concentration of 0. kg per liter of water, is flowing into the tank at a rate of 5 liters per minute and, simultaneousl, the mied solution is draining out of the tank at the same rate. a. Compute the amount of salt in the tank after 8 minutes. b. After how long will the amount of salt in the tank be twice the initial amount? 8) A rowboat is initiall towed at a rate of km/h. At time t 0 the cable is released and a man in the boat starts rowing in the direction of the motion, and applies a force of 0 newton to the boat. The mass of the boat & rower is 500 kg and the resistance (newton) is v, where v is the velocit of the boat in meters/sec. 9) Newton's Law of Cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and temperature of its surroundings. A substance with a temperature of 50 C is in a container which has the surrounding temperature of the air, a constant 30 C. The substance cools in accordance with Newton's Law of Cooling and, after half an hour, its temperature drops to 70 C. a. What is its temperature after an hour? b. After how long will its temperature be 40 C? 30) A spring of negligible weight is suspended verticall. A mass m is connected to its free end. If the mass is moving at a velocit 0 v m/sec when the spring is not etended, find the velocit v (in m/sec) as a function of the spring s etension (in m) Eas Education LLC All Rights Reserved 4

5 Final Answers: ) 3) 5) k ) k 4) 3 6) 3 7 a e e e ) 7) 9) k 0) ) a. a b. ) m c, 0 4) 3) k 5) k cosh C k k ln ln arctan c 6) 8) e e 4e e 7) e e 9) e e 4e e /4 e 0) () t 0e kt ) a. 7.8 b. 4.5 c. 6, ear 06 ) a t e b. 5.9 ears t t e b. 3.4 ears 3) a ) months 5) a gr b. 988 ears 6) a weeks b. 4.6 weeks 7) a kg b min 8) 0 m /sec 9) a b..3 hours 30) k v g v m Eas Education LLC All Rights Reserved 5