1. Find the exact length of the curve y = Inx 2. [10 marks] x2 -...

    

Transcribed Image Text:

1. Find the exact length of the curve y = Inx 2. [10 marks] x2 - 1x2. 2 4 Find the exact length of the curve x == 4y2' +421 y 2. 3. [10 marks] Find the exact value of the area of the surface obtained by rotating the curve y=+ x1 about the x-axis. , 1 4. [10 marks] Find the exact area of the surface generated by revolving the curve x = y+4, 0 y 2 around the x axis. 5. [10 marks] Given the demand curve D(x) = (x-6) and the supply curves(x) = x, find the consumer surplus and the producer surplus. Illustrate by sketching the supply and demand curves and identifying the surpluses as areas. 6. [10 marks] After an 8-mg injection of dye, the readings of dye concentration at 2-second intervals are as shown in the table. Use Simpson's Rule to estimate the cardiac output, giving the appropriate units. t 0 2 4 6 8 10 12 14 16 18 20 C(t) 0 2.4 5.1 7.8 7.6 5.4 3.9 2.3 1.6 0.7 0 7. [8 marks] Find an equation of a curve that passes through the point (1,-2) with the property that the slope of the curve at any point (x, y) is equal to x/y. Give answer in the form y = f(x). 8. [8 marks] Find the solution to the differential equation -xy + dy/dx = y/x for which y(1) 8e. Give your answer in the form y = f(x). your 9. [12 marks] litre A tank contains 100 litres of pure water. Brine that contains 0.1 kg of salt per enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes? 10. [12 marks] a. (6 marks) Show that, if P is a twice differentiable function of t and satisfies the logistic differential equation then b. (6 marks) dP dt d P =P(1-1) = k = kP 2P M 2P - kP M dr = kp (1-2) (1-2). M Suppose a population grows according to the logistic growth model and that the initial population size is positive but less than half the carrying capacity of the environment. Use the result from part a. of this question to show the growth rate of the population is maximal when the population size is exactly half the carrying capacity. 1. Find the exact length of the curve y = Inx 2. [10 marks] x2 - 1x2. 2 4 Find the exact length of the curve x == 4y2' +421 y 2. 3. [10 marks] Find the exact value of the area of the surface obtained by rotating the curve y=+ x1 about the x-axis. , 1 4. [10 marks] Find the exact area of the surface generated by revolving the curve x = y+4, 0 y 2 around the x axis. 5. [10 marks] Given the demand curve D(x) = (x-6) and the supply curves(x) = x, find the consumer surplus and the producer surplus. Illustrate by sketching the supply and demand curves and identifying the surpluses as areas. 6. [10 marks] After an 8-mg injection of dye, the readings of dye concentration at 2-second intervals are as shown in the table. Use Simpson's Rule to estimate the cardiac output, giving the appropriate units. t 0 2 4 6 8 10 12 14 16 18 20 C(t) 0 2.4 5.1 7.8 7.6 5.4 3.9 2.3 1.6 0.7 0 7. [8 marks] Find an equation of a curve that passes through the point (1,-2) with the property that the slope of the curve at any point (x, y) is equal to x/y. Give answer in the form y = f(x). 8. [8 marks] Find the solution to the differential equation -xy + dy/dx = y/x for which y(1) 8e. Give your answer in the form y = f(x). your 9. [12 marks] litre A tank contains 100 litres of pure water. Brine that contains 0.1 kg of salt per enters the tank at a rate of 10 L/min. The solution is kept thoroughly mixed and drains from the tank at the same rate. How much salt is in the tank after 6 minutes? 10. [12 marks] a. (6 marks) Show that, if P is a twice differentiable function of t and satisfies the logistic differential equation then b. (6 marks) dP dt d P =P(1-1) = k = kP 2P M 2P - kP M dr = kp (1-2) (1-2). M Suppose a population grows according to the logistic growth model and that the initial population size is positive but less than half the carrying capacity of the environment. Use the result from part a. of this question to show the growth rate of the population is maximal when the population size is exactly half the carrying capacity.