version Practice Math 34B Fall 2010 Prof D.A.R.Y.L. PRINT NAME Final no calculators Tardis Quality Bonus SCORE 2 71 Put final answers in boxes on this page. Show high quality work in the blue book for all answers. Points may be awarded for this. Number your solutions in the blue book. At the end of the exam place this page INSIDE the blue book, and then put the bluebook inside the envelope. Signature of TA who checked your photo ID (1) [ R /6] (x 3)2 dx = R2 R0t 0 k + 2e3t dt = sin(4x + 3) dx = (2) [ /6] (5) [ /6] A virus replicates itself at a rate proportional to the mass of virus already present. Intially there are 20 mg of and it is growing at a rate of 4 mg per hour. m(t) = mass in mg after t hours (a) What differential equation describes this. There should be no unknown constants. f (x) = (2x y)(x + 3y) + 7exy fx(x, y) = (b) What is the doubling time for this virus ? fy (x, y) = (6) [ fxy (x, y) = y /6] (a) Plot the slope field for y (t) = 2y t (3) [ /6] Find the equation of the tangent plane to z = x2 sin(2y) + 5x + 3 at (x, y) = (3, 0) Write it in the form z = mx + ny + b at each of the 25 points on the grid where t and y are 0, 0.5, 1 t (b) sketch the particular solution with y(1) = 1 (4) [ /6] A ball of molten rock is cooling to form a planet. (c) What is the long term behavior of this solution ? It starts at a temperature of 10, 000oK. After billions of years it has reached a steady temperature of 300oK . The initial rate of cooling is 10o K per thousand years. It cools according to Newton's law. Use y(t) for temperature t years after formation (a) What differential equation describes this. There should be no unknown constants. (b) How many years until it reaches a temperature of 500oK (7) [ /6 ] Find the local minimum and maximum of the function below. Use the second derivative test. f (x) = 2x3 3x2 36x + 7 x local min is at local max is at y f (x) is + or y practice (7) [ /6 ] t (a) What is the amplitude of the sine wave shown ? (b) What is the frequency ? (c) What is the equation for the graph in the form y = f(t) (9) [ /4] The number of people on planet Htrae was 6 billion in year 2000 D.C. and increasing at a rate of 60 million per year. The number of calories consumed by each Htrealing (= inhabitant of Htrae) was 2000 per day. This number is increasing at a rate of 10 per year (due to eating more pizza). Conveniently, there are 100 days in an Htraen year. How quickly was the total number of calories consumed by the entire planet growing that year ? Show how you worked this out here: (8) [ /4] The height of the point on a hill above the point (x,y) is (6x2 + 3xy 2y 2 + 15x 42y + 5000)/100 (a) At the point (2,3) which of the eight compass directions listed below would you walk to go down the hill as quickly as possible ? The x-axis points east and the y-axis points north. y=N NW NE (b) At which points (x,y) W would walking north initally not change your height ? x=E SE SW y= S (10) [ /3] Find the x and y values which give a local minimum of f (x, y) = 6x2 + 2xy + 8y 2 + 4x + 6y + 9 x= y= (11) [ /6] Beaker A contains 3 liters of which 10% is protein and the rest is fat. Beaker B contains 2 liters of which an unknown percentage is protein and the rest is fat. When the liquids are combined the result contain x% protein and the rest is fat. What is the percentage of protein in beaker B ? (12)[ /6] A water tank has a circular base of radius 4 meters and initally contains water to a depth of 5 meters. Then water drains out at an increasing rate so that after t ours the water is leaving at a rate of 6t m3 /hr How many hours until the tank is empty ? Tutors: please do not supply answers to these questions. I prefer you do similar problems so that students don't think because they can copy your answers they know how to do them. Thanks, Daryl