AP Exam Practice Questions 407 Section 2, Part A, Free Response, Technology Permitted 11. Consider the differential equation y' = (2x)/y with a 8. The rate of growth of the number of bacteria y is given particular solution in the form of y = f(x) that satisfies by dy/di = 0.5y, where I is the time in hours and / 2 0. the initial condition f(1) = 2. Initially, there are 200 bacteria. (a) Use Euler's Method, starting at x = I with two (a) Solve for y, the number of bacteria present, at any steps of equal size, to approximate f(1.4). Show the time / 2 0. work that leads to your answer. (b) Write and evaluate an expression to find the (b) Find the particular solution of the given differential average number of bacteria in the population for equation that passes through (1, 2) and state its O S I s 10. domain. 12. Consider the differential equation dy/dx = xy. (a) Let y = f(x) be the function that satisfies the Section 2, Part B, Free Response, No Technology differential equation with initial condition f(1) = 1. 9. Jet y = f(x) be a particular solution of the differential Use Euler's Method, starting at x = 1 with a step equation size of 0.1, to approximate f(1.2). Show the work that leads to your answer. dy 1 dx (b) Find day/dx2. Determine whether the approximation found in part (a) is less than or greater than f(1.2). with f(1) = 2. Justify your answer. (a) Find d'y/dx at the point (1, 2). (c) Find the particular solution of the given differential (b) Write an equation for the line tangent to the graph equation that passes through (1, 1). of f at (1, 2) and use it to approximate f(1.1). Is the 13. At any time / 2 0, the rate of the spread of a disease is approximation for f(1.1) greater than or less than modeled by the differential equation f(1.1)? Explain your reasoning. dy= (c) Find the solution of the given differential equation di 10 1000 that satisfies the initial condition f(1) = 2. 10. Consider the differential equation dy/dx = x-(1 - y). where y is the number of people who have the disease. In an isolated town of 1000 inhabitants, 100 people (a) On the axes provided, sketch a slope field for have the disease at the beginning of the week. the given differential equation at the nine points indicated. (a) Is the disease spreading faster when 100 people have the disease or when 200 people have the disease? Explain your reasoning. (b) Write a model of the form y = L/(1 + Ce-#) for the population y = f() at any time / 2 0. (c) What is lim y()? 14. Consider the differential equation dy/dx = x/y, where # 0. (a) On the axes provided, sketch a slope field for the given differential equation at the indicated points. (b) While the slope field in part (a) is drawn only at nine points, it is defined at every point in the xy-plane. Describe all points in the xy-plane for which the slopes are positive. (c) Find the particular solution in the form of y = f(x) to the given differential equation with the initial condition f(0) = 2. (d) Find day/dx in terms of x and y. Then determine if the particular solution from part (c) has a relative (b) Find day/dx in terms of x and y. minimum, a relative maximum, or neither at x = 0. (c) Find the particular solution of the given differential equation that satisfies the initial condition y(0) = 2.27. Sales The sales S (in thousands of units) of a new product after it has been on the market for 1' years is given by S = Ce\". (a) Find S as a function of I when 5000 units have been sold after 1 year and the saturation point for the market is 30,000 units (that is, lim S = 30). [105 (b) How many units will have been sold after 5 years? Finding a Particular Solution Using Separation of Variables In Exercises 33-36, find the particular 45. solution that satisfies the initial condition. Differential Equation Initial Condition 33. yy' - 3x = 0 y(2) = 2 34, yy' - 5e2x = 0 y(0) = -3 35. y3( x4 + 1 )y' - x3(y+ + 1) = 0 y(0) = 1 36. yy' - x cos x2 = 0 y(0) = -2