AP Calculus. Slope Fields and Differential Equations. Student Handout
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1 AP Calculus Slope Fields and Differential Equations Student Handout EDITION Use the following link or scan the QR code to complete the evaluation for the Stud Session Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at
2 Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at
3 Slope Fields and Differential Equations Students should be able to: Draw a slope field at a specified number of points b hand. Sketch a solution that passes through a given point on a slope field. Match a slope field to its differential equation. Match a slope field to its solution. Determine features of the solution to a differential equation based on its slope field and/or its solution. Solve separable differential equations. Determine a particular solution using an initial condition. Model a real world situation using a differential equation. Use derivatives to verif that a function is a solution to a given differential equation. Recognize that the model for eponential growth and deca that arises from the statement The rate of change of a quantit is proportional to the size of the quantit is d k dt kt and the solution to the differential equation is Ce. Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 1
4 Multiple Choice 1. (calculator not allowed) d If d, then could be (A) ln (B) (C) (D) (E) 7 e e e 1. (calculator not allowed) Which of the following is the solutiom to the differential equation where ()? d 4, d (A) for 0 (B) 6 for (C) 4 1 for (D) 4 1 for (E) 4 6 for 1.5 Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at
5 . (calculator not allowed) d Which of the following is a slope field for the differential equation? d Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at
6 4. (calculator not allowed) Shown above is a slope field for which of the following differential equations? (A) (B) (C) (D) (E) d d d d d d d d d d 5. (calculator not allowed) d with the initial d Which of the following is the solution to the differential equation cos condition? sin (A) sin (B) sin (C) (D) sin 5 Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 4
7 6. (calculator not allowed) t 0 f () t 4 1 Let f () t be a solution to the differential equation d k dt, where k is a constant. Values of f for selected values of t are given in the table above. Which of the following is an epression for f () t? (A) 4e t ln (B) (C) t ln 9 e t 4 (D) 4t 4 7. (calculator not allowed) The function e 6 is a solution to which of the following differential equations? (A) 1 0 (B) 1 0 (C) 6 0 (D) (calculator not allowed) d If and if 1 when 1, then when, d (A) (B) 1 (C) 0 (D) 1 (E) Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 5
8 9. (calculator not allowed) Which of the following is the solution to the differential equation condition (0) ln 7? (A) ln 7 (B) ln 6 (C) ln e 6 (D) ln 6e d d e with the initial 10. (calculator not allowed) The slope field for a certain differential equation is shown above. Which of the following could be a specific solution to that differential equation? (A) ln (B) 1 4 (C) sin (D) cos (E) Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 6
9 Free Response 11. (calculator not allowed) d Consider the differential equation ( ) cos d. Let f( ) be the particular solution to the differential equation with the initial condition f (0) 1. The function f is defined for all real numbers. (a) A portion of the slope field of the differential equation is given below. Sketch the solution curve through the point 0,1. (b) Write an equation for the line tangent to the solution curve in part (a) at the point 0,1. Use the equation to appropriate f (0.). (c) Find f( ), the particular solution to the differential equation with the initial condition f (0) 1. Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 7
10 1. (calculator not allowed) d Consider the differential equation d. (a) On the aes provided, sketch a slope field for the given differential equation at the si points indicated. d (b) Find in terms of and. Determine the concavit of all solution curves for the d given differential equation in Quadrant II. Give a reason for our answer. (c) Let f () be the particular solution to the differential equation with the initial condition f (). Does f have a relative minimum, a relative maimum, or neither at? Justif our answer. (d) Find the values of the constants m and b for which m b is a solution to the differential equation. Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 8
11 1. (calculator not allowed) At the beginning of 010, a landfill contained 1400 tons of solid waste. The increasing function W models the total amount of solid waste stored at the landfill. Planners estimate dw 1 that W will satisf the differential equation ( W 00) for the net 0 ears. W is dt 5 measured in tons, and t is measured in ears from the start of 010. (c) Find the particular solution W with initial condition W dw dt 1 5 W t to the differential equation W 00 Copright 016 National Math + Science Initiative, Dallas, Teas. All rights reserved. Visit us online at 9
Slope Fields and Differential Equations
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