Slope Fields and Differential Equations

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Student Stud Session 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. Recognize the solution to the differential equation model for when the rate of change of is directl proportional to : d kt k; Ce is the solution dt Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

Slope Fields and Differential Equations Student Stud Session Multiple Choice 1. (calculator not allowed) If d d, then could be (A) ln (B) (C) (D) 7 e e e 1. (calculator not allowed) Which of the following is the solutiom to the differential equation where ()? (A) for 0 (B) 6 for (C) (D) 4 1 for 4 1 for 4 6 for 1.5 d d 4, Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

Slope Fields and Differential Equations Student Stud Session. (calculator not allowed) d Which of the following is a slope field for the differential equation? d Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

4. (calculator not allowed) Slope Fields and Differential Equations Student Stud Session Shown above is a slope field for which of the following differential equations? (A) (B) (C) (D) d d d d d d d d d d Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

Slope Fields and Differential Equations Student Stud Session 5. (calculator not allowed) Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how man hours will the number of bacteria triple? (A) (B) (C) (D) ln ln ln ln ln ln 7 ln 9 ln 6. (calculator not allowed) d If and if 1 when t 0, what is the value of t for which 1 dt? (A) ln (B) 1 4 (C) ln (D) ln 7. (calculator not allowed) d If and if 1 when 1, then when, d (A) (B) 1 (C) 0 (D) 1 Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

Slope Fields and Differential Equations Student Stud Session 8. (calculator not allowed) At each point, on a certain curve, the slope of the curve is. If the curve contains the point 0,8, then its equation is (A) 8e (B) 8 (C) e 7 (D) ln( 1) 8 8 d 9. (calculator not allowed) If the graph of f ( ) contains the point (0, ), and f( ) 0 d e for all, then f ( ) (A) (B) (C) 1 (D) e e e e e 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 Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

Free Response Slope Fields and Differential Equations Student Stud Session 11. (calculator not allowed) Consider the differential equation d. d (a) On the aes provided, sketch a slope field for the given differential equation at the twelve points indicated. (b) Let f () be the particular solution to the differential equation with the initial condition f ( 1) 1. Write an equation for the line tangent to the graph of f at 1, 1 and use it to approimate f (1.1). (c) Find the particular solution f () to the given differential equation with the initial condition f ( 1) 1. Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

1. (calculator not allowed) Consider the differential equation 1 cos Slope Fields and Differential Equations Student Stud Session d. d (a) On the aes provided, sketch a slope field for the given differential equation at the nine points indicated. (b) There is a horizontal line with equation value of c. c that satisfies this differential equation. Find the (c) Find the particular solution f () to the differential equation with the initial condition f ( 1) 0. Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

Slope Fields and Differential Equations Student Stud Session 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 that W will satisf the dw 1 differential equation ( W 00) for the net 0 ears. W is measured in tons, and t is dt 5 measured in ears from the start of 010. (c) Find the particular solution W condition W 0 1400. dw 1 with initial dt 5 Wt to the differential equation W 00 14. (calculator not allowed) (00 BC 5) Consider the differential equation d 4. d (a) The slope field for the given differential equation is provided. Sketch the solution curve that passes through the point (0,1) and sketch the solution curve that passes through the point (0,1). (c) Find the value of b for which b is a solution to the given differential equation. Justif our answer. Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

15. (calculator not allowed) 1997 AB/BC 6 Slope Fields and Differential Equations Student Stud Session Let vt () be the velocit, in feet per second, of a skdiver at time t seconds, t 0. After her parachute dv opens, her velocit satisfies the differential equation v, with initial condition (0) 50 dt v. (a) Use separation of variables to find an epression for v in terms of t, where t is measured in seconds. (b) Terminal velocit in defined as lim vt ( ). Find the terminal velocit of the skdiver to the nearest foot per second. t (c) It is safe to land when her speed is 0 feet per second. At what time t does she reach this speed? Copright 014 National Math + Science Initiative, Dallas, TX. All rights reserved. Visit us online at www.nms.org

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