means Name a function whose derivative is 2x. x 4, and so forth.

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1 AP Slope Fields Worksheet Slope fields give us a great wa to visualize a famil of antiderivatives, solutions of differential equations. d Solving means Name a function whose derivative is. d Answers might include, 3, 4, and so forth. In general, C. If ou sketch several of these antiderivatives on the same graph grid ou see the famil of antiderivatives. Another wa to show the famil of antiderivatives is to draw a slope field d for d. The small segments simpl represent the slope of the functions at various points. It gives the shape of the possible solutions for the differential equations. Look at the slope field and visualize the famil of antiderivatives. You can also sketch the solution curve through a particular point. Drawing a Slope Field You are simpl drawing a short line, given a point and a slope. Intro. Draw a slope field for a differential equation such as d d. Pick a starting point on the grid and draw a tin line segment that passes through the point and that has a slope of. Move to another point, the slope will also be. Summarize & Analze: 1) Draw a slope field for the differential equation d d 1. Pick a starting point on the grid, sa (0, 0), find the slope d d 0 1 1, and draw a tin line segment that passes through the point and with the slope that ou found. Name other points that have the same slope. Draw the segments. Repeat the process with another point, until the slope field is filled. Summarize & Analze: ) Draw a slope field for d d Repeat the process established until the slope field is filled. Summarize & Analze: S. Stirling 014 Page 1 of 7

2 Simple Slope Field Patterns Summar: If all line segments on the slope field have the same slope, then the differential equation will be of the form d/d = constant. If all line segments in the vertical direction on a slope field have the same slope, then the differential equation does not contain a -term. The -coordinate determines the slope. If all segments in the horizontal direction on a slope field have the same slope, then the differential equation does not contain an -term. The -coordinate determines the slope. With differential equations that contain both an -term and a -term, such as d/d = +, look for points that have the same slope. Draw a slope field for each of the following differential equations. Each tick mark is one unit. 3) d d 4) d d 5) d d 1 6) d d / S. Stirling 014 Page of 7

3 Match the slope fields with their differential equations. 7) d d sin 8) d d 9) d d 10) d d Match the slope fields with their differential equations. 11) d d ) d d ) d d / 14) d d S. Stirling 014 Page 3 of 7

4 15) The slope field from a certain differential equation is shown above. Which of the following could be a specific solution to that differential equation? (A) (E) ln (B) e (C) e (D) cos 16) The slope field for a certain differential equation is shown. Which of the following could be a specific solution to that differential equation? (A) sin (B) cos (C) (D) S. Stirling 014 Page 4 of 7

5 d 17. Consider the differential equation given b d (A) On the aes provided, sketch a slope field for the given differential equation. (B) Let f be the function that satisfies the given differential equation. Write an equation for the tangent line to the curve f ( ) through the point (1, 1). Then use our tangent line equation to estimate the value of f (1.). (C) Find the particular solution g( ) to the differential equation with the initial condition g(1) 1. Use our solution to find g(1.). (D) Compare our estimate of f (1.) found in part (B) to the actual value of g(1.) found in part (C). Was our estimate from part (B) an underestimate or an overestimate? Use our slope field to eplain wh. S. Stirling 014 Page 5 of 7

6 18. Consider the differential equation given b d d (A) On the aes provided, sketch a slope field for the given differential equation. (B) Sketch a solution curve that passes through the point (0, 1) on our slope field. (C) Find the particular solution f ( ) to the differential equation with the initial condition f (0) =1. (D) Sketch a solution curve that passes through the point (0, 1) on our slope field. (E) Find the particular solution f ( ) to the differential equation with the initial condition f (0) = 1. S. Stirling 014 Page 6 of 7

7 006 AB 5 (No calculator) Consider the differential equation where 0. d 1, d (a) On the aes provided, sketch a slope field for the given differential equation at the eight points indicated. (scale is 1) (b) Find the particular solution f 1 1 and state its domain. f to the differential equation with the initial condition S. Stirling 014 Page 7 of 7