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Chapter 6 - Differential Equations and Mathematical Modeling

Chapter 6 - Differential Equations and Mathematical Modeling. 6.1 Slope Fields and Euler’s Method. An equation involving a derivative is called a differential equation . The order of a differential equation is the order of the highest derivative involved in the equation.

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Chapter 6 - Differential Equations and Mathematical Modeling

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  1. Chapter 6 - Differential Equations and Mathematical Modeling

  2. 6.1 Slope Fields and Euler’s Method An equation involving a derivative is called a differential equation. The order of a differential equation is the order of the highest derivative involved in the equation.

  3. First-order Differential Equation If the general solution to a first-order differential equation is continuous, the only additional information needed to find a unique solution is the value of the function at a single point, called an initial condition. A differential equation with an initial condition is called an initial-value problem. It has a unique solution, called the particular solution to the differential equation.

  4. Ex: Find the particular solution to the equation whose graph passes through the point (0,3).

  5. Slope Field The differential equation gives the slope at any point (x, y). This information can be used to draw a small piece of the linearization at that point, which approximates the solution curve that passes through that point. Repeating that process at many points yields an approximation called a slope field. Graph the family functions that solve the differential equation .

  6. Example Constructing a Slope Field

  7. We will now do some extra activities with slope fields after 6.4 and how they relate to the AP exam. Homework p.327 (1-39)odd

  8. 6.2 Antiderivation by Substitution Indefinite Integral

  9. Homework p. 337 (1-69)odd

  10. 7.2 Areas in the Plane Area Between Curves

  11. Area Between Curves

  12. How to find the area between two curves: Step 1: Graph the curves and determine which is on top Step 2: Find the limits of integration Step 3: Write a formula for Step 4: Integrate from a to b.

  13. Ex: Find the area of the region R in the first quadrant that is bounded above by and below by the x-axis and the line y = x - 2.

  14. Make a choice to find the area of the region enclosed by the graphs y = x3 and x = y2 – 2. Example Using Geometry

  15. Homework: p. 395 (1 – 49) odd

  16. 6.4 Exponential Growth and Decay Separable Differential Equation

  17. A special example:

  18. The Law of Exponential Change

  19. Continuously Compounded Interest

  20. Homework p. 357 (1-14)

  21. 6.5 Logistic Growth

  22. Homework p.369 (1-22)

  23. Volume of a Solid How to Find Volumes by the Method of Slicing 7.3 Volumes

  24. Example: Volume of a Wedge A curved wedge is cut from a cylinder of radius 3 by two planes. One plane is perpendicular to the axis of the cylinder. The second plane crosses the first plane at 450 angle at the center of the cylinder. Find the volume of the wedge.

  25. Solids of Revolution: Circular Cross Sections The most common application of the method of slicing is to solids of revolution. Solids of revolution are solids whose shapes can be generated by revolving plane regions about the axes. The Disk Method

  26. Example: About the x-axis The region between the curve , and the x-axis is revolved about the x-axis to generate a solid. Find its volume.

  27. Example: About the line y = 1 Find the volume of the solid generated by revolving the region bounded by and the lines y = 1, x = 4 about the line y = 1. How to find volume using the Disk Method: Step 1: Draw the region and identify the radius function Step 2: Square the radius function and multiply by pi Step 3: Integrate to find the volume

  28. Example: Revolution about the y-axis Find the volume of the solid generated by revolving the region between the y-axis and the curve , about the y-axis.

  29. Example: Revolution about a vertical axis Find the volume of the solid generated by revolving the region between the parabola and the line x = 3 about the line x = 3

  30. Washer Cross sections: If the region we revolve to generate a solid does not border on or cross the axis of revolution, the solid has a hold in it. The cross sections perpendicular to the axis of revolution are washers instead of disks. Outer radius: R(x) Inner radius: r(x) The washer’s area is:

  31. Example: Rotation about the x-axis The region bounded by the curve y = x2 + 1 and the line y = -x + 3 is revolved about the x-axis to generate a solid. Find the volume of the solid.

  32. How to find volume using washer cross sections: Draw the region and sketch a line segment across it perpendicular to the axis of revolution. Find the limits of integration Find the outer and inner radii Integrate Example: Rotation about the y-axis The region bounded by the parabola y = x2 and the line y =2x in the first quadrant is revolved about the y-axis to generate a solid. Find the volume of the solid.

  33. Homework: • 406 (1 – 42) • No shell problems

  34. 8.2 L’Hôpital’s Rule L’Hôpital’s Rule First Form

  35. L’Hôpital’s Rule Stronger Form

  36. Homework p. 450 (1-16)

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