Sketching Slope Fields - AP Calculus BC

1. What effect does a positive slope have on a slope field?

   Lines tilt upward from left to right. Positive slope creates upward-slanting segments.

2. How do you determine equilibrium solutions from a slope field?

   Look for horizontal line segments. Zero slope indicates equilibrium solutions.

3. What is the slope at $(2, 0)$ for $dy/dx = \frac{1}{y}$?

   Slope is undefined. Division by zero creates undefined slope.

4. How is a slope field constructed for $dy/dx = x + y$?

   Compute slopes at grid points using $m = x + y$. Substitute coordinates into the differential equation.

5. What information can you obtain from a slope field?

   The general shape and behavior of solution curves. Reveals overall patterns and equilibrium points.

6. How is $dy/dx = 0$ represented in a slope field?

   Horizontal line segments. Zero slope creates flat line segments.

7. What effect does a negative slope have on a slope field?

   Lines tilt downward from left to right. Negative slope creates downward-slanting segments.

8. Which direction do lines in a slope field point for $dy/dx = -1$?

   Lines point downward at 45 degrees. Constant negative slope creates uniform downward tilt.

9. How do you find the slope at $ (x, y) $ for $ dy/dx = $x^2$ + $y^2$ $?

   Compute $x^2 + y^2$ at $ (x, y) $. Evaluate the expression at each grid point.

10. What does a vertical line segment in a slope field indicate?

    The slope approaches infinity. Infinite slope means vertical tangent line.

11. What does it mean for a slope field to have symmetry?

    The slope pattern repeats across an axis. Pattern mirrors across coordinate axes.

12. What does each line segment in a slope field represent?

    The slope of the solution curve at a given point. Each segment shows the tangent direction at that point.

13. How do you find the slope at $(x, y)$ for $dy/dx = x^2 + y^2$?

    Compute $x^2 + y^2$ at $(x, y)$. Evaluate the expression at each grid point.

14. What is the slope at $(0, 1)$ for $dy/dx = \frac{1}{x}$?

    Slope is undefined. Division by zero creates undefined slope.

15. What does a horizontal line segment in a slope field indicate?

    The slope is zero at that point. Zero slope means no vertical change.

16. How do you sketch a slope field for $dy/dx = y$?

    Compute $m = y$ at grid points. Evaluate $y$ at each grid point for slope.

17. What is a key feature of slope fields for $dy/dx = y/x$?

    Lines are radial from the origin. Lines point away from origin along rays.

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