Sketching Slope Fields - AP Calculus AB

1. Find the slope at $(0, 1)$ for $dy/dx = y - x^2$.

   1. Substitute: $1 - 0^2 = 1$.

2. Find the slope at $(1, 1)$ for $dy/dx = x^2 - 2y$.

   -1. Substitute: $1^2 - 2(1) = -1$.

3. What is the slope at $(0, 2)$ for $dy/dx = 2x - y$?

   -2. Substitute: $2(0) - 2 = -2$.

4. What is the slope at $(3, 2)$ for $dy/dx = x - y^2$?

   -1. Substitute: $3 - 2^2 = -1$.

5. Identify the slope at $ (2, 2) $ for $ dy/dx = x + y $.

   4. Substitute: $ 2 + 2 = 4 $.

6. What is the slope at $(1, 3)$ for $dy/dx = y - 2x$?

   1. Substitute: $3 - 2(1) = 1$.

7. Identify the slope of the line segment at (1, 2) if $dy/dx = x + y$.

   3. Substitute: $1 + 2 = 3$.

8. Identify the slope at $(1, 0)$ for $dy/dx = 2x + y$.

   2. Substitute: $2(1) + 0 = 2$.

9. Identify the slope of the line segment at (0, 0) if $dy/dx = x^2 - y^2$.

   0. Substitute: $0^2 - 0^2 = 0$.

10. What is the slope of the field at point $(2, 3)$ for $dy/dx = x - y$?

    -1. Substitute: $2 - 3 = -1$.

11. Identify the slope at $(1, 2)$ for $dy/dx = y - x$.

    1. Substitute: $2 - 1 = 1$.

12. Identify the slope at $ (1, 2) $ for $ dy/dx = y - x $.

    1. Substitute: $2 - 1 = 1$.

13. Which mathematical tool helps visualize differential equation solutions?

    Slope fields. They show solution directions graphically.

14. What is the slope at $(0, -1)$ for $dy/dx = x^2 + y$?

    -1. Substitute: $0^2 + (-1) = -1$.

15. Identify the slope at $(2, 2)$ for $dy/dx = x + y$.

    4. Substitute: $2 + 2 = 4$.

16. What is the slope at $(2, 1)$ for $dy/dx = y^2 + x$?

    3. Substitute: $1^2 + 2 = 3$.

17. What is the slope at $(3, 1)$ for $dy/dx = x^2 - y^2$?

    8. Substitute: $3^2 - 1^2 = 8$.

18. What is the slope at $(0, 2)$ for $dy/dx = x^2 - y$?

    -2. Substitute: $0^2 - 2 = -2$.

19. What is the slope at $(1, 1)$ for $dy/dx = 2x - y$?

    1. Substitute: $2(1) - 1 = 1$.

20. What is the purpose of sketching a slope field?

    To visualize possible solutions of a differential equation. Helps understand solution behavior patterns.

21. What is the slope at $(1, 2)$ for $dy/dx = y + x^2$?

    3. Substitute: $2 + 1^2 = 3$.

22. What is the slope at $(2, 3)$ for $dy/dx = x - y$?

    -1. Substitute: $2 - 3 = -1$.

23. What is the slope at $(0, 1)$ for $dy/dx = x^2 + y^2$?

    1. Substitute: $0^2 + 1^2 = 1$.

24. Identify the slope at $(2, 1)$ for $dy/dx = x + y$.

    3. Substitute: $2 + 1 = 3$.

25. What is a slope field?

    A graphical representation of a differential equation's solutions. Shows direction of solutions at each point.

26. What is the slope at $(2, 0)$ for $dy/dx = x^2 - y$?

    4. Substitute: $2^2 - 0 = 4$.

27. Find the slope at $(1, 1)$ for $dy/dx = x^2 - y^2$.

    0. Substitute: $1^2 - 1^2 = 0$.

28. What does a slope field represent?

    The slopes of solution curves for a differential equation at given points. Each segment shows the derivative value.

29. What is the slope at $(3, 3)$ for $dy/dx = x - 2y$?

    -3. Substitute: $3 - 2(3) = -3$.

30. Identify the slope at $(0, 3)$ for $dy/dx = x + y^2$.

    9. Substitute: $0 + 3^2 = 9$.

31. What is the slope at $(0, 2)$ for $dy/dx = x^2 + 2y$?

    4. Substitute: $0^2 + 2(2) = 4$.

32. Identify the slope at $(0, 1)$ for $dy/dx = x^2 - 2y$.

    -2. Substitute: $0^2 - 2(1) = -2$.

33. Find the slope at $ (1, 1) $ for $ dy/dx = $x^2$ - $y^2$ $.

    0. Substitute: $1^2 - 1^2 = 0$.

34. What is the slope at $(1, 0)$ for the differential equation $dy/dx = xy$?

    0. Substitute: $1 \cdot 0 = 0$.

35. What is the slope at $(1, 0)$ for $dy/dx = y^2 - x$?

    -1. Substitute: $0^2 - 1 = -1$.

36. What is the slope at $(2, 2)$ for $dy/dx = x^2 + y$?

    6. Substitute: $2^2 + 2 = 6$.

37. What is the slope at $(0, 0)$ for $dy/dx = x + y^2$?

    0. Substitute: $0 + 0^2 = 0$.

38. What does each segment in a slope field indicate?

    The slope of the solution curve at that point. It's the tangent direction at that location.

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