Reasoning Using Slope Fields - AP Calculus AB

1. Describe slope field for $y' = x + y$ at the origin.

   Slope is $0$ at $(0, 0)$. $0 + 0 = 0$ at origin.

2. Identify the slope at $(0, 0)$ for $y' = y^3$.

   Slope is $0$ at $(0, 0)$. $0^3 = 0$ at origin.

3. What kind of slope does $y' = 3$ imply in a slope field?

   A constant slope of $3$ everywhere. Derivative is independent of $x$ and $y$.

4. What feature in a slope field indicates a solution curve is increasing?

   Positive slopes. Positive slope means function increasing.

5. What is the slope at $(1, -2)$ for $y' = 2x + y$?

   Slope is $0$ at $(1, -2)$. $2(1) + (-2) = 0$ when substituted.

6. How does a slope field for $y' = \frac{y}{x}$ look?

   Slopes equal $\frac{y}{x}$, resembling radial lines. Lines through origin with varying slopes.

7. What differential equation characterizes a slope field with all slopes zero?

   $y' = 0$. Derivative equals zero everywhere.

8. What does a slope field for $y' = x - y$ depict?

   Slopes decrease as y increases, and vice versa. Equilibrium line where $x = y$.

9. State the purpose of a slope field.

   To visualize solution curves of differential equations. Helps sketch solution curves graphically.

10. Identify slope at $(0, 2)$ for $y' = x^3 - y$.

    Slope is $-2$ at $(0, 2)$. $0^3 - 2 = -2$ when substituted.

11. Find the slope at $(2, 1)$ for $y' = x - y$.

    Slope is $1$ at $(2, 1)$. $2 - 1 = 1$ when substituted.

12. What is the slope at $(1, 0)$ for $y' = x^2 + y^2$?

    Slope is $1$ at $(1, 0)$. $1^2 + 0^2 = 1$ when substituted.

13. How can you verify a solution curve in a slope field?

    Check if curve follows the slope at each point. Tangent must match field direction.

14. What does a vertical line segment in a slope field suggest?

    Undefined slope or vertical tangent. Infinite slope creates vertical lines.

15. Describe the slope at $(0, 0)$ for $y' = x - y$.

    Slope is $0$ at $(0, 0)$. $0 - 0 = 0$ at origin.

16. Identify the slope at $(2, 2)$ for $y' = 3x + y$.

    Slope is $8$ at $(2, 2)$. $3(2) + 2 = 8$ when substituted.

17. What is the slope at $(3, 4)$ for $y' = x + 2y$?

    Slope is $11$ at $(3, 4)$. $3 + 2(4) = 11$ when substituted.

18. What does a slope field represent?

    A visual representation of a differential equation's solutions. Shows tangent directions at each point.

19. Identify the slope at $(2, 3)$ for $y' = 4x - y$.

    Slope is $5$ at $(2, 3)$. $4(2) - 3 = 5$ when substituted.

20. What characterizes a slope field for $y' = \text{constant}$?

    Parallel line segments. Same slope at every point.

21. Which line segment would represent $y' = 0$ in a slope field?

    A horizontal line segment. Zero derivative means no change.

22. Find slope at $(1, 1)$ for $y' = y^2 - x^2$.

    Slope is $0$ at $(1, 1)$. $1^2 - 1^2 = 0$ when substituted.

23. Determine the slope at $(1, 1)$ for $y' = x^2 - xy + y^2$.

    Slope is $1$ at $(1, 1)$. $1^2 - 1(1) + 1^2 = 1$.

24. Describe the slope field for $y' = \frac{1}{x}$.

    Vertical lines with slopes decreasing as $x$ increases. Undefined at $x = 0$, varies with $x$.

25. Which differential equation produces a slope field with concentric circles?

    $y' = \frac{-x}{y}$. Orthogonal trajectories to circles.

26. What is the characteristic of a slope field for $y' = y$?

    Exponential growth with positive y-values. Slopes proportional to $y$ values.

27. What is the slope at $(2, -1)$ for $y' = 2 - y$?

    Slope is $3$ at $(2, -1)$. $2 - (-1) = 3$ when substituted.

28. Describe the slope field for $y' = -y$.

    Slopes are negative y-values, decreasing as y increases. Exponential decay pattern emerges.

29. What is the slope at a point $(x, y)$ in a slope field defined by $y' = x + y$?

    Slope is $x + y$ at $(x, y)$. Substitute coordinates into the equation.

30. What slope does $y' = y$ produce at $(0, 1)$?

    Slope is $1$ at $(0, 1)$. $y' = 1$ when $y = 1$.

31. Find the slope at $(0, 0)$ for $y' = x^2 + y^2$.

    Slope is $0$ at $(0, 0)$. $0^2 + 0^2 = 0$ at origin.

32. What feature in a slope field indicates a solution curve is constant?

    Horizontal line segments. Zero slope means no change.

33. What is the slope field for $y' = 2x$ at the origin?

    Slope is $0$ at $(0, 0)$. $y' = 2(0) = 0$ at origin.

34. Identify the slope at $(1, 2)$ for $y' = 2x + 3y$.

    Slope is $8$ at $(1, 2)$. $2(1) + 3(2) = 8$ when substituted.

35. What slope does $y' = x^2$ produce at $(2, -1)$?

    Slope is $4$ at $(2, -1)$. $2^2 = 4$ regardless of $y$.

36. Determine the slope at $(3, 0)$ for $y' = x^2 - y^2$.

    Slope is $9$ at $(3, 0)$. $3^2 - 0^2 = 9$ when substituted.

37. Find slope at $(3, 3)$ for $y' = x - 2y$.

    Slope is $-3$ at $(3, 3)$. $3 - 2(3) = -3$ when substituted.

38. Describe the slope field for $y' = x - y^2$.

    Slopes decrease with increasing y. Quadratic term dominates behavior.

Study these flashcards