AP Calculus AB Flashcards: Reasoning Using Slope Fields

What this deck covers

This deck focuses on Reasoning Using Slope Fields, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: Describe the slope field for $y' = x - y^2$.

Answer: Slopes decrease with increasing y. Quadratic term dominates behavior.

Flashcard 2: What characterizes a slope field for $y' = \text{constant}$?

Answer: Parallel line segments. Same slope at every point.

Flashcard 3: Find the slope at $(2, 1)$ for $y' = x - y$.

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

Flashcard 4: What feature in a slope field indicates a solution curve is constant?

Answer: Horizontal line segments. Zero slope means no change.

Flashcard 5: Determine the slope at $(3, 0)$ for $y' = x^2 - y^2$.

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

Flashcard 6: Describe slope field for $y' = x + y$ at the origin.

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

Flashcard 7: What is the slope at $(3, 4)$ for $y' = x + 2y$?

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

Flashcard 8: What kind of slope does $y' = 3$ imply in a slope field?

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

Flashcard 9: What slope does $y' = x^2$ produce at $(2, -1)$?

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

Flashcard 10: Find slope at $(1, 1)$ for $y' = y^2 - x^2$.

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

Flashcard 11: What does a slope field for $y' = x - y$ depict?

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

Flashcard 12: What is the slope at $(1, -2)$ for $y' = 2x + y$?

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

Flashcard 13: Identify slope at $(0, 2)$ for $y' = x^3 - y$.

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

Flashcard 14: Describe the slope field for $y' = -y$.

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

Flashcard 15: What differential equation characterizes a slope field with all slopes zero?

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

Flashcard 16: What does a vertical line segment in a slope field suggest?

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

Flashcard 17: How does a slope field for $y' = \frac{y}{x}$ look?

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

Flashcard 18: Describe the slope at $(0, 0)$ for $y' = x - y$.

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

Flashcard 19: How can you verify a solution curve in a slope field?

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

Flashcard 20: Find the slope at $(0, 0)$ for $y' = x^2 + y^2$.

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

Flashcard 21: What feature in a slope field indicates a solution curve is increasing?

Answer: Positive slopes. Positive slope means function increasing.

Flashcard 22: Identify the slope at $(1, 2)$ for $y' = 2x + 3y$.

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

Flashcard 23: State the purpose of a slope field.

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

Flashcard 24: What is the slope field for $y' = 2x$ at the origin?

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

Flashcard 25: Which line segment would represent $y' = 0$ in a slope field?

Answer: A horizontal line segment. Zero derivative means no change.

Flashcard 26: What is the slope at a point $(x, y)$ in a slope field defined by $y' = x + y$?

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

Flashcard 27: Identify the slope at $(2, 3)$ for $y' = 4x - y$.

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

Flashcard 28: What does a slope field represent?

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

Flashcard 29: Which differential equation produces a slope field with concentric circles?

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

Flashcard 30: Determine the slope at $(1, 1)$ for $y' = x^2 - xy + y^2$.

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