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

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

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

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

Flashcard 3: 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 4: 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 5: What feature in a slope field indicates a solution curve is increasing?

Answer: Positive slopes. Positive slope means function increasing.

Flashcard 6: 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 7: 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 8: What differential equation characterizes a slope field with all slopes zero?

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

Flashcard 9: 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 10: Which line segment would represent $y' = 0$ in a slope field?

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

Flashcard 11: State the purpose of a slope field.

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

Flashcard 12: Identify slope at $(0, 2)$ for $y' = x^3 - y$.

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

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

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

Flashcard 14: 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 15: Find the slope at $(2, 1)$ for $y' = x - y$.

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

Flashcard 16: What is the characteristic of a slope field for $y' = y$?

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

Flashcard 17: Which differential equation produces a slope field with concentric circles?

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

Flashcard 18: What feature in a slope field indicates a solution curve is constant?

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

Flashcard 19: What feature in a slope field indicates a solution curve is increasing?

Answer: Positive slopes. Positive slope means function increasing.

Flashcard 20: What is the slope at $(1, 0)$ for $y' = x^2 + y^2$?

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

Flashcard 21: Identify the slope at $(2, 2)$ for $y' = 3x + y$.

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

Flashcard 22: 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 23: 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 24: What characterizes a slope field for $y' = \text{constant}$?

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

Flashcard 25: What does a vertical line segment in a slope field suggest?

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

Flashcard 26: Describe the slope at $(0, 0)$ for $y' = x - y$.

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

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

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

Flashcard 28: 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 29: What does a slope field represent?

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

Flashcard 30: 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 31: Find the slope at $(2, 1)$ for $y' = x - y$.

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

Flashcard 32: Identify the slope at $(2, 3)$ for $y' = 4x - y$.

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

Flashcard 33: 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 34: Identify the slope at $(2, 3)$ for $y' = 4x - y$.

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

Flashcard 35: 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 36: What characterizes a slope field for $y' = \text{constant}$?

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

Flashcard 37: Which line segment would represent $y' = 0$ in a slope field?

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

Flashcard 38: What does a vertical line segment in a slope field suggest?

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

Flashcard 39: What is the slope at $(1, 0)$ for $y' = x^2 + y^2$?

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

Flashcard 40: Describe the slope field for $y' = \frac{1}{x}$.

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

Flashcard 41: What slope does $y' = y$ produce at $(0, 1)$?

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

Flashcard 42: 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 43: Determine the slope at $(1, 1)$ for $y' = x^2 - xy + y^2$.

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

Flashcard 44: What differential equation characterizes a slope field with all slopes zero?

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

Flashcard 45: Describe slope field for $y' = x + y$ at the origin.

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

Flashcard 46: Describe the slope field for $y' = \frac{1}{x}$.

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

Flashcard 47: Identify the slope at $(0, 0)$ for $y' = y^3$.

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

Flashcard 48: Which differential equation produces a slope field with concentric circles?

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

Flashcard 49: What is the characteristic of a slope field for $y' = y$?

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

Flashcard 50: What is the slope at $(2, -1)$ for $y' = 2 - y$?

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

Flashcard 51: Describe the slope field for $y' = -y$.

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

Flashcard 52: 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 53: 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 54: What slope does $y' = y$ produce at $(0, 1)$?

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

Flashcard 55: 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 56: Describe the slope field for $y' = -y$.

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

Flashcard 57: 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 58: What feature in a slope field indicates a solution curve is constant?

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

Flashcard 59: 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 60: 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 61: Identify slope at $(0, 2)$ for $y' = x^3 - y$.

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

Flashcard 62: What slope does $y' = x^2$ produce at $(2, -1)$?

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

Flashcard 63: 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 64: 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 65: Describe the slope at $(0, 0)$ for $y' = x - y$.

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

Flashcard 66: What does a slope field represent?

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

Flashcard 67: Describe the slope field for $y' = x - y^2$.

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

Flashcard 68: What slope does $y' = x^2$ produce at $(2, -1)$?

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

Flashcard 69: What is the slope at $(2, -1)$ for $y' = 2 - y$?

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

Flashcard 70: 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 71: 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 72: 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 73: Find slope at $(3, 3)$ for $y' = x - 2y$.

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

Flashcard 74: Determine the slope at $(1, 1)$ for $y' = x^2 - xy + y^2$.

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

Flashcard 75: Describe the slope field for $y' = x - y^2$.

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

Flashcard 76: State the purpose of a slope field.

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