AP Calculus BC 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 BC.

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: Find the slope at point $(0, 0)$ for $y' = \frac{x}{y}$.

Answer: Slope is undefined. Division by zero at origin makes slope undefined.

Flashcard 2: What is the appearance of a slope field for $y' = x$?

Answer: Lines of increasing slope parallel to the $y$-axis. Slope depends only on $x$, creating vertical patterns.

Flashcard 3: What does a slope field show for a separable differential equation?

Answer: Slopes that can be separated into functions of $x$ and $y$. Variables separate into independent $f(x)$ and $g(y)$ functions.

Flashcard 4: What characterizes the slope field of $y' = -y$?

Answer: Slopes decrease as $y$ increases. Negative coefficient creates decreasing exponential behavior.

Flashcard 5: Find slope at $(0, 2)$ for $y' = e^x + y^2$.

Answer: Slope is $5$. At $(0,2)$: $y' = e^0 + 2^2 = 1 + 4 = 5$.

Flashcard 6: What does a consistent slope indicate in a slope field?

Answer: A linear solution. Uniform slope produces straight-line solutions.

Flashcard 7: Find the slope at $(0, -2)$ for $y' = x^2 - y$.

Answer: Slope is $2$. At $(0,-2)$: $y' = 0^2 - (-2) = 2$.

Flashcard 8: Predict the slope at $(0, 3)$ for $y' = \frac{x}{2}$.

Answer: Slope is $0$. At $(0,3)$: $y' = \frac{0}{2} = 0$.

Flashcard 9: Determine the slope at $(1, 1)$ for $y' = \frac{x}{y}$.

Answer: Slope is $1$. At $(1,1)$: $y' = \frac{1}{1} = 1$.

Flashcard 10: Find the slope at $(0, 0.5)$ for $y' = 2xy$.

Answer: Slope is $0$. At $(0,0.5)$: $y' = 2(0)(0.5) = 0$.

Flashcard 11: Identify slope at $(2, 1)$ for $y' = \frac{1}{x}$.

Answer: Slope is $\frac{1}{2}$. At $(2,1)$: $y' = \frac{1}{2}$.

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

Answer: Slope is $-3$. At $(3,3)$: $y' = 3 - 2(3) = -3$.

Flashcard 13: Find slope at $(2, 4)$ for $y' = 3x - y$.

Answer: Slope is $2$. At $(2,4)$: $y' = 3(2) - 4 = 2$.

Flashcard 14: What is the slope field for $y' = 1$?

Answer: All lines have slope $1$. Constant derivative creates uniform slope throughout field.

Flashcard 15: Identify the slope at point $(1, 1)$ for $y' = x + y$.

Answer: Slope is $2$. Substitute $(1,1)$: $y' = 1 + 1 = 2$.

Flashcard 16: What is the general solution form for $y' = kx$?

Answer: $y = \frac{k}{2}x^2 + C$. Integration of $kx$ with arbitrary constant $C$.

Flashcard 17: Identify slope at origin for $y' = x + y$.

Answer: Slope is $0$. At $(0,0)$: $y' = 0 + 0 = 0$.

Flashcard 18: What is the slope at $(2, 0)$ for $y' = y$?

Answer: Slope is $0$. When $y = 0$, derivative equals zero.

Flashcard 19: Which point has a slope of $0$ for $y' = x^2 - y^2$?

Answer: Points where $x^2 = y^2$. When $x^2 = y^2$, the derivative equals zero.

Flashcard 20: What is the appearance of a slope field for $y' = y$?

Answer: Exponential growth. Positive feedback creates exponentially increasing curves.

Flashcard 21: What is a solution curve in the context of slope fields?

Answer: Curve that follows the direction of the slopes. Path tangent to slope field segments at every point.

Flashcard 22: What is the slope at $(1, 1)$ for $y' = \frac{x^2}{y}$?

Answer: Slope is $1$. At $(1,1)$: $y' = \frac{1^2}{1} = 1$.

Flashcard 23: What does a slope field for $y' = \tan(x)$ look like?

Answer: Slopes oscillate between positive and negative. Tangent function creates periodic vertical asymptotes.

Flashcard 24: Identify the slope at $(2, 0)$ for $y' = 3y$.

Answer: Slope is $0$. When $y = 0$, derivative equals zero.

Flashcard 25: What is the slope at $(1, 2)$ for $y' = 2x + y$?

Answer: Slope is $4$. At $(1,2)$: $y' = 2(1) + 2 = 4$.

Flashcard 26: Determine slope at $(0, 0)$ for $y' = \frac{y}{x}$.

Answer: Undefined. Division by zero at origin makes slope undefined.

Flashcard 27: What does a slope field help visualize?

Answer: The behavior of differential equation solutions. Shows qualitative solution behavior without solving analytically.

Flashcard 28: Identify the slope at $(1, 2)$ for $y' = x + 2$.

Answer: Slope is $3$. At $(1,2)$: $y' = 1 + 2 = 3$.

Flashcard 29: What is the slope at $(0, -1)$ for $y' = x^3$?

Answer: Slope is $0$. At $(0,-1)$: $y' = 0^3 = 0$.

Flashcard 30: Identify the slope at $(0, 1)$ for $y' = \frac{1}{y}$.

Answer: Slope is $1$. At $(0,1)$: $y' = \frac{1}{1} = 1$.