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AP Calculus AB Flashcards: Sketching Slope Fields

Study Sketching Slope Fields in AP Calculus AB with focused flashcards that help you recognize the idea, recall the key rule, and apply it in practice-style prompts.

AP Calculus AB

Sketching Slope Fields

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QUESTION
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Find the slope at (0,1)(0, 1) for dy/dx=y−x2dy/dx = y - x^2.

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ANSWER
  1. Substitute: 1−02=11 - 0^2 = 1.

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What this deck covers

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

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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.

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Flashcard 1: Find the slope at (0,1)(0, 1) for dy/dx=y−x2dy/dx = y - x^2.

Answer:

  1. Substitute: 1−02=11 - 0^2 = 1.

Flashcard 2: Find the slope at (1,1)(1, 1) for dy/dx=x2−2ydy/dx = x^2 - 2y.

Answer: -1. Substitute: 12−2(1)=−11^2 - 2(1) = -1.

Flashcard 3: Identify the slope at (2,1)(2, 1) for dy/dx=x+ydy/dx = x + y.

Answer:

  1. Substitute: 2+1=32 + 1 = 3.

Flashcard 4: Find the slope at (1,1)(1, 1) for dy/dx=x2−2ydy/dx = x^2 - 2y.

Answer: -1. Substitute: 12−2(1)=−11^2 - 2(1) = -1.

Flashcard 5: What is the slope at (0,2)(0, 2) for dy/dx=2x−ydy/dx = 2x - y?

Answer: -2. Substitute: 2(0)−2=−22(0) - 2 = -2.

Flashcard 6: What is the slope at (0,2)(0, 2) for dy/dx=x2+2ydy/dx = x^2 + 2y?

Answer:

  1. Substitute: 02+2(2)=40^2 + 2(2) = 4.

Flashcard 7: What is the slope at (1,3)(1, 3) for dy/dx=y−2xdy/dx = y - 2x?

Answer:

  1. Substitute: 3−2(1)=13 - 2(1) = 1.

Flashcard 8: What is the slope at (3,2)(3, 2) for dy/dx=x−y2dy/dx = x - y^2?

Answer: -1. Substitute: 3−22=−13 - 2^2 = -1.

Flashcard 9: Identify the slope at (2,2)(2, 2) for dy/dx=x+ydy/dx = x + y.

Answer:

  1. Substitute: 2+2=42 + 2 = 4.

Flashcard 10: What is the slope at (1,3)(1, 3) for dy/dx=y−2xdy/dx = y - 2x?

Answer:

  1. Substitute: 3−2(1)=13 - 2(1) = 1.

Flashcard 11: Identify the slope of the line segment at (1, 2) if dy/dx=x+ydy/dx = x + y.

Answer:

  1. Substitute: 1+2=31 + 2 = 3.

Flashcard 12: Identify the slope at (1,0)(1, 0) for dy/dx=2x+ydy/dx = 2x + y.

Answer:

  1. Substitute: 2(1)+0=22(1) + 0 = 2.

Flashcard 13: What is the slope at (3,1)(3, 1) for dy/dx=x2−y2dy/dx = x^2 - y^2?

Answer:

  1. Substitute: 32−12=83^2 - 1^2 = 8.

Flashcard 14: Identify the slope of the line segment at (0, 0) if dy/dx=x2−y2dy/dx = x^2 - y^2.

Answer:

  1. Substitute: 02−02=00^2 - 0^2 = 0.

Flashcard 15: What is the slope at (0,1)(0, 1) for dy/dx=x2+y2dy/dx = x^2 + y^2?

Answer:

  1. Substitute: 02+12=10^2 + 1^2 = 1.

Flashcard 16: What is the slope of the field at point (2,3)(2, 3) for dy/dx=x−ydy/dx = x - y?

Answer: -1. Substitute: 2−3=−12 - 3 = -1.

Flashcard 17: Which mathematical tool helps visualize differential equation solutions?

Answer: Slope fields. They show solution directions graphically.

Flashcard 18: What is the slope at (3,2)(3, 2) for dy/dx=x−y2dy/dx = x - y^2?

Answer: -1. Substitute: 3−22=−13 - 2^2 = -1.

Flashcard 19: What is the slope at (1,0)(1, 0) for the differential equation dy/dx=xydy/dx = xy?

Answer:

  1. Substitute: 1⋅0=01 \cdot 0 = 0.

Flashcard 20: Identify the slope at (1,2)(1, 2) for dy/dx=y−xdy/dx = y - x.

Answer:

  1. Substitute: 2−1=12 - 1 = 1.

Flashcard 21: What is the slope at (0,2)(0, 2) for dy/dx=x2−ydy/dx = x^2 - y?

Answer: -2. Substitute: 02−2=−20^2 - 2 = -2.

Flashcard 22: Identify the slope at (1,2)(1, 2) for dy/dx=y−xdy/dx = y - x.

Answer:

  1. Substitute: 2−1=12 - 1 = 1.

Flashcard 23: Which mathematical tool helps visualize differential equation solutions?

Answer: Slope fields. They show solution directions graphically.

Flashcard 24: What is the slope at (0,−1)(0, -1) for dy/dx=x2+ydy/dx = x^2 + y?

Answer: -1. Substitute: 02+(−1)=−10^2 + (-1) = -1.

Flashcard 25: Identify the slope at (2,2)(2, 2) for dy/dx=x+ydy/dx = x + y.

Answer:

  1. Substitute: 2+2=42 + 2 = 4.

Flashcard 26: What is the slope of the field at point (2,3)(2, 3) for dy/dx=x−ydy/dx = x - y?

Answer: -1. Substitute: 2−3=−12 - 3 = -1.

Flashcard 27: What is the slope at (2,1)(2, 1) for dy/dx=y2+xdy/dx = y^2 + x?

Answer:

  1. Substitute: 12+2=31^2 + 2 = 3.

Flashcard 28: Identify the slope at (1,0)(1, 0) for dy/dx=2x+ydy/dx = 2x + y.

Answer:

  1. Substitute: 2(1)+0=22(1) + 0 = 2.

Flashcard 29: What is the slope at (3,1)(3, 1) for dy/dx=x2−y2dy/dx = x^2 - y^2?

Answer:

  1. Substitute: 32−12=83^2 - 1^2 = 8.

Flashcard 30: What does each segment in a slope field indicate?

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

Flashcard 31: What is the slope at (0,2)(0, 2) for dy/dx=x2−ydy/dx = x^2 - y?

Answer: -2. Substitute: 02−2=−20^2 - 2 = -2.

Flashcard 32: What is the slope at (0,−1)(0, -1) for dy/dx=x2+ydy/dx = x^2 + y?

Answer: -1. Substitute: 02+(−1)=−10^2 + (-1) = -1.

Flashcard 33: What is the slope at (1,1)(1, 1) for dy/dx=2x−ydy/dx = 2x - y?

Answer:

  1. Substitute: 2(1)−1=12(1) - 1 = 1.

Flashcard 34: What is the purpose of sketching a slope field?

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

Flashcard 35: What is the slope at (1,2)(1, 2) for dy/dx=y+x2dy/dx = y + x^2?

Answer:

  1. Substitute: 2+12=32 + 1^2 = 3.

Flashcard 36: What is the slope at (2,1)(2, 1) for dy/dx=y2+xdy/dx = y^2 + x?

Answer:

  1. Substitute: 12+2=31^2 + 2 = 3.

Flashcard 37: What is the slope at (2,3)(2, 3) for dy/dx=x−ydy/dx = x - y?

Answer: -1. Substitute: 2−3=−12 - 3 = -1.

Flashcard 38: What is the slope at (0,1)(0, 1) for dy/dx=x2+y2dy/dx = x^2 + y^2?

Answer:

  1. Substitute: 02+12=10^2 + 1^2 = 1.

Flashcard 39: What is a slope field?

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

Flashcard 40: What is the slope at (0,0)(0, 0) for dy/dx=x+y2dy/dx = x + y^2?

Answer: 00. Substitute: 0+02=00 + 0^2 = 0.

Flashcard 41: What is the slope at (1,2)(1, 2) for dy/dx=y+x2dy/dx = y + x^2?

Answer:

  1. Substitute: 2+12=32 + 1^2 = 3.

Flashcard 42: What is the slope at (1,0)(1, 0) for dy/dx=y2−xdy/dx = y^2 - x?

Answer: -1. Substitute: 02−1=−10^2 - 1 = -1.

Flashcard 43: What is the slope at (1,1)(1, 1) for dy/dx=2x−ydy/dx = 2x - y?

Answer:

  1. Substitute: 2(1)−1=12(1) - 1 = 1.

Flashcard 44: What is the purpose of sketching a slope field?

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

Flashcard 45: Identify the slope at (2,1)(2, 1) for dy/dx=x+ydy/dx = x + y.

Answer:

  1. Substitute: 2+1=32 + 1 = 3.

Flashcard 46: What is a slope field?

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

Flashcard 47: What is the slope at (2,2)(2, 2) for dy/dx=x2+ydy/dx = x^2 + y?

Answer:

  1. Substitute: 22+2=62^2 + 2 = 6.

Flashcard 48: What is the slope at (2,0)(2, 0) for dy/dx=x2−ydy/dx = x^2 - y?

Answer:

  1. Substitute: 22−0=42^2 - 0 = 4.

Flashcard 49: Find the slope at (1,1)(1, 1) for dy/dx=x2−y2dy/dx = x^2 - y^2.

Answer:

  1. Substitute: 12−12=01^2 - 1^2 = 0.

Flashcard 50: What does a slope field represent?

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

Flashcard 51: What is the slope at (3,3)(3, 3) for dy/dx=x−2ydy/dx = x - 2y?

Answer: -3. Substitute: 3−2(3)=−33 - 2(3) = -3.

Flashcard 52: Identify the slope at (0,3)(0, 3) for dy/dx=x+y2dy/dx = x + y^2.

Answer:

  1. Substitute: 0+32=90 + 3^2 = 9.

Flashcard 53: Identify the slope at (0,3)(0, 3) for dy/dx=x+y2dy/dx = x + y^2.

Answer:

  1. Substitute: 0+32=90 + 3^2 = 9.

Flashcard 54: Identify the slope at (0,1)(0, 1) for dy/dx=x2−2ydy/dx = x^2 - 2y.

Answer: -2. Substitute: 02−2(1)=−20^2 - 2(1) = -2.

Flashcard 55: What is the slope at (0,2)(0, 2) for dy/dx=x2+2ydy/dx = x^2 + 2y?

Answer:

  1. Substitute: 02+2(2)=40^2 + 2(2) = 4.

Flashcard 56: Identify the slope at (0,1)(0, 1) for dy/dx=x2−2ydy/dx = x^2 - 2y.

Answer: -2. Substitute: 02−2(1)=−20^2 - 2(1) = -2.

Flashcard 57: What is the slope at (3,3)(3, 3) for dy/dx=x−2ydy/dx = x - 2y?

Answer: -3. Substitute: 3−2(3)=−33 - 2(3) = -3.

Flashcard 58: Find the slope at (1,1)(1, 1) for dy/dx=x2−y2dy/dx = x^2 - y^2.

Answer:

  1. Substitute: 12−12=01^2 - 1^2 = 0.

Flashcard 59: Identify the slope of the line segment at (0, 0) if dy/dx=x2−y2dy/dx = x^2 - y^2.

Answer:

  1. Substitute: 02−02=00^2 - 0^2 = 0.

Flashcard 60: What is the slope at (1,0)(1, 0) for the differential equation dy/dx=xydy/dx = xy?

Answer:

  1. Substitute: 1⋅0=01 \cdot 0 = 0.

Flashcard 61: Identify the slope of the line segment at (1, 2) if dy/dx=x+ydy/dx = x + y.

Answer:

  1. Substitute: 1+2=31 + 2 = 3.

Flashcard 62: What is the slope at (1,0)(1, 0) for dy/dx=y2−xdy/dx = y^2 - x?

Answer: -1. Substitute: 02−1=−10^2 - 1 = -1.

Flashcard 63: What is the slope at (2,2)(2, 2) for dy/dx=x2+ydy/dx = x^2 + y?

Answer:

  1. Substitute: 22+2=62^2 + 2 = 6.

Flashcard 64: What is the slope at (0,2)(0, 2) for dy/dx=2x−ydy/dx = 2x - y?

Answer: -2. Substitute: 2(0)−2=−22(0) - 2 = -2.

Flashcard 65: What is the slope at (2,0)(2, 0) for dy/dx=x2−ydy/dx = x^2 - y?

Answer:

  1. Substitute: 22−0=42^2 - 0 = 4.

Flashcard 66: What is the slope at (2,3)(2, 3) for dy/dx=x−ydy/dx = x - y?

Answer: -1. Substitute: 2−3=−12 - 3 = -1.

Flashcard 67: Find the slope at (0,1)(0, 1) for dy/dx=y−x2dy/dx = y - x^2.

Answer:

  1. Substitute: 1−02=11 - 0^2 = 1.

Flashcard 68: What is the slope at (0,0)(0, 0) for dy/dx=x+y2dy/dx = x + y^2?

Answer:

  1. Substitute: 0+02=00 + 0^2 = 0.

Flashcard 69: What does each segment in a slope field indicate?

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