Sketching Slope Fields - AP Calculus AB
Card 1 of 30
What is the slope at $(1, 3)$ for $dy/dx = y - 2x$?
What is the slope at $(1, 3)$ for $dy/dx = y - 2x$?
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- Substitute: $3 - 2(1) = 1$.
- Substitute: $3 - 2(1) = 1$.
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Find the slope at $(1, 1)$ for $dy/dx = x^2 - 2y$.
Find the slope at $(1, 1)$ for $dy/dx = x^2 - 2y$.
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-1. Substitute: $1^2 - 2(1) = -1$.
-1. Substitute: $1^2 - 2(1) = -1$.
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What is a slope field?
What is a slope field?
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A graphical representation of a differential equation's solutions. Shows direction of solutions at each point.
A graphical representation of a differential equation's solutions. Shows direction of solutions at each point.
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What is the slope at $(1, 0)$ for $dy/dx = y^2 - x$?
What is the slope at $(1, 0)$ for $dy/dx = y^2 - x$?
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-1. Substitute: $0^2 - 1 = -1$.
-1. Substitute: $0^2 - 1 = -1$.
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What is the purpose of sketching a slope field?
What is the purpose of sketching a slope field?
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To visualize possible solutions of a differential equation. Helps understand solution behavior patterns.
To visualize possible solutions of a differential equation. Helps understand solution behavior patterns.
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What is the slope at $(2, 1)$ for $dy/dx = y^2 + x$?
What is the slope at $(2, 1)$ for $dy/dx = y^2 + x$?
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- Substitute: $1^2 + 2 = 3$.
- Substitute: $1^2 + 2 = 3$.
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Which mathematical tool helps visualize differential equation solutions?
Which mathematical tool helps visualize differential equation solutions?
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Slope fields. They show solution directions graphically.
Slope fields. They show solution directions graphically.
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Identify the slope at $(0, 1)$ for $dy/dx = x^2 - 2y$.
Identify the slope at $(0, 1)$ for $dy/dx = x^2 - 2y$.
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-2. Substitute: $0^2 - 2(1) = -2$.
-2. Substitute: $0^2 - 2(1) = -2$.
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What is the slope at $(3, 2)$ for $dy/dx = x - y^2$?
What is the slope at $(3, 2)$ for $dy/dx = x - y^2$?
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-1. Substitute: $3 - 2^2 = -1$.
-1. Substitute: $3 - 2^2 = -1$.
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Identify the slope at $(1, 0)$ for $dy/dx = 2x + y$.
Identify the slope at $(1, 0)$ for $dy/dx = 2x + y$.
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- Substitute: $2(1) + 0 = 2$.
- Substitute: $2(1) + 0 = 2$.
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What is the slope at $(2, 3)$ for $dy/dx = x - y$?
What is the slope at $(2, 3)$ for $dy/dx = x - y$?
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-1. Substitute: $2 - 3 = -1$.
-1. Substitute: $2 - 3 = -1$.
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Find the slope at $(0, 1)$ for $dy/dx = y - x^2$.
Find the slope at $(0, 1)$ for $dy/dx = y - x^2$.
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- Substitute: $1 - 0^2 = 1$.
- Substitute: $1 - 0^2 = 1$.
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What is the slope at $(0, 0)$ for $dy/dx = x + y^2$?
What is the slope at $(0, 0)$ for $dy/dx = x + y^2$?
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- Substitute: $0 + 0^2 = 0$.
- Substitute: $0 + 0^2 = 0$.
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Identify the slope at $(0, 3)$ for $dy/dx = x + y^2$.
Identify the slope at $(0, 3)$ for $dy/dx = x + y^2$.
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- Substitute: $0 + 3^2 = 9$.
- Substitute: $0 + 3^2 = 9$.
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What is the slope at $(3, 1)$ for $dy/dx = x^2 - y^2$?
What is the slope at $(3, 1)$ for $dy/dx = x^2 - y^2$?
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- Substitute: $3^2 - 1^2 = 8$.
- Substitute: $3^2 - 1^2 = 8$.
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What is the slope at $(0, 2)$ for $dy/dx = x^2 - y$?
What is the slope at $(0, 2)$ for $dy/dx = x^2 - y$?
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-2. Substitute: $0^2 - 2 = -2$.
-2. Substitute: $0^2 - 2 = -2$.
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What is the slope at $(0, -1)$ for $dy/dx = x^2 + y$?
What is the slope at $(0, -1)$ for $dy/dx = x^2 + y$?
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-1. Substitute: $0^2 + (-1) = -1$.
-1. Substitute: $0^2 + (-1) = -1$.
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Identify the slope of the line segment at (0, 0) if $dy/dx = x^2 - y^2$.
Identify the slope of the line segment at (0, 0) if $dy/dx = x^2 - y^2$.
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- Substitute: $0^2 - 0^2 = 0$.
- Substitute: $0^2 - 0^2 = 0$.
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What is the slope of the field at point $(2, 3)$ for $dy/dx = x - y$?
What is the slope of the field at point $(2, 3)$ for $dy/dx = x - y$?
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-1. Substitute: $2 - 3 = -1$.
-1. Substitute: $2 - 3 = -1$.
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What is the slope at $(3, 3)$ for $dy/dx = x - 2y$?
What is the slope at $(3, 3)$ for $dy/dx = x - 2y$?
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-3. Substitute: $3 - 2(3) = -3$.
-3. Substitute: $3 - 2(3) = -3$.
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What is the slope at $(0, 2)$ for $dy/dx = 2x - y$?
What is the slope at $(0, 2)$ for $dy/dx = 2x - y$?
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-2. Substitute: $2(0) - 2 = -2$.
-2. Substitute: $2(0) - 2 = -2$.
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Identify the slope at $(2, 1)$ for $dy/dx = x + y$.
Identify the slope at $(2, 1)$ for $dy/dx = x + y$.
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- Substitute: $2 + 1 = 3$.
- Substitute: $2 + 1 = 3$.
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What is the slope at $(1, 0)$ for the differential equation $dy/dx = xy$?
What is the slope at $(1, 0)$ for the differential equation $dy/dx = xy$?
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- Substitute: $1 \cdot 0 = 0$.
- Substitute: $1 \cdot 0 = 0$.
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Identify the slope of the line segment at (1, 2) if $dy/dx = x + y$.
Identify the slope of the line segment at (1, 2) if $dy/dx = x + y$.
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- Substitute: $1 + 2 = 3$.
- Substitute: $1 + 2 = 3$.
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What is the slope at $(0, 1)$ for $dy/dx = x^2 + y^2$?
What is the slope at $(0, 1)$ for $dy/dx = x^2 + y^2$?
Tap to reveal answer
- Substitute: $0^2 + 1^2 = 1$.
- Substitute: $0^2 + 1^2 = 1$.
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Find the slope at $(1, 1)$ for $dy/dx = x^2 - y^2$.
Find the slope at $(1, 1)$ for $dy/dx = x^2 - y^2$.
Tap to reveal answer
- Substitute: $1^2 - 1^2 = 0$.
- Substitute: $1^2 - 1^2 = 0$.
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What is the slope at $(1, 1)$ for $dy/dx = 2x - y$?
What is the slope at $(1, 1)$ for $dy/dx = 2x - y$?
Tap to reveal answer
- Substitute: $2(1) - 1 = 1$.
- Substitute: $2(1) - 1 = 1$.
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What is the slope at $(2, 2)$ for $dy/dx = x^2 + y$?
What is the slope at $(2, 2)$ for $dy/dx = x^2 + y$?
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- Substitute: $2^2 + 2 = 6$.
- Substitute: $2^2 + 2 = 6$.
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What is the slope at $(2, 0)$ for $dy/dx = x^2 - y$?
What is the slope at $(2, 0)$ for $dy/dx = x^2 - y$?
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- Substitute: $2^2 - 0 = 4$.
- Substitute: $2^2 - 0 = 4$.
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What does each segment in a slope field indicate?
What does each segment in a slope field indicate?
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The slope of the solution curve at that point. It's the tangent direction at that location.
The slope of the solution curve at that point. It's the tangent direction at that location.
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