Sketching Slope Fields - AP Calculus BC
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How do you sketch a slope field for $dy/dx = y$?
How do you sketch a slope field for $dy/dx = y$?
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Compute $m = y$ at grid points. Evaluate $y$ at each grid point for slope.
Compute $m = y$ at grid points. Evaluate $y$ at each grid point for slope.
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What information can you obtain from a slope field?
What information can you obtain from a slope field?
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The general shape and behavior of solution curves. Reveals overall patterns and equilibrium points.
The general shape and behavior of solution curves. Reveals overall patterns and equilibrium points.
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What is a key feature of slope fields for $dy/dx = y/x$?
What is a key feature of slope fields for $dy/dx = y/x$?
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Lines are radial from the origin. Lines point away from origin along rays.
Lines are radial from the origin. Lines point away from origin along rays.
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How is a slope field constructed for $dy/dx = x + y$?
How is a slope field constructed for $dy/dx = x + y$?
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Compute slopes at grid points using $m = x + y$. Substitute coordinates into the differential equation.
Compute slopes at grid points using $m = x + y$. Substitute coordinates into the differential equation.
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What does it mean for a slope field to have symmetry?
What does it mean for a slope field to have symmetry?
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The slope pattern repeats across an axis. Pattern mirrors across coordinate axes.
The slope pattern repeats across an axis. Pattern mirrors across coordinate axes.
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What effect does a positive slope have on a slope field?
What effect does a positive slope have on a slope field?
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Lines tilt upward from left to right. Positive slope creates upward-slanting segments.
Lines tilt upward from left to right. Positive slope creates upward-slanting segments.
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What does each line segment in a slope field represent?
What does each line segment in a slope field represent?
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The slope of the solution curve at a given point. Each segment shows the tangent direction at that point.
The slope of the solution curve at a given point. Each segment shows the tangent direction at that point.
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How is $dy/dx = 0$ represented in a slope field?
How is $dy/dx = 0$ represented in a slope field?
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Horizontal line segments. Zero slope creates flat line segments.
Horizontal line segments. Zero slope creates flat line segments.
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How do you determine equilibrium solutions from a slope field?
How do you determine equilibrium solutions from a slope field?
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Look for horizontal line segments. Zero slope indicates equilibrium solutions.
Look for horizontal line segments. Zero slope indicates equilibrium solutions.
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What is the slope at $(2, 0)$ for $dy/dx = \frac{1}{y}$?
What is the slope at $(2, 0)$ for $dy/dx = \frac{1}{y}$?
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Slope is undefined. Division by zero creates undefined slope.
Slope is undefined. Division by zero creates undefined slope.
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How do you find the slope at $(x, y)$ for $dy/dx = x^2 + y^2$?
How do you find the slope at $(x, y)$ for $dy/dx = x^2 + y^2$?
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Compute $x^2 + y^2$ at $(x, y)$. Evaluate the expression at each grid point.
Compute $x^2 + y^2$ at $(x, y)$. Evaluate the expression at each grid point.
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What is the slope at $(0, 1)$ for $dy/dx = \frac{1}{x}$?
What is the slope at $(0, 1)$ for $dy/dx = \frac{1}{x}$?
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Slope is undefined. Division by zero creates undefined slope.
Slope is undefined. Division by zero creates undefined slope.
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What does a vertical line segment in a slope field indicate?
What does a vertical line segment in a slope field indicate?
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The slope approaches infinity. Infinite slope means vertical tangent line.
The slope approaches infinity. Infinite slope means vertical tangent line.
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What effect does a negative slope have on a slope field?
What effect does a negative slope have on a slope field?
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Lines tilt downward from left to right. Negative slope creates downward-slanting segments.
Lines tilt downward from left to right. Negative slope creates downward-slanting segments.
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Which direction do lines in a slope field point for $dy/dx = -1$?
Which direction do lines in a slope field point for $dy/dx = -1$?
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Lines point downward at 45 degrees. Constant negative slope creates uniform downward tilt.
Lines point downward at 45 degrees. Constant negative slope creates uniform downward tilt.
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What is the slope at $(2, 0)$ for $dy/dx = \frac{1}{y}$?
What is the slope at $(2, 0)$ for $dy/dx = \frac{1}{y}$?
Tap to reveal answer
Slope is undefined. Division by zero creates undefined slope.
Slope is undefined. Division by zero creates undefined slope.
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What information can you obtain from a slope field?
What information can you obtain from a slope field?
Tap to reveal answer
The general shape and behavior of solution curves. Reveals overall patterns and equilibrium points.
The general shape and behavior of solution curves. Reveals overall patterns and equilibrium points.
← Didn't Know|Knew It →
What does each line segment in a slope field represent?
What does each line segment in a slope field represent?
Tap to reveal answer
The slope of the solution curve at a given point. Each segment shows the tangent direction at that point.
The slope of the solution curve at a given point. Each segment shows the tangent direction at that point.
← Didn't Know|Knew It →
How do you sketch a slope field for $dy/dx = y$?
How do you sketch a slope field for $dy/dx = y$?
Tap to reveal answer
Compute $m = y$ at grid points. Evaluate $y$ at each grid point for slope.
Compute $m = y$ at grid points. Evaluate $y$ at each grid point for slope.
← Didn't Know|Knew It →
How do you find the slope at $ (x, y) $ for $ dy/dx = x^2 + y^2 $?
How do you find the slope at $ (x, y) $ for $ dy/dx = x^2 + y^2 $?
Tap to reveal answer
Compute $x^2 + y^2$ at $ (x, y) $. Evaluate the expression at each grid point.
Compute $x^2 + y^2$ at $ (x, y) $. Evaluate the expression at each grid point.
← Didn't Know|Knew It →
How is a slope field constructed for $dy/dx = x + y$?
How is a slope field constructed for $dy/dx = x + y$?
Tap to reveal answer
Compute slopes at grid points using $m = x + y$. Substitute coordinates into the differential equation.
Compute slopes at grid points using $m = x + y$. Substitute coordinates into the differential equation.
← Didn't Know|Knew It →
Which direction do lines in a slope field point for $dy/dx = -1$?
Which direction do lines in a slope field point for $dy/dx = -1$?
Tap to reveal answer
Lines point downward at 45 degrees. Constant negative slope creates uniform downward tilt.
Lines point downward at 45 degrees. Constant negative slope creates uniform downward tilt.
← Didn't Know|Knew It →
What effect does a negative slope have on a slope field?
What effect does a negative slope have on a slope field?
Tap to reveal answer
Lines tilt downward from left to right. Negative slope creates downward-slanting segments.
Lines tilt downward from left to right. Negative slope creates downward-slanting segments.
← Didn't Know|Knew It →
What does a vertical line segment in a slope field indicate?
What does a vertical line segment in a slope field indicate?
Tap to reveal answer
The slope approaches infinity. Infinite slope means vertical tangent line.
The slope approaches infinity. Infinite slope means vertical tangent line.
← Didn't Know|Knew It →
What is a key feature of slope fields for $dy/dx = y/x$?
What is a key feature of slope fields for $dy/dx = y/x$?
Tap to reveal answer
Lines are radial from the origin. Lines point away from origin along rays.
Lines are radial from the origin. Lines point away from origin along rays.
← Didn't Know|Knew It →
What effect does a positive slope have on a slope field?
What effect does a positive slope have on a slope field?
Tap to reveal answer
Lines tilt upward from left to right. Positive slope creates upward-slanting segments.
Lines tilt upward from left to right. Positive slope creates upward-slanting segments.
← Didn't Know|Knew It →
What does it mean for a slope field to have symmetry?
What does it mean for a slope field to have symmetry?
Tap to reveal answer
The slope pattern repeats across an axis. Pattern mirrors across coordinate axes.
The slope pattern repeats across an axis. Pattern mirrors across coordinate axes.
← Didn't Know|Knew It →
How is $dy/dx = 0$ represented in a slope field?
How is $dy/dx = 0$ represented in a slope field?
Tap to reveal answer
Horizontal line segments. Zero slope creates flat line segments.
Horizontal line segments. Zero slope creates flat line segments.
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What does a horizontal line segment in a slope field indicate?
What does a horizontal line segment in a slope field indicate?
Tap to reveal answer
The slope is zero at that point. Zero slope means no vertical change.
The slope is zero at that point. Zero slope means no vertical change.
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What is the slope at $(0, 1)$ for $dy/dx = \frac{1}{x}$?
What is the slope at $(0, 1)$ for $dy/dx = \frac{1}{x}$?
Tap to reveal answer
Slope is undefined. Division by zero creates undefined slope.
Slope is undefined. Division by zero creates undefined slope.
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