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Flip to reveal answersIn a slope field for dy/dx = f(x, y), how do you find the gradient of the segment at a point (x, y)?
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All 8 Flashcards — Slope fields
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Question
In a slope field for dy/dx = f(x, y), how do you find the gradient of the segment at a point (x, y)?
Answer
Substitute the point into f: the gradient there is f(x, y).
Question
What is an isocline?
Answer
The set of points f(x, y) = c where every segment has the same gradient c (all parallel).
Question
What does f(x, y) = 0 tell you about a slope field?
Answer
Those points have flat (horizontal) segments — solution curves have their maximum/minimum (turning) points there.
Question
How do you sketch a solution curve through a given point?
Answer
Start at the point and glide so the curve is always tangent to the nearby segments; it never crosses a segment.
Question
In a slope field, where is a solution curve increasing?
Answer
Wherever f(x, y) > 0 (the segments tilt upward); it decreases where f(x, y) < 0.
Question
For dy/dx = x + 0.5y, what is the gradient at (2, 4)?
Answer
f(2, 4) = 2 + 0.5(4) = 4 (a steep, rising segment).
Question
For the cooling model dy/dx = −0.2(y − 20), where are the segments flat?
Answer
On the line y = 20 (room temperature): set −0.2(y − 20) = 0 ⇒ y = 20, the equilibrium.
Question
Is a GDC allowed when working with slope fields in AI HL?
Answer
Yes — a GDC is allowed on every AI paper (P1, P2, P3); use it to evaluate f at points and solve f = c for isoclines.
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Slope fields (HL only)
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