Practice Slope Fields in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

A graphical representation of a first-order DE dydx=f(x,y)\frac{dy}{dx} = f(x, y). At each point (x,y)(x, y) in the plane, draw a short line segment with slope f(x,y)f(x, y). The resulting pattern of segments shows the direction solutions must follow.

Imagine a field with tiny arrows showing which way a river flows at each point. A slope field is the same idea: the DE tells you the slope (direction) at every point, and solution curves are paths that follow these directions everywhere. Drop a 'particle' anywhere and follow the arrowsβ€”that's a solution.

Showing a random 20 of 50 problems.

Example 1

medium
For dydx=x2βˆ’y\frac{dy}{dx}=x^2-y, find the slope at (2,1)(2,1).

Example 2

hard
Compare the slope fields of dydx=y\frac{dy}{dx} = y and dydx=βˆ’y\frac{dy}{dx} = -y. Describe how their solutions differ.

Example 3

easy
For dydx=x+y\frac{dy}{dx}=x+y, what slope is drawn at the origin?

Example 4

easy
For dydx=y2\frac{dy}{dx} = y^2, find the slope at (0,βˆ’2)(0, -2).

Example 5

challenge
For dydx=βˆ’xy\frac{dy}{dx}=-\frac{x}{y}, show the slope field is consistent with circular solutions.

Example 6

hard
For dydx=(yβˆ’2)(y+1)\frac{dy}{dx} = (y - 2)(y + 1), identify the equilibria and classify their stability.

Example 7

medium
For dydx=yβˆ’x\frac{dy}{dx} = y - x, what slope is drawn at (4,2)(4, 2)?

Example 8

challenge
For dydx=x+y\frac{dy}{dx}=x+y, sketch how the slope of solutions changes moving up a vertical line x=1x=1.

Example 9

medium
For dydx=x+y\frac{dy}{dx}=x+y, compare slopes at (0,1)(0,1) and (1,0)(1,0).

Example 10

medium
For dydx=βˆ’xy\frac{dy}{dx} = -\frac{x}{y} (yβ‰ 0y \neq 0), what family of curves are the solutions?

Example 11

easy
For dydx=2\frac{dy}{dx}=2, describe the slope field.

Example 12

easy
True or false: solution curves of dydx=f(x,y)\frac{dy}{dx} = f(x, y) must be tangent to the slope field segments.

Example 13

medium
For dydx=xβˆ’y\frac{dy}{dx} = x - y, find the isocline of slope 11.

Example 14

medium
The DE dydx=y(1βˆ’y)\frac{dy}{dx} = y(1 - y) models logistic growth. Find all equilibrium lines and classify each as stable or unstable.

Example 15

easy
For dydx=x+y\frac{dy}{dx}=x+y, what slope is drawn at (1,βˆ’1)(1,-1)?

Example 16

medium
For dy/dx=xβˆ’ydy/dx = x-y, find the zero-slope isocline and describe long-term behavior of solutions.

Example 17

challenge
Sketch-style: for dydx=sin⁑(x+y)\frac{dy}{dx} = \sin(x + y), find every isocline of slope 00.

Example 18

easy
For dy/dx=x2dy/dx = x^2, compute slopes at (0,0)(0,0), (1,0)(1,0), (βˆ’1,3)(-1,3), (2,5)(2,5).

Example 19

easy
For dydx=x\frac{dy}{dx}=x, what is the slope along the yy-axis (x=0x=0)?

Example 20

medium
For dydx=1+y2\frac{dy}{dx} = 1 + y^2, what is the smallest slope drawn anywhere in the field?
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