Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

13. Intro to Differential Equations

Slope Fields

Calculus

13. Intro to Differential Equations

Slope Fields

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Multiple Choice

Shown above is a slope field for which of the following differential equations?

\frac{d y}{d x} = y - x

\frac{d y}{d x} = x + y

\frac{d y}{d x} = x y

\frac{d y}{d x} = x - y

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Verified step by step guidance

Step 1: Understand the problem. A slope field visually represents the behavior of a differential equation by showing the slopes of tangent lines at various points (x, y). The goal is to determine which differential equation corresponds to the given slope field.

Step 2: Analyze the given differential equations. Each equation represents a relationship between the derivative dy/dx and the variables x and y. For example, dy/dx = y - x means the slope at any point (x, y) is calculated by subtracting x from y.

Step 3: Compare the slope field to the behavior of each equation. For each equation, consider how the slope changes at specific points (e.g., when x = 0 or y = 0). For instance, if dy/dx = x - y, the slope at (0, 0) would be 0 - 0 = 0, and at (1, 0) it would be 1 - 0 = 1.

Step 4: Test the behavior of dy/dx = x * y. For this equation, the slope at any point (x, y) is the product of x and y. For example, at (1, 1), the slope would be 1 * 1 = 1, and at (0, 1), the slope would be 0 * 1 = 0. Compare these results to the slope field to see if they match.

Step 5: Eliminate incorrect options. If the slope field does not match the behavior of dy/dx = y - x, dy/dx = x + y, or dy/dx = x - y, then the correct equation is dy/dx = x * y. Confirm this by testing additional points and ensuring consistency with the slope field.