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

Which of the following differential equations could produce a slope field where the slope at each point $(x, y)$ is given by $y^{3} - x$ ?

\frac{d y}{d x} = y - x^{3}

\frac{d y}{d x} = y^{3} - x

\frac{d y}{d x} = x^{3} - y

\frac{d y}{d x} = x^{3} + y

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Verified step by step guidance

Step 1: Understand the problem. The question asks which differential equation corresponds to a slope field where the slope at each point (x, y) is given by y³ - x. This means the slope at any point (x, y) is determined by substituting x and y into the expression y³ - x.

Step 2: Recall that a slope field is a graphical representation of a differential equation. Each small line segment in the field represents the slope of the solution curve at a given point (x, y). The slope is determined by the derivative dy/dx.

Step 3: Compare the given differential equations to the slope expression y³ - x. The correct differential equation must have dy/dx equal to y³ - x, as this matches the slope at each point (x, y).

Step 4: Analyze each option: (a) dy/dx = y - x³ does not match y³ - x because the terms are arranged differently and involve x³ instead of x. (b) dy/dx = y³ - x matches the slope expression exactly. (c) dy/dx = x³ - y and (d) dy/dx = x³ + y do not match y³ - x either.

Step 5: Conclude that the correct differential equation is dy/dx = y³ - x, as it directly corresponds to the given slope expression.