Slope fields Knowledge-Based Practice

Questions

1. 1.

   In a slope field, the slopes of the tangent segments depend only on the $y$-coordinate of each point and are equal to zero if and only if $y = -1$ or $y = 4$. Which of the following differential equations could represent this slope field?

   • A) $\frac{dy}{dx} = (x + 1)(x - 4)$
   • B) $\frac{dy}{dx} = (y - 1)(y + 4)$
   • C) $\frac{dy}{dx} = x(y + 1)(y - 4)$
   • D) $\frac{dy}{dx} = (y + 1)(y - 4)$
2. 2.

   Consider the slope field for the differential equation $\frac{dy}{dx} = (y - 2)(5 - y)$. If $y = f(x)$ is the particular solution with the initial condition $f(0) = 3$, what is the value of $\lim_{x \to \infty} f(x)$?

   • A) 2
   • B) 5
   • C) 3
   • D) $\infty$
3. 3.

   Consider the slope field for the differential equation $\frac{dy}{dx} = x^2 - 2y$. At which of the following points does a solution curve have a horizontal tangent segment and is concave upward?

   • A) $(-2, 2)$
   • B) $(-1, \frac{1}{2})$
   • C) $(1, 1)$
   • D) $(2, 2)$
4. 4.

   In the slope field for the differential equation $\frac{dy}{dx} = 3x - y$, along which line are all tangent segments parallel to the line itself?

   • A) $y = 3x - 3$
   • B) $y = 3x + 3$
   • C) $y = 3x$
   • D) $y = -3x + 3$
5. 5.

   Which of the following differential equations produces a slope field in which all line segments along the line $y = 2x$ are horizontal, and all line segments in the region $y > 2x$ have negative slope?

   • A) $\frac{dy}{dx} = y - 2x$
   • B) $\frac{dy}{dx} = 2x - y$
   • C) $\frac{dy}{dx} = 2x + y$
   • D) $\frac{dy}{dx} = x - 2y$

Answers and step-by-step solutions appear after you finish the test.