AP Calculus AB: Slope Fields — Drill 1

About This Drill

AP Calculus AB: Slope Fields — Drill 1 is a Math practice drill covering Slope Fields. It contains 5 original questions created by Brian Stewart, a Barron's test prep author with over 20 years of tutoring experience.

Practice interpreting slope fields, matching differential equations to their slope fields, sketching solution curves, and analyzing equilibrium and long-run behavior. These AP Calculus AB differential equations skills appear on the multiple-choice and free-response sections of the AP exam.

Questions in This Drill

1. The slope field for a differential equation is described as follows: at every point \( (x, y) \), the slope equals \( x - y \). What is the slope of the tangent line to a solution curve passing through the point \( (3, 1) \)?
2. A slope field has the following properties: along the line \( y = x \), all slopes equal \( 0 \); for points where \( y > x \), slopes are negative; for points where \( y < x \), slopes are positive. Which differential equation matches this slope field?
3. The slope field for \( \dfrac{dy}{dx} = y \) is given. A student sketches the solution curve through the point \( (0, 2) \). Which of the following best describes the long-run behavior of this curve as \( x \to \infty \)?
4. Consider the slope field for \( \dfrac{dy}{dx} = y(y - 3) \). Which of the following correctly identifies all equilibrium solutions and describes the behavior of a solution curve through \( (0, 1) \)?
5. A slope field has all of the following properties: slopes are zero along the y-axis (\( x = 0 \)); slopes are positive for \( x > 0 \) and negative for \( x < 0 \); the slope at \( (2, 5) \) equals the slope at \( (2, -3) \); for a fixed \( x \), slope does not change as \( y \) varies. Which differential equation is consistent with all four properties?