7.4 practice for Calculus AB, Cheat Sheet of Mathematics

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Name: Ahmed Almenhali and Khaled Alhossani Date: 4/2/

📈 Solution Synthesis: Slope Fields

Part 1: Sketching Particular Solutions ✏

A slope field represents the general solution to a differential equation. Given an initial condition, you can trace the path of the specific solution curve through the field.

1. On the slope field shown to the left, sketch the graph of a particular solution that contains the point (0, 1). Label this point as Point A. The curve through (0,1) follows the slope field, curving downward to the left and rising rapidly to the right. 2. Sketch the graph of a second particular solution that contains the point (-1, - 1). Label this point as Point B.

Part 2: Analyzing Field Properties 📈

Consider the differential equation dy/dx = x + y and its associated slope field. Answer the following questions based on the properties of this equation.

3. Describe all points (x, y) in the coordinate plane for which the slope is exactly zero. The slope is 0 at all points where x+y=0. This happens along the line y=-x. At every point on this line, the slope field shows horizontal line segments, meaning the slope is exactly 0.For example, points like (1,-1), (2,-2), and (-1,1) all have a slope of 0. 4. When are the slopes in the field strictly positive? The slopes are positive at all points where x+y>0. This occurs in the region above the line y=-x. In this area, the slope field shows line segments that tilt upward from left to right, indicating positive slopes. For example, at points like (0,1), (2,1), and (-1,2), the slopes are positive. On the graph, the line segments at these points tilt upward from left to right. The curve through (-1, - 1) follows the slope field, decreasing to the right and increasing to the left.

Part 3: Multiple Choice Matching 📈

5. The slope field for a certain differential equation shows horizontal tangent lines along the y-axis (where x = 0). Which of the following could be a solution to the differential equation with the initial condition y(0) = 2? A) y = x³ + 2 B) y = 2eˣ C) y = 2x + 2 D) y = ln(x + 2) 6. Shown is a slope field for the differential equation dy/dx = y(3 - y). If y = f(x) is the solution to the differential equation with initial condition f(0) = 1, then what is the limit of f(x) as x approaches infinity? A) 0 B) 1 C) 3 D) ∞

Part 4: Critical Reasoning 📈

A student is given the differential equation dy/dx = x² / y. They are shown a slope field that has horizontal segments along the x-axis.

7. Explain why this slope field CANNOT represent the given differential equation. The slope field cannot match

=

𝑥^2

because along the x-axis, y=0, and that would make the equation undefined since you cannot divide by zero. A slope field for this differential equation should not have valid horizontal segments on the x-axis. Also, slopes are zero only when x 2 =0, which happens when x=0, which is on the y-axis, not the x-axis.