Let's solve each part step by step. --- ### **(a) Calculate the slopes at each of the nine points** The differential equation is: \[ \frac{dy}{dx} = -2x + y \] The nine points on the grid are \((-1,2)\), \((0,2)\), \((1,2)\), \((-1,1)\), \((0,1)\), \((1,1)\), \((-1,0)\), \((0,0)\), \((1,0)\). Calculate the slope at each point: 1. **At (-1, 2):** \[ \frac{dy}{dx} = -2(-1) + 2 = 2 + 2 = \boxed{4} \] 2. **At (0, 2):** \[ \frac{dy}{dx} = -2(0) + 2 = 0 + 2 = \boxed{2} \] 3. **At (1, 2):** \[ \frac{dy}{dx} = -2(1) + 2 = -2 + 2 = \boxed{0} \] 4. **At (-1, 1):** \[ \frac{dy}{dx} = -2(-1) + 1 = 2 + 1 = \boxed{3} \] 5. **At (0, 1):** \[ \frac{dy}{dx} = -2(0) + 1 = 0 + 1 = \boxed{1} \] 6. **At (1, 1):** \[ \frac{dy}{dx} = -2(1) + 1 = -2 + 1 = \boxed{-1} \] 7. **At (-1, 0):** \[ \frac{dy}{dx} = -2(-1) + 0 = 2 + 0 = \boxed{2} \] 8. **At (0, 0):** \[ \frac{dy}{dx} = -2(0) + 0 = 0 + 0 = \boxed{0} \] 9. **At (1, 0):** \[ \frac{dy}{dx} = -2(1) + 0 = -2 + 0 = \boxed{-2} \] --- ### **(b) Draw a slope field** Each point will have a small line segment with the calculated slope: | Point | Slope | |------------|-------| | (-1, 2) | 4 | | (0, 2) | 2 | | (1, 2) | 0 | | (-1, 1) | 3 | | (0, 1) | 1 | | (1, 1) | -1 | | (-1, 0) | 2 | | (0, 0) | 0 | | (1, 0) | -2 | To draw: At each grid point, draw a small line segment with the given slope number (e.g., for slope 0, draw horizontally; for large positive slopes, draw steeply up; for negative slopes, draw downward). --- ### **(c) Concavity at (-1, 1)** - **Find \( \frac{d^2y}{dx^2} \):** Start with \[ \frac{dy}{dx} = -2x + y \] Differentiate both sides with respect to \(x\): \[ \frac{d^2y}{dx^2} = \frac{d}{dx}(-2x + y) = -2 + \frac{dy}{dx} \] At \((-1, 1)\): - From above, \(\frac{dy}{dx} = 3\) at \((-1, 1)\). So, \[ \frac{d^2y}{dx^2} = -2 + 3 = 1 \] - Since \( \frac{d^2y}{dx^2} > 0 \), the function is **concave up** at \((-1, 1)\). --- ## **Summary of answers** ### (a) Slopes at each point: \[ \begin{matrix} 4 & 2 & 0 \\ 3 & 1 & -1 \\ 2 & 0 & -2 \\ \end{matrix} \] ### (b) Slope field: Draw short segments at each point with slopes as above. ### (c) Concavity at \((-1, 1)\): \[ \frac{d^2y}{dx^2} = 1 > 0 \implies \text{Concave up at } (-1, 1) \] **Work:** \[ \frac{d^2y}{dx^2} = -2 + \frac{dy}{dx} = -2 + 3 = 1 > 0 \] Let me know if you need the slope field drawn out!

Certainly! Here are three lines focusing purely on theory related to the differential equation and its properties: 1. **The slope field visually represents the direction of the solution curves of the differential equation at each point in the plane, with small line segments indicating the slope \(\frac{dy}{dx}\).** 2. **The second derivative \(\frac{d^2 y}{dx^2}\) determines the concavity of the solution \(y = f(x)\); a positive second derivative indicates the graph is concave up, while a negative second derivative indicates concave down.** 3. **For a linear differential equation like \(\frac{dy}{dx} = -2x + y\), the second derivative can be found by differentiating the first derivative with respect to \(x\), leading to insights about the concavity and the behavior of solution curves without solving the equation explicitly.**