This calculus worksheet provides practice for sketching slope fields for four different differential equations, complete with handwritten work and graphs.

Let’s solve each problem step by step. We’re drawing slope fields — that means at each point (x, y) on the grid, we draw a tiny line segment with slope equal to dy/dx given by the equation.

We’ll go one problem at a time.

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Problem 1: dy/dx = x + y

We need to compute the slope at each integer coordinate from x = -2 to 2 and y = -2 to 2.

Let’s make a table:

At (-2, -2): slope = -2 + (-2) = -4 → very steep down
At (-2, -1): slope = -2 + (-1) = -3 → steep down
At (-2, 0): slope = -2 + 0 = -2 → down
At (-2, 1): slope = -2 + 1 = -1 → slight down
At (-2, 2): slope = -2 + 2 = 0 → flat

At (-1, -2): slope = -1 + (-2) = -3 → steep down
At (-1, -1): slope = -1 + (-1) = -2 → down
At (-1, 0): slope = -1 + 0 = -1 → slight down
At (-1, 1): slope = -1 + 1 = 0 → flat
At (-1, 2): slope = -1 + 2 = 1 → slight up

At (0, -2): slope = 0 + (-2) = -2 → down
At (0, -1): slope = 0 + (-1) = -1 → slight down
At (0, 0): slope = 0 + 0 = 0 → flat
At (0, 1): slope = 0 + 1 = 1 → slight up
At (0, 2): slope = 0 + 2 = 2 → up

At (1, -2): slope = 1 + (-2) = -1 → slight down
At (1, -1): slope = 1 + (-1) = 0 → flat
At (1, 0): slope = 1 + 0 = 1 → slight up
At (1, 1): slope = 1 + 1 = 2 → up
At (1, 2): slope = 1 + 2 = 3 → steep up

At (2, -2): slope = 2 + (-2) = 0 → flat
At (2, -1): slope = 2 + (-1) = 1 → slight up
At (2, 0): slope = 2 + 0 = 2 → up
At (2, 1): slope = 2 + 1 = 3 → steep up
At (2, 2): slope = 2 + 2 = 4 → very steep up

Now, looking at the hand-drawn graph for #1 — it matches this pattern! For example:
- At (0,0), slope is 0 → horizontal line ✔️
- At (1,1), slope is 2 → steep upward ✔️
- At (-2,-2), slope is -4 → steep downward ✔️

So the sketch is correct.

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Problem 2: dy/dx = 2y

This one only depends on y, not x. So all points with same y have same slope.

Let’s compute:

If y = -2 → slope = 2*(-2) = -4 → steep down
If y = -1 → slope = 2*(-1) = -2 → down
If y = 0 → slope = 0 → flat
If y = 1 → slope = 2*1 = 2 → up
If y = 2 → slope = 4 → steep up

And since it doesn’t depend on x, every row has identical slopes across.

Looking at the graph for #2:
- All points at y=0 are flat ✔️
- All points at y=1 have same upward slope ✔️
- All points at y=-1 have same downward slope ✔️

Perfect match.

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Problem 3: dy/dx = x/y

Important: undefined when y=0 (division by zero). So no slope segments on the x-axis.

Compute for other points:

At (-2, -2): slope = (-2)/(-2) = 1 → up
At (-2, -1): slope = (-2)/(-1) = 2 → steep up
At (-2, 1): slope = (-2)/(1) = -2 → steep down
At (-2, 2): slope = (-2)/(2) = -1 → down

At (-1, -2): slope = (-1)/(-2) = 0.5 → slight up
At (-1, -1): slope = (-1)/(-1) = 1 → up
At (-1, 1): slope = (-1)/(1) = -1 → down
At (-1, 2): slope = (-1)/(2) = -0.5 → slight down

At (0, -2): slope = 0/(-2) = 0 → flat
At (0, -1): slope = 0/(-1) = 0 → flat
At (0, 1): slope = 0/1 = 0 → flat
At (0, 2): slope = 0/2 = 0 → flat

At (1, -2): slope = 1/(-2) = -0.5 → slight down
At (1, -1): slope = 1/(-1) = -1 → down
At (1, 1): slope = 1/1 = 1 → up
At (1, 2): slope = 1/2 = 0.5 → slight up

At (2, -2): slope = 2/(-2) = -1 → down
At (2, -1): slope = 2/(-1) = -2 → steep down
At (2, 1): slope = 2/1 = 2 → steep up
At (2, 2): slope = 2/2 = 1 → up

Check the graph for #3:
- On x-axis (y=0), nothing drawn → good, undefined ✔️
- At (0,1), (0,2), etc., flat lines → because x=0 → slope=0 ✔️
- At (1,1), slope=1 → diagonal up ✔️
- At (-2,1), slope=-2 → steep down ✔️

Looks correct.

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Problem 4: dy/dx = (x+1)/(y-2)

Undefined when y=2 (denominator zero). So no segments on the line y=2.

Also, note: if numerator is zero (x = -1), then slope = 0 (as long as denominator ≠ 0).

Let’s check the tables provided in the image — they seem to be computing values correctly.

For example:

At x=-3, y=3: slope = (-3+1)/(3-2) = (-2)/1 = -2 ✔️
At x=-3, y=2: undefined ✔️
At x=-3, y=1: (-3+1)/(1-2) = (-2)/(-1) = 2 ✔️
At x=-3, y=0: (-3+1)/(0-2) = (-2)/(-2) = 1 ✔️

At x=-2, y=3: (-2+1)/(3-2) = (-1)/1 = -1 ✔️
At x=-2, y=2: undefined ✔️
At x=-2, y=1: (-2+1)/(1-2) = (-1)/(-1) = 1 ✔️
At x=-2, y=0: (-2+1)/(0-2) = (-1)/(-2) = 0.5 ✔️

At x=-1, y=3: (-1+1)/(3-2) = 0/1 = 0 → flat ✔️
At x=-1, y=1: 0/(1-2)=0 → flat ✔️
At x=-1, y=0: 0/(0-2)=0 → flat ✔️

At x=0, y=3: (0+1)/(3-2)=1/1=1 ✔️
At x=0, y=2: undefined ✔️
At x=0, y=1: (0+1)/(1-2)=1/(-1)=-1 ✔️
At x=0, y=0: (0+1)/(0-2)=1/(-2)=-0.5 ✔️

At x=1, y=3: (1+1)/(3-2)=2/1=2 ✔️
At x=1, y=2: undefined ✔️
At x=1, y=1: (1+1)/(1-2)=2/(-1)=-2 ✔️
At x=1, y=0: (1+1)/(0-2)=2/(-2)=-1 ✔️

The hand-drawn slope field matches these calculations:
- Horizontal lines where x=-1 (slope=0) ✔️
- No lines at y=2 ✔️
- Steep positive slopes where both num and den positive or both negative ✔️
- Negative slopes where signs differ ✔️

All four graphs are correctly sketched based on their differential equations.

Final Answer: All four slope fields are correctly drawn according to their respective differential equations.