Home Practice Questions AP Calculus AB Differential Equations Question No. 1

Answer is: option2

Solution:

At (1, 1): slope is negative
At (–1, 1): slope is positive
At (–1, –1): slope is negative
At (1, –1): slope is positive

1. (1, 1): \( \frac{1}{1} = 1 \rightarrow \text{positive} \ ❌\)
2. (–1, 1): \( \frac{-1}{1} = -1 \rightarrow \text{negative} \ ❌\)
3. (–1, –1): \( \frac{-1}{-1} = 1 \rightarrow \text{positive} \ ❌\)
4. (1, –1): \( \frac{1}{-1} = -1 \rightarrow \text{negative} \ ❌\)

Conclusion: ❌ Not a match. All four signs are opposite of observed slopes.

1. (1, 1): \( \frac{-1}{1} = -1 \rightarrow \text{negative} \ ✅\)
2. (–1, 1): \( \frac{1}{1} = 1 \rightarrow \text{positive} \ ✅\)
3. (–1, –1): \( \frac{1}{-1} = -1 \rightarrow \text{negative} \ ✅\)
4. (1, –1): \( \frac{-1}{-1} = 1 \rightarrow \text{positive} \ ✅\)

Conclusion: ✅ This is the correct match.

Note: \( x^2 \geq 0 \), so numerator is always non-negative.

1. (1, 1): \( \frac{1}{1} = 1 \rightarrow \text{positive} \ ❌\)
2. (–1, 1): \( \frac{1}{1} = 1 \rightarrow \text{positive} \ ✅\)
3. (–1, –1): \( \frac{1}{-1} = -1 \rightarrow \text{negative} \ ✅\)
4. (1, –1): \( \frac{1}{-1} = -1 \rightarrow \text{negative} \ ❌\)

Conclusion: ❌ Not a match. Only 2 out of 4 match.

Note: \( x^2 \geq 0 \), so numerator is always non-positive. Sign depends only on \( y \).

1. (1, 1): \( \frac{-1}{1} = -1 \rightarrow \text{negative} \ ✅\)
2. (–1, 1): \( \frac{-1}{1} = -1 \rightarrow \text{negative} \ ❌ \text{(observed positive)}\)
3. (–1, –1): \( \frac{-1}{-1} = 1 \rightarrow \text{positive} \ ✅\)
4. (1, –1): \( \frac{-1}{-1} = 1 \rightarrow \text{positive} \ ✅\)

Conclusion: ❌ Not a match. Fails at (–1, 1).

✅ Final Answer: (B) \( \frac{dy}{dx} = \frac{-x}{y} \)