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AP Calculus AB/Unit 7: Differential Equations
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Ordering X-Values by Second Derivative

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Consider the particular solution y=4x2+25y=\sqrt{4x^2+25} to the differential equation dydx=4xy\frac{dy}{dx}=\frac{4x}{y}. The second derivative is d2ydx2=100(4x2+25)3\frac{d^2y}{dx^2}=\frac{100}{(\sqrt{4x^2+25})^3}. The approximate values of the second derivative for selected values of xx are as follows:

  • At x=−3x = -3, d2ydx2≈0.210\frac{d^2y}{dx^2} \approx 0.210
  • At x=−1x = -1, d2ydx2≈0.642\frac{d^2y}{dx^2} \approx 0.642
  • At x=1x = 1, d2ydx2≈0.642\frac{d^2y}{dx^2} \approx 0.642
  • At x=3x = 3, d2ydx2≈0.210\frac{d^2y}{dx^2} \approx 0.210

Based on this information, which of the following correctly orders the given xx-values from least to greatest according to their corresponding second derivative values? If two second derivative values are equal, the smaller xx-value should be listed first.

A

1,-1,3,-3

B

-3,3,-1,1

C

-1,1,-3,3

D

3,-3,-1,1

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