Question

If \( x = A\cos 4t + B\sin 4t \), then \( \frac{d^2x}{dt^2} = \)

• A. \( x \)
• B. \( -16x \)
• C. \( 15x \)
• D. \( 16x \)
• E. \( -15x \)

Hint:

Second derivative of \( \sin(ax),\cos(ax) \) gives factor \( -a^2 \).

Answer 1

The Correct Option is B

Concept: Second derivative of trig functions.

Step 1: Differentiate once: \[ \frac{dx}{dt} = -4A\sin4t + 4B\cos4t \]

Step 2: Differentiate again: \[ \frac{d^2x}{dt^2} = -16A\cos4t -16B\sin4t \]

Step 3: Factor: \[ = -16(A\cos4t + B\sin4t) \]

Step 4: Substitute original \( x \).

Step 5: Final: \[ \frac{d^2x}{dt^2} = -16x \]