Case Study Questions on Differentiability and Continuity Class 12

For the function \( \theta(t) = \ln(\cos t) \), which of the following time intervals ensures that \( \theta(t) \) is defined and continuous?

Answer: (B)
Solution: \( \cos t > 0 \) is required for \( \ln(\cos t) \) to be defined. This occurs in the interval \( \left( 0, \frac{\pi}{2} \right) \).

What is the derivative \( \frac{d\theta}{dt} \) if \( \theta(t) = \ln(\cos t) \)?

Answer: (B)
Solution: Using chain rule, \( \frac{d}{dt}[\ln(\cos t)] = \frac{1}{\cos t} \cdot (-\sin t) = -\tan t \).

Which of the following functions is differentiable for all real numbers?

Answer: (B)
Solution: \( \tan^{-1} x \) is differentiable for all real \( x \) because it has no domain restrictions. The other functions are undefined at certain points.

Let \( f(x) = \ln(\cos x) \) and \( g(x) = 2x + 5 \). What is \( \frac{d}{dx}[f(g(x))] \)?

Answer: (A)
Solution: Using chain rule: \( \frac{d}{dx}[\ln(\cos(2x + 5))] = \frac{-\sin(2x + 5)}{\cos(2x + 5)} \cdot 2 = -2\tan(2x + 5) \).

Which of the following statements is true regarding \( f(x) = \ln(\cos x) \)?

Answer: (B)
Solution: \( \ln(\cos x) \) is undefined (and hence discontinuous) wherever \( \cos x \leq 0 \), specifically at \( x = \frac{\pi}{2} + n\pi \).