Calculus and Trigonometric Functions Quiz
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Questions and Answers

What is the derivative of $f(x)=\frac{\sin x}{1+\cos x}\cdot \tan^3x$?

  • $f'(x)=\frac{\sin x}{1+\cos x}\cdot (3\tan^2x-\tan x)$
  • $f'(x)=\frac{\sin x}{1+\cos x}\cdot \tan^2x$
  • $f'(x)=\frac{2\sin^2x}{(1+\cos x)^2}$ (correct)
  • $f'(x)=\frac{\sin x}{1+\cos x}\cdot (3\tan^2x+\tan x)$

    • At which points is the function $f(x)=\frac{\sin x}{1+\cos x}\cdot \tan^3x$ undefined?

  • $x=\pi+k\pi, \: k\in \mathbb{Z}$
  • $x=2\pi+k\pi, \: k\in \mathbb{Z}$
  • $x=\frac{3\pi}{2}+k\pi, \: k\in \mathbb{Z}$
  • $x=\frac{\pi}{2}+k\pi, \: k\in \mathbb{Z}$ (correct)

    • What is the limit of $f(x)=\frac{\sin x}{1+\cos x}\cdot \tan^3x$ as $x$ approaches $\frac{\pi}{2}$?

  • The limit does not exist
  • The limit is $\infty$ (correct)
  • The limit is $0$
  • The limit is $1$

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