Learn how to find domain and range of tangent, trig, sine inverse function. Practice derivative of inverse cosine and tangent, Sine.
The graph of \(sin^{-1}x\) v/s \(x\) is as shown below.
A \(x\in[1, 2]\)
B \(x\in(-\infty, 4]\)
C \(x\in[-1, 3]\)
D \(x\in[3, \infty)\)
A \(\dfrac {-cos\,x}{2\sqrt {sin\,x}\sqrt {1-cos\,x}}\)
B \(\dfrac {1}{\sqrt {1-sin\,x}}\)
C \(\dfrac {-sin\,x}{2\sqrt {cos\,x}\sqrt {1-sin\,x}}\)
D \(cos^2\,x-sin\,x+C\)
\(=\dfrac {1}{1+(f(x))^2}×f'(x)\)
\(=\dfrac {f'(x)}{1+(f(x))^2}\)
A \(\dfrac {sin\,x}{e^x}\)
B \(\dfrac {e^{3x}}{1+e^{6x}}\)
C \(\dfrac {e^{5x}}{x+4}\)
D \(\dfrac {3e^{3x}}{1+e^{6x}}\)
The inverse function of this restricted tangent function is called inverse tangent function or \(tan^{-1}\,x\) or arc\((tan\,x)\).
\(\therefore \;\; tan^{-1}(-1)=\dfrac {-\pi}{4}\)
A \(\left [ \dfrac {-\pi}{2},\;\dfrac {\pi}{2} \right]\)
B \(\left ( \dfrac {-\pi}{2},\;\dfrac {\pi}{2} \right)\)
C \((-\infty, \;\infty)\)
D \(\left ( \dfrac {-5\pi}{2},\;\dfrac {5\pi}{2} \right)\)
The domain of \(cos^{-1}(g(x))\) is obtained by solving \(-1\leq g(x) \leq 1\).
A \(x\in\left [ \dfrac {5}{4},\;7 \right]\)
B \(x\in\left [ \dfrac {-2}{3},\;\dfrac {1}{3} \right]\)
C \(x\in\left [ \dfrac {1}{3},\;1 \right]\)
D \(x\in\left [ 2,\;\infty \right)\)
\(=\dfrac {-f'(x)}{\sqrt {1-(f(x))^2}}\)
A \(\dfrac {-3}{\sqrt {6x-9x^2}}\)
B \(\dfrac {-4}{\sqrt {4x-x^2}}\)
C \(2\,sin\,x+cos^{-1}\,x\)
D \(2\,cos\,x+sin^{-1}\,x\)