Generating...                               s20quiz10_n16

  1. Suppose that $\displaystyle g\left( {{\pi}\over{3}} \right) = 1 $ and $\displaystyle g'\left( {{\pi}\over{3}} \right) = 2 $. Then find the derivative $\displaystyle f'\left( {{\pi}\over{3}} \right)$ for $f(x) = \displaystyle g\left(x\right)\,\cos x $ .

    $\displaystyle {{\sqrt{3}}\over{2}}+1$ $\displaystyle 1-{{\sqrt{3}}\over{2}}$ $\displaystyle -{{\sqrt{3}}\over{2}}-1$ 2 $\displaystyle {{1}\over{2}}$

  2. Find the derivative $f'(x)$ for $f(x) =\displaystyle {{\sin x}\over{x^2}} $ .

    $f'(x) =\displaystyle -{{2\,\sin x-x\,\cos x}\over{x^3}} $ $f'(x) =\displaystyle {{x\,\sin x+2\,\cos x}\over{x^3}} $ $f'(x) =\displaystyle {{2\,\sin x-x\,\cos x}\over{x^3}} $ $f'(x) =\displaystyle -{{x\,\sin x+2\,\cos x}\over{x^3}} $

  3. Choose the correct trigonometric identity.

    $\displaystyle \sin x+\cos x\,\cot x=2\,\sin x-\csc x$

    $\displaystyle \sin x+\cos x\,\cot x=-\csc x$

    $\displaystyle \sin x+\cos x\,\cot x=\csc x$

    $\displaystyle \sin x+\cos x\,\cot x={{2\,\csc x}\over{\left(\sec x\right)^2}}- \csc x$

  4. Find the values of $\displaystyle \sin t$ and $\displaystyle \cos t$ for $\displaystyle t={{4\,\pi}\over{3}}$ .

    $\displaystyle \left[ \sin t={{\sqrt{3}}\over{2}} , \cos t={{1}\over{2}} \right] $

    $\displaystyle \left[ \sin t={{\sqrt{3}}\over{2}} , \cos t=-{{1}\over{2}} \right] $

    $\displaystyle \left[ \sin t=-{{\sqrt{3}}\over{2}} , \cos t={{1}\over{2}} \right] $

    $\displaystyle \left[ \sin t=-{{\sqrt{3}}\over{2}} , \cos t=-{{1}\over{2}} \right] $

  5. Choose the correct trigonometric identity.

    $\sin( t+{{3\,\pi}\over{2}}) = -\sin t$

    $\sin( t+{{3\,\pi}\over{2}}) = -\cos t$

    $\sin( t+{{3\,\pi}\over{2}}) = \cos t$

    $\sin( t+{{3\,\pi}\over{2}}) = \sin t$

  6. Choose the correct trigonometric identity.

    $\displaystyle \sin x\,\tan x+\cos x=2\,\cos x-\sec x$

    $\displaystyle \sin x\,\tan x+\cos x=\sec x$

    $\displaystyle \sin x\,\tan x+\cos x={{2\,\sec x}\over{\left(\csc x\right)^2}}- \sec x$

    $\displaystyle \sin x\,\tan x+\cos x=-\sec x$

  7. Choose the correct trigonometric identity.

    $\displaystyle -\left(\csc t\right)^2-\left(\cot t\right)^2=1$

    $\displaystyle -\left(\csc t\right)^2-\left(\cot t\right)^2={{\sin ^2t-2}\over{ \sin ^2t}}$

    $\displaystyle -\left(\csc t\right)^2-\left(\cot t\right)^2=-{{\sin ^2t-2}\over{ \sin ^2t}}$

    $\displaystyle -\left(\csc t\right)^2-\left(\cot t\right)^2=-1$

  8. Choose the correct trigonometric identity.

    $\displaystyle {{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=\cot t$

    $\displaystyle {{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=-\cot t$

    $\displaystyle {{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=\tan t$

    $\displaystyle {{\sin \left(2\,t\right)}\over{1-\cos \left(2\,t\right)}}=-\tan t$

  9. Suppose that $\frac{3\pi}{2} < t < 2\pi$, and that $\displaystyle \csc t=-5$ . Find the values of the trigonometric functions $\sin(t)$, $\cos(t)$, and $\tan(t)$.

    $\displaystyle \left[ \sin t=-{{1}\over{5}} , \cos t={{2\,\sqrt{6}}\over{5}} , \tan t=-{{1}\over{2\,\sqrt{6}}} \right] $

    $\displaystyle \left[ \sin t={{2\,\sqrt{6}}\over{5}} , \cos t=-{{1}\over{5}} , \tan t=-2\,\sqrt{6} \right] $

    $\displaystyle \left[ \sin t=-{{1}\over{5}} , \cos t=-{{1}\over{24}} , \tan t={{24 }\over{5}} \right] $

    $\displaystyle \left[ \sin t=-{{1}\over{24}} , \cos t=-{{1}\over{5}} , \tan t={{5 }\over{24}} \right] $

  10. Find the derivative $f'(x)$ for $f(x) =\displaystyle \csc x\,\tan x $ .

    $f'(x) =\displaystyle \sec x\,\tan x $ $f'(x) =\displaystyle \csc x $ $f'(x) =\displaystyle -\cot x\,\csc x $ $f'(x) =\displaystyle -\sec x\,\tan x $ $f'(x) =\displaystyle \cot x\,\csc x $


Department of Mathematics
Last modified: 2025-05-04