Chapter 3: Answers 4 Jack K. Cohen Colorado School of Mines

  1. Use and if you still don't see the light, ask your friends or the instructor for help.

  2. Use the fact that D sin x = cos x to find the derivatives of the following functions:
    1. D sin(x3) = cos( ... )⋅D( ... ) = cos(x3)⋅Dx3 = 3x2cos x3.
    2. D(sin x)3 = 3( ... )2⋅D( ... ) = 3(sin x)2⋅D sin x = 3 sin2x cos x.
    3. D sin(sin x) = cos( ... )⋅D( ... ) = cos(sin x)⋅D sin x = cos(sin x)cos x.
    4. D sin(sin(sin x)) = cos( ... )⋅D( ... ) = cos(sin(sin x))⋅D sin(sin x) = cos(sin(sin x))cos( ... )⋅D( ... ) = cos(sin(sin x))cos(sin x)⋅D sin x = cos(sin(sin x))cos(sin x)cos x. (Whew!)
  3. The area of a circle in terms of the radius is A = πr2.
    1. ${\frac{{dA}}{{dt}}}$ = ${\frac{{dA}}{{dr}}}$${\frac{{dr}}{{dt}}}$ = 2πr${\frac{{dr}}{{dt}}}$.
    2. (3.3.49) ${\frac{{dA}}{{dt}}}$ = 2πr${\frac{{dr}}{{dt}}}$ = 2π⋅10⋅2 = 40π cm2/sec.
    3. From part (a), ${\frac{{dr}}{{dt}}}$ = ${\frac{{1}}{{2 \pi r}}}$${\frac{{dA}}{{dt}}}$. Use r = $\sqrt{{A/ \pi}}$ and simplify to get ${\frac{{dr}}{{dt}}}$ = ${\frac{{1}}{{2 \sqrt{\pi A}}}}$${\frac{{dA}}{{dt}}}$.
    4. (3.3.50) - $\sqrt{{3}}$/15 cm/sec.