Page:Calculus Made Easy.pdf/282

(14) ${\displaystyle {\dfrac {dy}{d\theta }}=3\theta ^{2}+3\cos \left(\theta +3\right)-\log _{\epsilon }3\left(\cos \theta \times 3^{\sin \theta }+3\theta \right)}$.

(15) ${\displaystyle \theta =\cot \theta ;\theta =\pm 0.86}$; is max. for ${\displaystyle +\theta }$, min. for ${\displaystyle -\theta }$.

(1) ${\displaystyle x^{3}-6x^{2}y-2y^{2}}$; ${\displaystyle {\frac {1}{3}}-2x^{3}-4xy}$.

(2) ${\displaystyle 2xyz+y^{2}z+z^{2}y+2xy^{2}z^{2}}$;

${\displaystyle 2xyz+x^{2}z+xz^{2}+2x^{2}yz^{2}}$;

${\displaystyle 2xyz+x^{2}y+xy^{2}+2x^{2}y^{2}z}$.

(3) ${\displaystyle {\dfrac {1}{r}}\{\left(x-a\right)+\left(y-b\right)+\left(z-c\right)\}={\dfrac {\left(x+y+z\right)-\left(a+b+c\right)}{r}}}$; ${\displaystyle {\dfrac {3}{r}}}$.

(4) ${\displaystyle dy=vu^{v-1}\,du+u^{v}\log _{\epsilon }u\,dv}$.

(5) ${\displaystyle dy=3\sin vu^{2}\,du+u^{3}\cos v\,dv}$,

${\displaystyle dy=u\sin x^{u-1}\cos x\,dx+(\sin x)^{u}\log _{\epsilon }\sin xdu}$,

${\displaystyle dy={\dfrac {1}{v}}\,{\dfrac {1}{u}}\,du-\log _{\epsilon }u{\dfrac {1}{v^{2}}}\,dv}$.

(7) Minimum for ${\displaystyle x=y=-{\tfrac {1}{2}}}$.

(8) (a) Length ${\displaystyle 2}$ feet, width = depth = ${\displaystyle 1}$ foot, vol. = ${\displaystyle 2}$ cubic feet.

(b) Radius = ${\displaystyle 2\pi }$ feet = ${\displaystyle 7.46}$ in., length = ${\displaystyle 2}$ feet, vol. = ${\displaystyle 2.54}$.

(9) All three parts equal; the product is maximum.

(10) Minimum for ${\displaystyle x=y=1}$.

(11) Min.: ${\displaystyle x={\tfrac {1}{2}}}$ and ${\displaystyle y=2}$.

(12) Angle at apex ${\displaystyle =90^{\circ }}$; equal sides = length =${\displaystyle {\sqrt[{3}]{2V}}}$.

(1) ${\displaystyle 1{\tfrac {1}{3}}}$.

(2) ${\displaystyle 0.6344}$.

(3) ${\displaystyle 0.2624}$.

(4) (a) ${\displaystyle y={\tfrac {1}{8}}x^{2}+C}$;

(b) ${\displaystyle y=\sin x+C}$.

(5) ${\displaystyle y=x^{2}+3x+C}$.