Page:Calculus Made Easy.pdf/282

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

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

 

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Exercises XV. (p. 180.)

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

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

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

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

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

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

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

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

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

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

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

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

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

(10) Minimum for $x=y=1$.

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

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

 

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Exercises XVI. (p. 190.)

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

(2) $0.6344$.

(3) $0.2624$.

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

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

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

 

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