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 $$-\theta$$.

(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) $$dy = 3\sin v u^2\, du + u^3 \cos v\, dv$$,
 * $$dy = u \sin x^{u-1} \cos x\, dx + (\sin x)^u \log_\epsilon \sin x du$$,
 * $$dy = \dfrac{1}{v}\, \dfrac{1}{u}\, du - \log_\epsilon u \dfrac{1}{v^2}\, dv$$.

(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 =$$\sqrt[3]{2V}$$.

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