Page:Calculus Made Easy.pdf/275

 (10) $$\dfrac{\text{Rate of change of P when t varies}} {\text{Rate of change of P when D varies}} = - \dfrac{D}{t}$$.

(11) $$2\pi$$, $$2\pi r$$, $$\pi l$$, $$\tfrac{2}{3}\pi rh$$, $$8\pi r$$, $$4\pi r^2$$.

(12) $$\dfrac{dD}{dT} = \dfrac{0.000012l_t}{\pi}$$.

(1) (a) $$1 + x + \dfrac{x^2}{2} + \dfrac{x^3}{6} + \dfrac{x^4}{24} + \ldots$$.


 * (b) $$2ax + b$$.


 * (c) $$2x + 2a$$.


 * (d) $$3x^2 + 6ax + 3a^2$$.

(2) $$\dfrac{dw}{dt} = a - bt$$.

(3) $$\dfrac{dy}{dx} = 2x$$.

(4) $$14110x^4 - 65404x^3 - 2244x^2 + 8192x + 1379$$.

(5) $$\dfrac{dx}{dy} = 2y + 8$$.

(6) $$185.9022654x^2 + 154.36334$$.

(7) $$\dfrac{-5}{(3x + 2)^2}$$.

(8) $$\dfrac{6x^4 + 6x^3 + 9x^2}{(1 + x + 2x^2)^2}$$.

(9) $$\dfrac{ad - bc}{(cx + d)^2}$$.

(10) $$\dfrac{anx^{-n-1} + bnx^{n-1} + 2nx^{-1}}{(x^{-n} + b)^2}$$.

(11) $$b + 2ct$$.

(12) $$R_0(a + 2bt)$$, $$R_0 \left(a + \dfrac{b}{2\sqrt{t}}\right)$$, $$-\dfrac{R_0(a + 2bt)}{(1 + at + bt^2)^2}$$ or $$\dfrac{R^2 (a + 2bt)}{R_0}$$.

(13) $$1.4340(0.000014t - 0.001024)$$, $$-0.00117$$, $$-0.00107$$, $$-0.00097$$.

(14) $$\dfrac{dE}{dl} = b + \dfrac{k}{i}$$, $$\dfrac{dE}{di} = -\dfrac{c + kl}{i^2}$$.

(1) $$17+24x$$; $$24$$.

(2) $$\dfrac{x^2 + 2ax - a}{(x + a)^2}$$; $$\dfrac{2a(a + 1)}{(x + a)^3}$$

(3) $$1 + x + \dfrac{x^2}{1 \times 2} + \dfrac{x^3}{1 \times 2 \times 3}$$.

(4) (Exercises III.):

(1) (a) $$\dfrac{d^2 y}{dx^2} = \dfrac{d^3 y}{dx^3} = 1 + x + \frac{1}{2}x^2 + \frac{1}{6} x^3 + \ldots$$


 * (b) $$2a$$, $$0$$.


 * (c) $$2, 0$$.


 * (d) $$6x+6a, 6$$.