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) ${\dfrac {dD}{dT}}={\dfrac {0.000012l_{t}}{\pi }}$.

 

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Exercises III. (p. 46.)

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

 

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Exercises IV. (p. 51.)

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