Page:Calculus Made Easy.pdf/276

 (2) $$-b$$, $$0$$.

(3) $$2$$, $$0$$.

(4) $$56440x^3 - 196212x^2 - 4488x + 8192$$. $$\quad \quad \quad 169320x^2 - 392424x - 4488$$.

(5) $$2$$, $$0$$.

(6) $$371.80453x$$, $$371.80453$$.

(7) $$\dfrac{30}{(3x + 2)^3}$$, $$-\dfrac{270}{(3x + 2)^4}$$.

(Examples, p. 41):

(1) $$\dfrac{6a}{b^2} x$$, $$\dfrac{6a}{b^2}$$.

(2) $$\dfrac{3a \sqrt{b}} {2 \sqrt{x}} - \dfrac{6b \sqrt[3]{a}}{x^3}$$, $$\dfrac{18b \sqrt[3]{a}}{x^4} - \dfrac{3a \sqrt{b}}{4 \sqrt{x^3}}$$.

(3) $$\dfrac{2}{\sqrt[3]{\theta^8}} - \dfrac{1.056}{\sqrt[5]{\theta^{11}}}$$, $$\dfrac{2.3232}{\sqrt[5]{\theta^{16}}} - \dfrac{16}{3 \sqrt[3]{\theta^{11}}}$$.

(4) $$810t^4 - 648t^3 + 479.52t^2 - 139.968t + 26.64$$. $$\quad \quad \quad 3240t^3 - 1944t^2 + 959.04t - 139.968$$.

(5) $$12x+2$$, $$12$$.

(6) $$6x^2-9x$$, $$12x-9$$.

(7) $$\begin{aligned}[t] &\dfrac{3}{4} \left(\dfrac{1}{\sqrt{\theta}} + \dfrac{1}{\sqrt{\theta^5}}\right) +\dfrac{1}{4} \left(\dfrac{15}{\sqrt{\theta^7}} - \dfrac{1}{\sqrt{\theta^3}}\right). \\ &\dfrac{3}{8} \left(\dfrac{1}{\sqrt{\theta^5}} - \dfrac{1}{\sqrt{\theta^3}}\right) -\dfrac{15}{8}\left(\dfrac{7}{\sqrt{\theta^9}} + \dfrac{1}{\sqrt{\theta^7}}\right). \end{aligned}$$

(2) $$64$$; $$147.2$$; and $$0.32$$ feet per second.

(3) $$\dot{x} = a - gt$$; $$\ddot{x}=-g$$.

(4) $$45.1$$ feet per second.

(5) $$12.4$$ feet per second per second. Yes.

(6) Angular velocity $$=11.2$$ radians per second; angular acceleration $$=9.6$$ radians per second per second.

(7) $$v=20.4t^2-10.8$$. $$a=40.8t$$. $$172.8 in./sec.$$, $$122.4 in./sec^2$$.

(8) $$v = \dfrac{1}{30 \sqrt[3]{(t - 125)^2}}$$, $$a=-\dfrac{1}{45\sqrt[3]{(t-125)^5}}$$.

(9) $$v=0.8-\dfrac{8t}{(4 + t^2)^2}$$, $$a= \dfrac{24t^2-32}{(4 + t^2)^3}$$, $$0.7926$$ and $$0.00211$$.

(10) $$n=2$$, $$n=11$$.